This course develops the basic language and geometry of complex manifolds, then uses it to study holomorphic vector bundles and the Hermitian structures that connect local complex analysis with global geometric invariants. It begins with the notion of a complex manifold and holomorphic map, then introduces differential forms of type $(p,q)$ and the $\bar\partial$-operator, which together organize the analytic side of complex geometry. From there, Dolbeault cohomology provides the natural global setting for holomorphic functions and their obstructions, while the theory of holomorphic bundles extends these ideas from functions to vector-valued objects. The main themes are complex structure, holomorphicity, and curvature. After setting up bundles and the $\bar\partial$-operator on them, the course turns to Hermitian metrics, connections, and the Chern connection, which is the distinguished connection compatible with both the complex and metric structures. Curvature tensors then measure how these structures twist, and Chern-Weil theory shows how curvature produces topological invariants. The later chapters apply this framework to holomorphic line bundles, divisors, and meromorphic data, tying analytic objects to global algebraic and geometric information. The chapters are arranged to move from local foundations to global synthesis. Early chapters establish the differential and cohomological tools needed to formulate problems precisely. Midway through, these tools are applied to bundles, metrics, and connections, building the machinery needed for curvature calculations. The final chapters bring the pieces together: line bundles and divisors illustrate how local holomorphic data encodes global geometry, and the course closes by showing how the analytic and metric viewpoints combine into a coherent picture of complex geometry. # Introduction This opening chapter fixes the scope of the course and the background language that will be used throughout. Complex geometry sits between several subjects: complex analysis supplies holomorphic functions, differential geometry supplies manifolds and bundles, and topology supplies the invariants that constrain global behaviour. The aim of the first part of the course is to make these languages compatible on complex manifolds, before Hermitian metrics and curvature enter later as geometric structures on the same objects. The guiding question is how much of one-variable complex analysis survives when the domain is replaced by a manifold locally modelled on $\mathbb C^n$. In several variables, holomorphicity is more rigid than smoothness, and global topology can prevent local holomorphic data from extending. The course therefore moves constantly between local coordinate calculations and intrinsic constructions. ## The Objects of the Course What kind of space should carry complex analytic geometry when there is no preferred global coordinate system? The first answer is a manifold whose coordinate changes are holomorphic, so that the word holomorphic is independent of the chart in which it is tested. [definition: Complex Manifold] A complex manifold of complex dimension $n$ is a Hausdorff, second-countable [topological space](/page/Topological%20Space) $X$ equipped with an atlas of charts $(U_i,\varphi_i)$, where $\varphi_i:U_i\to \varphi_i(U_i)\subseteq \mathbb C^n$ is a homeomorphism onto an [open set](/page/Open%20Set), such that every transition map \begin{align*} \varphi_j\circ \varphi_i^{-1}:\varphi_i(U_i\cap U_j)\to \varphi_j(U_i\cap U_j) \end{align*} is holomorphic. [/definition] This definition is local, but it has immediate global consequences: the topology of $X$ and the compatibility of the charts both matter. The simplest examples come from open subsets of $\mathbb C^n$, but the course quickly needs examples with nontrivial topology and no single global coordinate chart. [example: Basic Complex Manifolds] For a domain $\Omega\subset \mathbb C^n$, take the single chart $(\Omega,\operatorname{id}_\Omega)$. Its only transition map is $\operatorname{id}_\Omega\circ \operatorname{id}_\Omega^{-1}=\operatorname{id}_\Omega$, and the identity map on an open subset of $\mathbb C^n$ is holomorphic coordinate by coordinate. For $\mathbb{CP}^n$, let \begin{align*} U_i=\{[z_0:\cdots:z_n]:z_i\ne 0\}. \end{align*} Define the affine chart $\varphi_i:U_i\to \mathbb C^n$ by \begin{align*} \varphi_i([z_0:\cdots:z_n])=\left(\frac{z_0}{z_i},\ldots,\frac{z_{i-1}}{z_i},\frac{z_{i+1}}{z_i},\ldots,\frac{z_n}{z_i}\right). \end{align*} This is well-defined because replacing $(z_0,\ldots,z_n)$ by $(\lambda z_0,\ldots,\lambda z_n)$ leaves every quotient $\lambda z_k/\lambda z_i=z_k/z_i$ unchanged. On $U_i\cap U_j$, write affine coordinates in the $i$-chart as $w_k=z_k/z_i$ for $k\ne i$. Since $j\ne i$ and $z_j\ne 0$ on the overlap, we have $w_j=z_j/z_i\ne 0$. The $j$-chart coordinates are \begin{align*} \frac{z_k}{z_j}=\frac{z_k/z_i}{z_j/z_i}=\frac{w_k}{w_j}\quad\text{for }k\ne i,j, \end{align*} and \begin{align*} \frac{z_i}{z_j}=\frac{1}{z_j/z_i}=\frac{1}{w_j}. \end{align*} Thus every component of $\varphi_j\circ\varphi_i^{-1}$ is either $1/w_j$ or $w_k/w_j$, which is holomorphic on the overlap because $w_j\ne 0$ there. If $\Lambda\subset \mathbb C^n$ is a lattice, the quotient map $q:\mathbb C^n\to \mathbb C^n/\Lambda$ identifies $z$ with $z+\lambda$ for $\lambda\in\Lambda$. Choose an open ball $B\subset\mathbb C^n$ small enough that $(B-B)\cap\Lambda=\{0\}$. Then $q|_B$ is injective: if $q(z)=q(w)$ for $z,w\in B$, then $z-w\in\Lambda$, while $z-w\in B-B$, so $z-w=0$ and hence $z=w$. The chart on $q(B)$ is $(q|_B)^{-1}:q(B)\to B\subset\mathbb C^n$. If two such balls $B$ and $B'$ give overlapping quotient charts, then on each connected piece of the overlap the two lifts differ by a fixed lattice element $\lambda\in\Lambda$, so the transition map has the form $z\mapsto z+\lambda$, which is holomorphic. These three constructions give the basic local models: domains have global coordinates, projective space is glued by rational holomorphic coordinate changes, and complex tori are glued by translations. [/example] These examples indicate the range of phenomena: affine domains are local models, projective space is compact and curved from the start, and complex tori combine flat local geometry with nontrivial global topology. To compare such spaces, the correct maps are those that preserve the holomorphic coordinate structure. [definition: Holomorphic Map Between Complex Manifolds] Let $X$ and $Y$ be complex manifolds. A continuous map $f:X\to Y$ is holomorphic if, for every point $x\in X$, there are charts $(U,\varphi)$ on $X$ with $x\in U$ and $(V,\psi)$ on $Y$ with $f(x)\in V$ such that $f(U)\subset V$ and the coordinate expression \begin{align*} \psi\circ f\circ \varphi^{-1}:\varphi(U)\to \psi(V) \end{align*} is holomorphic. [/definition] The continuity assumption can often be absorbed into the coordinate condition, but including it keeps the definition aligned with the manifold structure. A biholomorphism is a holomorphic map with a holomorphic inverse, and it is the appropriate notion of sameness for complex manifolds. [example: Projective Linear Automorphisms] Let $A=(a_{\mu\nu})_{\mu,\nu=0}^n\in GL(n+1,\mathbb C)$. For a nonzero vector $z=(z_0,\ldots,z_n)$, the vector $Az$ is nonzero because $A$ is injective, so $[Az]$ is a point of $\mathbb{CP}^n$. If $z$ is replaced by $\lambda z$ with $\lambda\in\mathbb C^*$, then \begin{align*} A(\lambda z)=\lambda Az. \end{align*} Thus $[\lambda Az]=[Az]$, so the rule $F_A([z])=[Az]$ is well-defined on projective classes. If $c\in\mathbb C^*$, then \begin{align*} F_{cA}([z])=[cAz]=[Az]=F_A([z]), \end{align*} so scalar multiples of $A$ define the same projective map, and the construction descends to $PGL(n+1,\mathbb C)=GL(n+1,\mathbb C)/\mathbb C^*$. We now compute the coordinate expression. On the affine chart $U_i=\{z_i\ne 0\}$, write $w_k=z_k/z_i$ for $k\ne i$ and set $w_i=1$. Then $z=z_i w$, where $w=(w_0,\ldots,w_n)$ with $w_i=1$. Hence, for each component $\mu$, \begin{align*} (Az)_\mu=\sum_{\nu=0}^n a_{\mu\nu}z_\nu=z_i\sum_{\nu=0}^n a_{\mu\nu}w_\nu. \end{align*} On the part of $U_i$ where $F_A([z])\in U_j$, the $j$-th component of $Az$ is nonzero, so \begin{align*} \sum_{\nu=0}^n a_{j\nu}w_\nu\ne 0. \end{align*} For $k\ne j$, the $k$-th affine coordinate in the $j$-chart is therefore \begin{align*} \frac{(Az)_k}{(Az)_j}=\frac{z_i\sum_{\nu=0}^n a_{k\nu}w_\nu}{z_i\sum_{\nu=0}^n a_{j\nu}w_\nu}=\frac{\sum_{\nu=0}^n a_{k\nu}w_\nu}{\sum_{\nu=0}^n a_{j\nu}w_\nu}. \end{align*} Each coordinate is a quotient of two affine-linear holomorphic functions, with denominator nonzero on the chart domain, so $F_A$ is holomorphic in affine coordinates. Finally, \begin{align*} F_{A^{-1}}(F_A([z]))=F_{A^{-1}}([Az])=[A^{-1}Az]=[z], \end{align*} and similarly $F_A(F_{A^{-1}}([z]))=[z]$. The same coordinate calculation applied to $A^{-1}$ shows that $F_{A^{-1}}$ is holomorphic, so $F_A$ is a biholomorphism of $\mathbb{CP}^n$. [/example] ## Local Analysis and Global Geometry Why does the course begin with local theorems from complex analysis rather than with global invariants? The reason is that the local normal forms control the definitions of submanifolds, bundles, and tangent directions, while global questions are assembled from these local pieces. [quotetheorem:4950] [citeproof:4950] This theorem says that holomorphic maps with nondegenerate derivative are local coordinate changes, so it explains when a map can be used as a new chart. The invertibility hypothesis is essential: the map $f:\mathbb C\to\mathbb C$ given by $f(z)=z^2$ has derivative $0$ at $0$, and near $0$ it identifies $z$ and $-z$, so no neighbourhood of $0$ is mapped biholomorphically onto its image. The theorem is also only local; it does not say that a holomorphic map with everywhere invertible Jacobian is globally injective, since global topology and covering behaviour may still intervene. The next local problem is different: instead of inverting a map, we want to understand the zero set of several holomorphic equations and decide when that zero set is itself a complex manifold. [quotetheorem:7002] [citeproof:7002] The [implicit function theorem](/theorems/52) is the source of analytic submanifolds and local normal forms. Its rank hypothesis is the condition that the equations cut transversely in the variables being solved for; without it, singularities can appear. For instance, the zero set of $F(z,w)=w^2-z^3$ in $\mathbb C^2$ has a cusp at $(0,0)$, and the derivative of $F$ vanishes there, so it cannot be written near the origin as the graph of a [holomorphic function](/page/Holomorphic%20Function) in the way the theorem promises. The theorem also does not classify singular zero loci; it isolates the regular points where the zero set has the expected manifold structure. [example: A Smooth Quadric] Consider the hypersurface \begin{align*} Q=\{[z_0:\cdots:z_{n+1}]\in\mathbb{CP}^{n+1}:z_0^2+\cdots+z_{n+1}^2=0\}. \end{align*} Fix a point $p=[z_0:\cdots:z_{n+1}]\in Q$. Choose an index $r$ with $z_r\ne 0$ and work on the affine chart $U_r$. Put $w_k=z_k/z_r$ for $k\ne r$ and set $w_r=1$. Dividing the homogeneous equation by $z_r^2$ gives the affine equation \begin{align*} \frac{z_0^2+\cdots+z_{n+1}^2}{z_r^2}=1+\sum_{k\ne r}w_k^2=0. \end{align*} Thus $Q\cap U_r$ is cut out by the holomorphic function \begin{align*} f_r(w)=1+\sum_{k\ne r}w_k^2. \end{align*} At the chosen point, not all $w_k$ with $k\ne r$ can be zero, because then the equation would read $1=0$. Hence there is an index $s\ne r$ with $w_s\ne 0$. The derivative of $f_r$ in the $w_s$-direction is \begin{align*} \frac{\partial f_r}{\partial w_s}=2w_s, \end{align*} since the constant term has derivative $0$, the term $w_s^2$ has derivative $2w_s$, and every term $w_k^2$ with $k\ne s$ is independent of $w_s$. Therefore $\partial f_r/\partial w_s\ne 0$ at $p$. By the *[Holomorphic Implicit Function Theorem](/theorems/7002)*, near $p$ the equation $f_r=0$ solves holomorphically for $w_s$ in terms of the remaining $n$ affine coordinates. Thus $Q$ is locally biholomorphic to an open subset of $\mathbb C^n$, so it is a complex submanifold of $\mathbb{CP}^{n+1}$ of complex dimension $n$. [/example] ## Tangent Directions and Type Decompositions Once a complex manifold is also viewed as a smooth real manifold, what extra structure remembers multiplication by $i$? The tangent bundle carries an endomorphism squaring to $-1$, and after complexification this separates tangent directions into holomorphic and antiholomorphic parts. [definition: Almost Complex Structure] An almost complex structure on a smooth manifold $M$ is a smooth bundle endomorphism $J:TM\to TM$ such that \begin{align*} J^2=-\operatorname{id}_{TM}. \end{align*} [/definition] For a complex manifold, multiplication by $i$ in holomorphic coordinates defines such an endomorphism on the underlying real tangent bundle. To use complex-linear algebra on tangent vectors and later on differential forms, we enlarge the real tangent bundle by allowing complex coefficients and then separate the two eigendirections of $J$. [definition: Complexified Tangent Bundle] Let $M$ be a smooth manifold. The complexified tangent bundle is \begin{align*} T_{\mathbb C}M:=TM\otimes_{\mathbb R}\mathbb C. \end{align*} [/definition] If $J$ is an almost complex structure, its complex-linear extension to $T_{\mathbb C}M$ has eigenbundles $T^{1,0}M$ and $T^{0,1}M$ with eigenvalues $i$ and $-i$. The subscripts record the two type components of the complexified tangent bundle. This splitting is one of the organising devices of the whole course. It raises the integrability question: when does an abstract almost complex structure come from holomorphic coordinates, rather than merely from a pointwise complex structure on tangent spaces? [quotetheorem:7003] The theorem is used as a diagnostic tool: to check that a proposed almost complex structure is genuinely complex, it is enough to test closure of the antiholomorphic tangent directions under Lie brackets. The condition is necessary because in local holomorphic coordinates the vector fields $\partial/\partial \bar z_i$ generate $T_{0,1}M$ and their brackets stay inside the same span. The theorem is not a method for producing global holomorphic functions or global coordinates. It gives local complex charts compatible with $J$ once the bracket condition holds; separate topological and analytic arguments are still needed for questions about embeddings, compactness, and global sections. [example: A Non-Integrable Local Model] On a small neighbourhood of $0$ in $\mathbb C^2$, define complex vector fields \begin{align*} L_1=\frac{\partial}{\partial \bar z_1},\qquad L_2=\frac{\partial}{\partial \bar z_2}+\bar z_1\frac{\partial}{\partial z_1}. \end{align*} They are pointwise linearly independent: if $aL_1+bL_2=0$, then \begin{align*} a\frac{\partial}{\partial \bar z_1}+b\frac{\partial}{\partial \bar z_2}+b\bar z_1\frac{\partial}{\partial z_1}=0. \end{align*} Comparing the coefficient of $\partial/\partial \bar z_2$ gives $b=0$, and then comparing the coefficient of $\partial/\partial \bar z_1$ gives $a=0$. Thus they span a complex rank-two subbundle $E\subset T_{\mathbb C}\mathbb C^2$. The complex conjugate subbundle is spanned by \begin{align*} \bar L_1=\frac{\partial}{\partial z_1},\qquad \bar L_2=\frac{\partial}{\partial z_2}+z_1\frac{\partial}{\partial \bar z_1}. \end{align*} To check transversality, suppose \begin{align*} aL_1+bL_2=c\bar L_1+d\bar L_2. \end{align*} Expanding both sides in the coordinate frame gives \begin{align*} b\bar z_1\frac{\partial}{\partial z_1}+a\frac{\partial}{\partial \bar z_1}+b\frac{\partial}{\partial \bar z_2} = c\frac{\partial}{\partial z_1}+d\frac{\partial}{\partial z_2}+dz_1\frac{\partial}{\partial \bar z_1}. \end{align*} Comparing the coefficients of $\partial/\partial \bar z_2$ and $\partial/\partial z_2$ gives $b=0$ and $d=0$. Then the coefficients of $\partial/\partial \bar z_1$ and $\partial/\partial z_1$ give $a=0$ and $c=0$. Hence $E\cap \bar E=\{0\}$, so $E$ can be used as the proposed $T_{0,1}$ subbundle of an almost complex structure. Now compute the bracket on a smooth [test function](/page/Test%20Function) $u$. Since mixed coordinate derivatives commute, \begin{align*} L_1(L_2u)=\frac{\partial}{\partial \bar z_1}\left(\frac{\partial u}{\partial \bar z_2}+\bar z_1\frac{\partial u}{\partial z_1}\right)=\frac{\partial^2u}{\partial \bar z_1\partial \bar z_2}+\frac{\partial u}{\partial z_1}+\bar z_1\frac{\partial^2u}{\partial \bar z_1\partial z_1}. \end{align*} Also, \begin{align*} L_2(L_1u)=\frac{\partial^2u}{\partial \bar z_2\partial \bar z_1}+\bar z_1\frac{\partial^2u}{\partial z_1\partial \bar z_1}. \end{align*} Subtracting the second displayed expression from the first cancels the two mixed-derivative terms and leaves \begin{align*} [L_1,L_2]u=\frac{\partial u}{\partial z_1}. \end{align*} Therefore \begin{align*} [L_1,L_2]=\frac{\partial}{\partial z_1}. \end{align*} This vector field is not a section of $E$: if $\partial/\partial z_1=aL_1+bL_2$, then comparing the coefficient of $\partial/\partial \bar z_2$ gives $b=0$, and then the right-hand side has no $\partial/\partial z_1$ component. Thus the bracket of two sections of the proposed $T_{0,1}$ subbundle leaves that subbundle. By the *[Newlander Nirenberg Theorem](/theorems/7003)*, this almost complex structure is not induced by holomorphic coordinates. [/example] This local model isolates the obstruction in the smallest possible setting: the tangent spaces carry complex structures pointwise, but the antiholomorphic directions do not form a bracket-closed system of directions. That failure is exactly what the [integrability criterion](/theorems/193) detects, so the bracket condition is not a cosmetic reformulation of the definition of a complex atlas. [remark: Integrability as a Bracket Condition] The bracket condition says that antiholomorphic tangent directions are closed under taking commutators of vector fields. The example shows that a pointwise complex structure on tangent spaces is weaker than a complex atlas: the missing ingredient is compatibility of those tangent directions under differentiation. [/remark] ## Forms, Bundles, Metrics, and Curvature The second major question of the course is how differential-geometric objects interact with the type decomposition. Ordinary differential forms split into bidegrees, and the [exterior derivative](/theorems/1525) separates into operators that detect holomorphic and antiholomorphic variation. [definition: Form of Type P Q] Let $X$ be a complex manifold. A smooth complex-valued differential form on $X$ has type $(p,q)$ if it has total degree $p+q$ and, locally, it is a linear combination of forms \begin{align*} dz_{i_1}\wedge\cdots\wedge dz_{i_p}\wedge d\bar z_{j_1}\wedge\cdots\wedge d\bar z_{j_q}. \end{align*} The subspace of smooth complex-valued differential forms of type $(p,q)$ is denoted $A^{p,q}(X)$. [/definition] After this definition, the de Rham complex becomes more structured because every form now has a bidegree rather than only a total degree. The next question is whether the exterior derivative respects this decomposition in a useful way, or whether it destroys the $(p,q)$ information. [quotetheorem:7004] [citeproof:7004] The operator $\bar\partial$ is the differential operator that encodes holomorphicity: a smooth function is holomorphic exactly when its $\bar\partial$ vanishes. The fact that $X$ is a complex manifold is doing real work here. On a merely almost complex manifold, $d$ still has components of different types, but non-integrability can introduce extra terms and the clean identity $\bar\partial^2=0$ need not hold on the would-be antiholomorphic complex; the local model spanned by $L_1$ and $L_2$ above is the concrete warning sign, since its antiholomorphic directions fail to close under brackets. These identities therefore give a cochain complex only after the integrability supplied by the complex atlas. They still do not solve $\bar\partial u=\alpha$ or compute cohomology by themselves; those are later analytic questions. This suggests the next layer of structure, where the objects are not functions but sections of vector bundles whose transition functions are holomorphic. [definition: Holomorphic Vector Bundle] A holomorphic vector bundle of rank $r$ over a complex manifold $X$ is a complex vector bundle $\pi:E\to X$ with local trivialisations $\pi^{-1}(U_i)\cong U_i\times\mathbb C^r$ such that the transition functions \begin{align*} g_{ij}:U_i\cap U_j\to GL(r,\mathbb C) \end{align*} are holomorphic. [/definition] Holomorphic bundles provide the natural coefficients for complex geometry. The holomorphic tangent bundle $T^{1,0}X$, line bundles on projective space, and bundles defined by divisors are recurring examples. [example: The Hyperplane Line Bundle] On $U_i=\{z_i\ne 0\}$, the tautological line $\mathcal O(-1)$ is generated by the local section \begin{align*} s_i([z_0:\cdots:z_n])=\frac{1}{z_i}(z_0,\ldots,z_n). \end{align*} This representative has $i$-th coordinate equal to $1$, so it depends only on the projective point: replacing $z$ by $\lambda z$ changes the numerator and denominator by the same factor. On $U_i\cap U_j$, the two local generators satisfy \begin{align*} s_j([z])=\frac{1}{z_j}z=\frac{z_i}{z_j}\frac{1}{z_i}z=\frac{z_i}{z_j}s_i([z]). \end{align*} Thus the transition function for $\mathcal O(-1)$ from the $i$-frame to the $j$-frame is $z_i/z_j$. If $e_i$ denotes the dual frame of $\mathcal O(1)$, defined by $e_i(s_i)=1$, then \begin{align*} e_j=\frac{z_j}{z_i}e_i, \end{align*} because \begin{align*} \left(\frac{z_j}{z_i}e_i\right)(s_j)=\frac{z_j}{z_i}e_i\left(\frac{z_i}{z_j}s_i\right)=e_i(s_i)=1. \end{align*} Therefore the transition function for $\mathcal O(1)$ is $z_j/z_i$ on $U_i\cap U_j$. In the affine coordinates of $U_i$, this transition function is exactly the coordinate $w_j=z_j/z_i$. On the overlap $U_i\cap U_j$ one has $z_i\ne 0$ and $z_j\ne 0$, so $w_j$ is holomorphic and never zero there. For tensor powers, the frame $e_i^{\otimes k}$ transforms by \begin{align*} e_j^{\otimes k}=\left(\frac{z_j}{z_i}\right)^k e_i^{\otimes k}. \end{align*} Thus $\mathcal O(k)$ has transition functions $(z_j/z_i)^k$, giving the standard family of holomorphic line bundles on projective space. [/example] This example shows that holomorphic bundles can carry global information even when they are locally products. To measure sizes of sections, define adjoints, and form curvature tensors, the course must add a metric structure compatible with the complex vector spaces in each fibre. [definition: Hermitian Metric on a Complex Vector Bundle] Let $E\to X$ be a complex vector bundle. A Hermitian metric on $E$ is a smooth map \begin{align*} h:E\times_X E\to\mathbb C \end{align*} such that, for each $x\in X$, the restriction $h_x:E_x\times E_x\to\mathbb C$ is a positive-definite Hermitian [inner product](/page/Inner%20Product). [/definition] This definition is the entry point to Hermitian geometry: it lets the course discuss lengths and orthogonality on a holomorphic bundle. Later chapters will add connections, then single out the Chern connection as the one compatible with both the holomorphic structure and the Hermitian metric. The two hypotheses serve different roles. The holomorphic structure specifies the intrinsic $\bar\partial_E$ operator, while the Hermitian metric supplies the compatibility condition that determines the remaining $(1,0)$ part. The resulting connection is usually not flat; its curvature is the invariant that later produces Chern forms and positivity. ## How the Course Develops The course is organised so that each new structure answers a limitation of the previous one. Complex atlases define holomorphic spaces and maps, type decompositions refine smooth calculus, holomorphic bundles introduce coefficients, and Hermitian metrics turn these bundles into geometric objects with curvature. [explanation: Roadmap of the Lectures] The opening lectures construct complex manifolds, holomorphic maps, analytic submanifolds, and the holomorphic tangent bundle. The next part develops forms of type $(p,q)$, the operators $\partial$ and $\bar\partial$, and the resulting local algebra of complex differential forms. Chapters 4 and 5 introduce holomorphic vector bundles, sheaves of holomorphic sections, and bundle-valued Dolbeault operators. Chapters 6 through 10 add Hermitian metrics, the Chern connection, curvature, and Chern forms, preparing the transition to Kähler geometry, where the Hermitian metric satisfies an additional closedness condition and interacts strongly with cohomology. [/explanation] The course does not assume algebraic geometry, but many examples come from projective space and holomorphic line bundles because they are the testing ground for later theories of positivity and moduli. It also does not develop elliptic theory in full, though several results point toward Hodge theory. [remark: Standing Background] Students are expected to know smooth manifolds, differential forms, de Rham cohomology, basic point-set topology, introductory complex analysis in one variable, multilinear algebra, vector bundles, and connections. Whenever a construction has both a smooth and a holomorphic version, the course will compare them rather than treating the holomorphic version as a formal change of adjective. [/remark] The opening chapter has fixed the vocabulary and the [comparison principle](/theorems/4870) that will guide the rest of the course. The next step is to see which smooth manifolds actually admit complex coordinate charts, and how the holomorphic transition maps encode that extra structure. # 1. Complex Manifolds and Holomorphic Maps Complex geometry begins by asking which smooth manifolds can be described using coordinates valued in $\mathbb C^n$ rather than only in $\mathbb R^{2n}$. The extra requirement is not the existence of such coordinates alone, but that the coordinate changes be holomorphic. This chapter sets up that language, studies the maps compatible with it, and connects the coordinate viewpoint to the tangent-bundle decomposition that will drive the later theory of $(p,q)$-forms and Hermitian metrics. ## Complex Atlases and First Examples The first problem is to decide what kind of atlas remembers complex analysis rather than only smooth topology. A smooth $2n$-manifold may admit many real coordinate systems, but complex geometry restricts the allowed coordinate changes to maps satisfying the [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) in several variables. We begin by naming the class of coordinate changes that will be allowed. [definition: Holomorphic Map Between Domains] Let $U \subseteq \mathbb C^m$ and $V \subseteq \mathbb C^n$ be open sets. A map $f:U \to V$ is holomorphic if, writing $f=(f_1,\dots,f_n)$, each component $f_j:U \to \mathbb C$ is holomorphic in the several-variable sense. [/definition] This definition supplies the admissible coordinate changes. The next need is to package many such coordinate systems on a single topological space, while requiring compatibility on every overlap. That compatibility is exactly what makes locally defined complex coordinates glue into a global geometric structure. [definition: Complex Atlas] Let $X$ be a topological space. A complex atlas of complex dimension $n$ on $X$ is a collection of pairs $(U_i,\varphi_i)$ such that: 1. The sets $U_i \subseteq X$ are open and cover $X$. 2. Each $\varphi_i:U_i \to \varphi_i(U_i) \subseteq \mathbb C^n$ is a homeomorphism onto an open subset of $\mathbb C^n$. 3. For all $i,j$ with $U_i \cap U_j \ne \varnothing$, the transition map \begin{align*} \varphi_j \circ \varphi_i^{-1}:\varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j) \end{align*} is holomorphic. [/definition] The atlas condition is symmetric because the inverse transition map is also a transition map. We still need to decide when two atlases describe the same complex geometry, since adding further compatible charts should not change the object. This leads to the maximal-atlas formulation of a complex manifold. [definition: Complex Manifold] A complex manifold of complex dimension $n$ is a Hausdorff, second-countable topological space $X$ equipped with a maximal complex atlas of complex dimension $n$. [/definition] The Hausdorff and second-countability assumptions keep the resulting space within the standard category of manifolds. The definition is local, so the first test case should be an open subset of the model space itself. This checks that the formalism recovers ordinary several-variable complex analysis before introducing global examples. [example: Domains In Complex Euclidean Space] Let $U \subseteq \mathbb C^n$ be open. The pair $(U,\operatorname{id}_U)$ is a chart because $\operatorname{id}_U:U\to U\subseteq \mathbb C^n$ is a homeomorphism onto the open set $U$. Since this atlas has only one chart, the only self-overlap is $U\cap U=U$, and the transition map is \begin{align*} \operatorname{id}_U\circ \operatorname{id}_U^{-1}=\operatorname{id}_U. \end{align*} For $z=(z_1,\dots,z_n)\in U$, this map is \begin{align*} \operatorname{id}_U(z)=(z_1,\dots,z_n), \end{align*} so each component is the coordinate function $z_j$, hence holomorphic. Thus the atlas defines the usual complex manifold structure of complex dimension $n$ on $U$. If $(V,\psi)$ is another chart with $V\subseteq U$, compatibility with $(U,\operatorname{id}_U)$ means that the transition map \begin{align*} \psi\circ \operatorname{id}_U^{-1}=\psi \end{align*} is holomorphic on $V$, and the reverse transition map \begin{align*} \operatorname{id}_U\circ \psi^{-1}=\psi^{-1} \end{align*} is holomorphic on $\psi(V)$. Therefore compatible extra charts are exactly biholomorphic coordinate systems on open subsets of $U$, so this construction recovers ordinary several-variable complex analysis as the local model. [/example] Domains are the local models, but complex geometry is not limited to global domains in $\mathbb C^n$. We next need an example built by gluing several affine coordinate systems. Projective space is the standard source of such examples because homogeneous coordinates have explicit holomorphic transition maps. [definition: Complex Projective Space] Complex projective space $\mathbb{CP}^n$ is the set of complex lines through $0$ in $\mathbb C^{n+1}$. Its points are equivalence classes \begin{align*} [z_0:\dots:z_n] \end{align*} where $(z_0,\dots,z_n) \in \mathbb C^{n+1}\setminus\{0\}$ and $(z_0,\dots,z_n) \sim (\lambda z_0,\dots,\lambda z_n)$ for every $\lambda \in \mathbb C^\times$. [/definition] This quotient description is compact and geometric, but it does not yet display local coordinates. To verify that $\mathbb{CP}^n$ is a complex manifold, we must cover it by charts and compute the coordinate changes. The affine charts come from normalising one homogeneous coordinate to be $1$. [example: Affine Charts On Projective Space] For $0 \le i \le n$, set \begin{align*} U_i=\{[z_0:\dots:z_n]\in \mathbb{CP}^n:z_i\ne 0\}. \end{align*} Define $\varphi_i:U_i\to \mathbb C^n$ by \begin{align*} \varphi_i([z_0:\dots:z_n])=\left(\frac{z_0}{z_i},\dots,\frac{z_{i-1}}{z_i},\frac{z_{i+1}}{z_i},\dots,\frac{z_n}{z_i}\right). \end{align*} This is well-defined on homogeneous coordinates because replacing $z$ by $\lambda z$ gives \begin{align*} \frac{\lambda z_k}{\lambda z_i}=\frac{z_k}{z_i} \end{align*} for every $k\ne i$. The inverse map sends $(w_0,\dots,w_{i-1},w_{i+1},\dots,w_n)\in\mathbb C^n$ to the homogeneous point \begin{align*} [w_0:\dots:w_{i-1}:1:w_{i+1}:\dots:w_n]. \end{align*} Indeed, if $[z_0:\dots:z_n]\in U_i$, then applying this inverse to $\varphi_i([z])$ gives \begin{align*} \left[\frac{z_0}{z_i}:\dots:\frac{z_{i-1}}{z_i}:1:\frac{z_{i+1}}{z_i}:\dots:\frac{z_n}{z_i}\right]=[z_0:\dots:z_n], \end{align*} because multiplying all displayed homogeneous coordinates by $z_i$ recovers $(z_0,\dots,z_n)$. Now fix $i\ne j$ and work on $U_i\cap U_j$. In the $i$-chart, write the affine coordinates as $w_k=z_k/z_i$ for $k\ne i$, with the omitted coordinate corresponding to $z_i/z_i=1$. Since the point also lies in $U_j$, we have \begin{align*} w_j=\frac{z_j}{z_i}\ne 0. \end{align*} The $j$-chart coordinates are $z_k/z_j$ for $k\ne j$. For $k\ne i,j$, \begin{align*} \frac{z_k}{z_j}=\frac{z_k/z_i}{z_j/z_i}=\frac{w_k}{w_j}. \end{align*} The coordinate corresponding to $z_i/z_j$ is \begin{align*} \frac{z_i}{z_j}=\frac{1}{z_j/z_i}=\frac{1}{w_j}. \end{align*} Thus every component of $\varphi_j\circ\varphi_i^{-1}$ is either $w_k/w_j$ or $1/w_j$, on the open set where $w_j\ne 0$. These are holomorphic functions of the affine coordinates, so the affine charts define a complex atlas on $\mathbb{CP}^n$. Since each chart has $n$ affine coordinates, this gives $\mathbb{CP}^n$ the structure of a complex manifold of complex dimension $n$. [/example] A second global construction identifies points by translations rather than by scalar multiplication. For the quotient to have local charts inherited from $\mathbb C^n$, nearby representatives must not be identified by arbitrarily small nonzero translations. The subgroup of translations therefore has to be discrete, and to make the quotient compact in all real directions it must span the full underlying real [vector space](/page/Vector%20Space). This is the role of a lattice. [definition: Lattice In Complex Euclidean Space] A lattice in $\mathbb C^n$ is a subgroup $\Lambda \subset \mathbb C^n$ generated by an $\mathbb R$-basis of the real vector space underlying $\mathbb C^n$. [/definition] Since $\mathbb C^n$ has real dimension $2n$, such a lattice is isomorphic to $\mathbb Z^{2n}$. The point of imposing this condition is that small balls in $\mathbb C^n$ meet at most one translate of a chosen point, so local complex coordinates can descend to the quotient. This motivates the following definition of the complex torus associated to a lattice. [definition: Complex Torus] Let $\Lambda \subset \mathbb C^n$ be a lattice. The quotient \begin{align*} \mathbb C^n/\Lambda \end{align*} with the [quotient topology](/page/Quotient%20Topology) and the complex charts induced by the projection $\mathbb C^n \to \mathbb C^n/\Lambda$ is called a complex torus. [/definition] The quotient is compact and topologically a real $2n$-torus, but its complex structure comes from translation-invariant charts. We should check that the transition maps really are holomorphic, because this is the point where the quotient construction interacts with the atlas definition. The calculation also shows why translations are harmless in complex coordinates. [example: Local Coordinates On A Complex Torus] Let $\pi:\mathbb C^n\to \mathbb C^n/\Lambda$ be the quotient map, so $\pi(z)=\pi(w)$ exactly when $w-z\in\Lambda$. Since $\Lambda$ is generated by a real basis of $\mathbb C^n$, it is discrete, so for each $a\in\mathbb C^n$ we may choose $r>0$ such that \begin{align*} (B(a,r)-B(a,r))\cap \Lambda=\{0\}. \end{align*} If $z,w\in B(a,r)$ and $\pi(z)=\pi(w)$, then $w-z\in\Lambda$. Also $w-z\in B(a,r)-B(a,r)$, so the displayed condition gives $w-z=0$, hence $z=w$. Thus $\pi|_{B(a,r)}$ is injective. The image $\pi(B(a,r))$ is open because \begin{align*} \pi^{-1}(\pi(B(a,r)))=\bigcup_{\lambda\in\Lambda}(B(a,r)+\lambda), \end{align*} which is a union of open balls. Therefore the inverse of $\pi|_{B(a,r)}$ gives a coordinate chart \begin{align*} \chi_a:\pi(B(a,r))\to B(a,r)\subseteq \mathbb C^n. \end{align*} Now take two such charts $\chi_a$ and $\chi_b$. If $x\in \pi(B(a,r))\cap \pi(B(b,s))$, write \begin{align*} z=\chi_a(x)\in B(a,r) \end{align*} and \begin{align*} w=\chi_b(x)\in B(b,s). \end{align*} Since both represent the same quotient point, $\pi(z)=x=\pi(w)$, so there is a unique $\lambda\in\Lambda$ with \begin{align*} w=z+\lambda. \end{align*} Hence, on the part of the overlap where this same $\lambda$ occurs, the transition map is \begin{align*} \chi_b\circ \chi_a^{-1}(z)=z+\lambda. \end{align*} Its $j$th component is $z_j+\lambda_j$, a polynomial of degree $1$ in the affine coordinates, hence holomorphic. Thus the quotient charts have holomorphic transition maps, and translations by lattice elements are exactly the coordinate changes on the complex torus. [/example] Complex tori make the topological content of complex manifolds visible. They have the same underlying topological type as products of circles, but their holomorphic behaviour depends on the lattice $\Lambda$, not only on the real torus. In later topics this distinction becomes the source of periods, theta functions, and the difference between arbitrary complex tori and those that can be embedded into projective space. After seeing open sets, projective space, and tori, one recurring case needs its own name: a complex manifold with only one complex coordinate locally. In this dimension the local theory is one-variable complex analysis, but the global object can have nontrivial topology and need not sit inside one coordinate plane. The following definition isolates that setting so later results can refer to the one-dimensional theory without changing the chart-based definition of complex manifold. [definition: Riemann Surface] A Riemann surface is a connected complex manifold of complex dimension $1$. [/definition] Thus open subsets of $\mathbb C$, the Riemann sphere $\mathbb{CP}^1$, and one-dimensional complex tori are Riemann surfaces. The important lesson from all these examples is that local holomorphic coordinates can coexist with nontrivial global topology. This prepares the next section, where we ask which maps preserve such structures. ## Holomorphic Maps and Local Normal Forms Once the objects are defined by holomorphic coordinate changes, the next question is which maps preserve the structure. The answer must be local in charts, and it must not depend on the chosen charts. We therefore define holomorphic maps by reducing them to holomorphic maps between domains. [definition: Holomorphic Map Between Complex Manifolds] Let $X$ and $Y$ be complex manifolds of complex dimensions $m$ and $n$. A continuous map $F:X \to Y$ is holomorphic if for every $p \in X$, every chart $(U,\varphi)$ on $X$ with $p \in U$, and every chart $(V,\psi)$ on $Y$ with $F(p) \in V$, the coordinate representation \begin{align*} \psi \circ F \circ \varphi^{-1}:\varphi(U \cap F^{-1}(V)) \to \psi(V) \end{align*} is holomorphic. [/definition] The intersection $U \cap F^{-1}(V)$ is open by continuity of $F$, so the displayed map is a map between domains in complex Euclidean spaces. This local formulation is chart-independent because the transition maps on both sides are holomorphic. Having defined the morphisms, we next need the corresponding notion of isomorphism. In complex geometry, preserving the structure in both directions is the correct notion of sameness. [definition: Biholomorphism] Let $X$ and $Y$ be complex manifolds. A biholomorphism from $X$ to $Y$ is a holomorphic map $F:X \to Y$ that is bijective and whose inverse $F^{-1}:Y \to X$ is holomorphic. [/definition] A biholomorphism identifies two complex manifolds as the same object in the holomorphic category. In one complex variable, the local version asks when a holomorphic function has a holomorphic inverse near a point. The answer is controlled by the ordinary complex derivative. [quotetheorem:4950] [citeproof:4950] The nonzero-derivative hypothesis is essential: the map $f:\mathbb C \to \mathbb C$ given by $f(z)=z^2$ has derivative $0$ at $0$, and no restriction to a neighbourhood of $0$ is injective because $z$ and $-z$ have the same image. The theorem is also only local; it does not assert that a function with nonzero derivative everywhere is globally injective or has a single-valued inverse on its image. What it gives is the one-dimensional model for a local holomorphic coordinate change. This local-global distinction is already visible in one complex variable. The exponential map provides the standard illustration: every point has a local logarithm, but there is no global logarithm on the punctured plane compatible with the entire exponential map. [example: A Local Biholomorphism] Consider $f:\mathbb C \to \mathbb C$ given by $f(z)=e^z$. For any $a\in\mathbb C$, the complex Jacobian is the $1\times 1$ matrix whose only entry is \begin{align*} f'(a)=e^a. \end{align*} This entry is nonzero because \begin{align*} e^a e^{-a}=e^{a-a}=e^0=1. \end{align*} Thus $f'(a)$ is nonzero, so by the *[Holomorphic Inverse Function Theorem](/theorems/4950)* there are neighbourhoods $a\in U_a\subseteq\mathbb C$ and $e^a\in V_a\subseteq\mathbb C$ such that $e^z|_{U_a}:U_a\to V_a$ is a biholomorphism. The same map is not globally injective. For every $z\in\mathbb C$, \begin{align*} e^{z+2\pi i}=e^z e^{2\pi i}=e^z(\cos 2\pi+i\sin 2\pi)=e^z(1+0i)=e^z. \end{align*} Since $z+2\pi i\ne z$, two distinct points have the same image. Hence the exponential map is locally biholomorphic everywhere but not a global biholomorphism onto its image. [/example] For several complex variables, local normal forms require matrix conditions rather than a single derivative. Many geometric subsets arise as zero sets of maps to a lower-dimensional target. To study those zero sets, we need the holomorphic implicit function theorem. [quotetheorem:7002] [citeproof:7002] The invertibility of the $w$-Jacobian is the condition that the equations can actually be solved for the $w$-variables. If it fails, the zero set may have a singularity rather than be a graph: for instance $F(z,w)=w^2-z^3$ has vanishing $w$-derivative at $(0,0)$, and its zero set has a cusp at the origin. The theorem is also local in both the source and target variables; it gives a graph near $(a,b)$, not a global parametrisation of the entire zero set. The implicit theorem shows that suitable holomorphic equations produce local holomorphic graphs. We now need an intrinsic name for subsets that locally have this graph or coordinate-plane form. This is the complex analogue of an embedded submanifold, with holomorphic coordinate changes built in. [definition: Analytic Submanifold] Let $X$ be a complex manifold of complex dimension $n$. A subset $Y \subseteq X$ is an analytic submanifold of complex dimension $k$ if for every $p \in Y$ there is a holomorphic chart $(U,\varphi)$ of $X$ around $p$ such that \begin{align*} \varphi(U \cap Y)=\varphi(U) \cap (\mathbb C^k \times \{0\}) \end{align*} inside $\mathbb C^k \times \mathbb C^{n-k}$. [/definition] The definition uses a normal form rather than a preferred set of equations, so it is stable under biholomorphic coordinate changes. The next example shows how the implicit function theorem verifies the condition in projective geometry. A smooth hypersurface in projective space is the basic model. [example: Quadric Hypersurface In Projective Space] In $\mathbb{CP}^{n+1}$, consider the zero set \begin{align*} Q=\{[z_0:\dots:z_{n+1}]:z_0^2+\cdots+z_{n+1}^2=0\}. \end{align*} Let \begin{align*} P(z)=z_0^2+\cdots+z_{n+1}^2. \end{align*} For each $k$, \begin{align*} \frac{\partial P}{\partial z_k}(z)=2z_k. \end{align*} Hence \begin{align*} \nabla P(z)=(2z_0,\dots,2z_{n+1}). \end{align*} If $\nabla P(z)=0$, then $2z_k=0$ for every $k$, so $z_k=0$ for every $k$, contradicting the requirement that homogeneous coordinates are represented by a nonzero vector. Thus $\nabla P(z)\ne 0$ at every representative of every point of $Q$. Now work in the affine chart $U_\ell=\{z_\ell\ne 0\}$. Write \begin{align*} w_k=\frac{z_k}{z_\ell} \end{align*} for $k\ne \ell$, so the normalized homogeneous representative has $\ell$th coordinate $1$. Since $z_k=w_kz_\ell$ for $k\ne \ell$, the equation $P(z)=0$ becomes \begin{align*} 0=z_0^2+\cdots+z_{n+1}^2=z_\ell^2\left(1+\sum_{k\ne \ell}w_k^2\right). \end{align*} Because $z_\ell\ne 0$, this is equivalent to \begin{align*} F_\ell(w):=1+\sum_{k\ne \ell}w_k^2=0. \end{align*} At a point $w$ satisfying $F_\ell(w)=0$, not all $w_k$ can be zero, since then $F_\ell(w)=1$. Choose an index $m\ne \ell$ with $w_m\ne 0$. Then \begin{align*} \frac{\partial F_\ell}{\partial w_m}(w)=2w_m\ne 0. \end{align*} By the *Holomorphic Implicit Function Theorem*, near this point the equation $F_\ell(w)=0$ solves holomorphically for $w_m$ as a function of the remaining $n$ affine coordinates. Therefore $Q\cap U_\ell$ is locally an analytic hypersurface in $U_\ell\cong\mathbb C^{n+1}$. Since the affine charts cover $\mathbb{CP}^{n+1}$, the quadric $Q$ is a complex manifold of complex dimension $n$. [/example] This example is the first point where the course touches algebraic geometry: homogeneous polynomial equations in projective space give projective varieties, and the condition $\nabla P\ne 0$ singles out the smooth ones. The complex-manifold structure obtained from the implicit function theorem is therefore the differential-geometric face of a projective algebraic construction. The same [rank principle](/theorems/4796) also describes maps rather than subsets. A map with maximal rank onto the target should locally forget some coordinates. To make this precise, we first name the holomorphic maps with surjective differential. [definition: Holomorphic Submersion] Let $F:X \to Y$ be a holomorphic map between complex manifolds. The map $F$ is a holomorphic submersion at $p \in X$ if, in local holomorphic coordinates around $p$ and $F(p)$, the complex Jacobian matrix has rank $\dim_{\mathbb C}Y$ at $p$. [/definition] The rank condition does not depend on the chosen charts because changing charts multiplies the Jacobian by invertible complex matrices. What remains to be justified is stronger than invariance of rank: if the differential is surjective, the apparent extra variables in the source should be removable by holomorphic coordinates, so that the map really becomes a projection near the point. Without such a normal form, the word submersion would not yet explain the local shape of its fibres. [quotetheorem:7005] [citeproof:7005] The rank hypothesis is what prevents the fibre from acquiring a singular or higher-dimensional tangent direction. For example, $F:\mathbb C \to \mathbb C$ given by $F(z)=z^2$ is not a submersion at $0$, and the fibre over $0$ is not locally modelled by the fibre of a projection with nonzero derivative. The theorem is local near the chosen point; it does not say that a global submersion is globally a product projection, since monodromy and topology can obstruct global coordinates. This normal form is the coordinate reason that fibres of submersions are manifolds of the expected dimension. The dual rank condition asks for injectivity on tangent spaces instead of surjectivity onto the target. This motivates the corresponding notion of holomorphic immersion. [definition: Holomorphic Immersion] Let $F:X \to Y$ be a holomorphic map between complex manifolds. The map $F$ is a holomorphic immersion at $p \in X$ if, in local holomorphic coordinates around $p$ and $F(p)$, the complex Jacobian matrix has rank $\dim_{\mathbb C}X$ at $p$. [/definition] Immersions are the maps expected to look locally like inclusions of coordinate planes, but injective differential alone is a pointwise condition. The missing issue is whether holomorphic coordinates can straighten the map itself, not just its derivative, so that no cusp or folded local parametrisation remains near the point. The local normal form supplies exactly this coordinate model while leaving global self-intersections as a separate question. [quotetheorem:7006] [citeproof:7006] Injective rank is necessary because a map with a critical point cannot be straightened into an inclusion. The map $F:\mathbb C \to \mathbb C^2$ given by $F(z)=(z^2,z^3)$ has derivative $0$ at $0$, and near that point its image has a cusp rather than the local shape of a coordinate line. Even when the immersion condition holds everywhere, the conclusion is still local: an immersion may have self-intersections globally, so its image need not be an embedded submanifold without an additional injectivity or embedding hypothesis. These local forms are the holomorphic counterparts of the constant-rank theorem, but the proof relies on holomorphic inverse and implicit function theorems rather than only smooth calculus. They will reappear in Chapter 4, where vector bundles and holomorphic subbundles are described by transition functions. Before reaching vector bundles, we need to translate holomorphic coordinates into tangent-space language. ## Holomorphic Tangent Directions The final problem in this chapter is to express the coordinate definition intrinsically. Complex manifolds have an underlying smooth manifold of real dimension $2n$, but their tangent spaces carry extra structure: multiplication by $i$ in holomorphic coordinates. We begin by separating the underlying smooth manifold from the additional complex structure. [definition: Underlying Smooth Manifold] Let $X$ be a complex manifold of complex dimension $n$. The underlying smooth manifold $X_{\mathbb R}$ is the real smooth manifold of dimension $2n$ obtained by viewing each holomorphic chart as a smooth chart valued in $\mathbb R^{2n}$. [/definition] Holomorphic transition maps are smooth, so this construction is compatible on overlaps. But a complex manifold contains more information than the real smooth manifold alone: it has a preferred notion of multiplication by $i$ on tangent vectors. This motivates the following definition of the almost complex structure associated to a complex manifold. [definition: Almost Complex Structure Associated To A Complex Manifold] Let $X$ be a complex manifold and let $X_{\mathbb R}$ be its underlying smooth manifold. The associated almost complex structure is the bundle endomorphism \begin{align*} J:TX_{\mathbb R} \to TX_{\mathbb R} \end{align*} which, in holomorphic coordinates $z_j=x_j+iy_j$, satisfies \begin{align*} J\left(\frac{\partial}{\partial x_j}\right)=\frac{\partial}{\partial y_j} \end{align*} and \begin{align*} J\left(\frac{\partial}{\partial y_j}\right)=-\frac{\partial}{\partial x_j}. \end{align*} [/definition] The holomorphicity of coordinate changes ensures that this local formula defines a global tensor. The next task is to split tangent directions into holomorphic and antiholomorphic parts, which cannot be done over the real tangent bundle because the relevant eigenvalues are $i$ and $-i$. This motivates complexifying the tangent bundle first. [definition: Complexified Tangent Bundle] Let $X$ be a smooth manifold. The complexified tangent bundle is \begin{align*} T_\mathbb C X := TX \otimes_{\mathbb R} \mathbb C. \end{align*} [/definition] Complexification allows the real endomorphism $J$ to be diagonalised. The obstruction over the real tangent bundle was that the eigenvalues $i$ and $-i$ are not real; after tensoring with $\mathbb C$, the two eigenspaces become honest complex subbundles. Naming these subbundles is what lets later constructions distinguish holomorphic tangent directions from antiholomorphic ones. [definition: Tangent Bundles Of Type One Zero And Zero One] Let $X$ be a complex manifold with associated almost complex structure $J:TX_{\mathbb R}\to TX_{\mathbb R}$. Extend $J$ $\mathbb C$-linearly to a bundle endomorphism \begin{align*} J:T_\mathbb C X \to T_\mathbb C X. \end{align*} The subbundles of $T_\mathbb C X$ are \begin{align*} T^{1,0}X=\{v \in T_\mathbb C X:Jv=iv\} \end{align*} and \begin{align*} T^{0,1}X=\{v \in T_\mathbb C X:Jv=-iv\}. \end{align*} [/definition] The notation anticipates the decomposition of complex differential forms into type $(p,q)$. To make the splitting concrete, we should compute the eigenvectors in a holomorphic coordinate chart. These local frames are the vector fields that will later pair with $dz_j$ and $d\bar z_j$. [example: Coordinate Frames For The Splitting] On a holomorphic chart with $z_j=x_j+iy_j$, work in the complexified tangent bundle and define \begin{align*} \frac{\partial}{\partial z_j}=\frac{1}{2}\left(\frac{\partial}{\partial x_j}-i\frac{\partial}{\partial y_j}\right) \end{align*} and \begin{align*} \frac{\partial}{\partial \bar z_j}=\frac{1}{2}\left(\frac{\partial}{\partial x_j}+i\frac{\partial}{\partial y_j}\right). \end{align*} Since $J$ is extended $\mathbb C$-linearly and satisfies $J(\partial/\partial x_j)=\partial/\partial y_j$ and $J(\partial/\partial y_j)=-\partial/\partial x_j$, we have \begin{align*} J\left(\frac{\partial}{\partial z_j}\right)=\frac{1}{2}\left(\frac{\partial}{\partial y_j}+i\frac{\partial}{\partial x_j}\right). \end{align*} Also \begin{align*} i\frac{\partial}{\partial z_j}=\frac{1}{2}\left(i\frac{\partial}{\partial x_j}+\frac{\partial}{\partial y_j}\right), \end{align*} so $J(\partial/\partial z_j)=i\,\partial/\partial z_j$. Thus $\partial/\partial z_j$ is a local section of $T^{1,0}X$. Similarly, \begin{align*} J\left(\frac{\partial}{\partial \bar z_j}\right)=\frac{1}{2}\left(\frac{\partial}{\partial y_j}-i\frac{\partial}{\partial x_j}\right). \end{align*} On the other hand, \begin{align*} -i\frac{\partial}{\partial \bar z_j}=\frac{1}{2}\left(-i\frac{\partial}{\partial x_j}+\frac{\partial}{\partial y_j}\right), \end{align*} so $J(\partial/\partial \bar z_j)=-i\,\partial/\partial \bar z_j$. Hence $\partial/\partial \bar z_j$ is a local section of $T^{0,1}X$. The real coordinate vectors are recovered from these two complex vectors by \begin{align*} \frac{\partial}{\partial x_j}=\frac{\partial}{\partial z_j}+\frac{\partial}{\partial \bar z_j} \end{align*} and \begin{align*} \frac{\partial}{\partial y_j}=i\left(\frac{\partial}{\partial z_j}-\frac{\partial}{\partial \bar z_j}\right). \end{align*} Therefore any complexified tangent vector \begin{align*} v=\sum_{j=1}^n\left(a_j\frac{\partial}{\partial x_j}+b_j\frac{\partial}{\partial y_j}\right) \end{align*} with $a_j,b_j\in\mathbb C$ becomes \begin{align*} v=\sum_{j=1}^n(a_j+ib_j)\frac{\partial}{\partial z_j}+\sum_{j=1}^n(a_j-ib_j)\frac{\partial}{\partial \bar z_j}. \end{align*} The first sum lies in $T^{1,0}X$ and the second lies in $T^{0,1}X$. The decomposition is unique because the displayed recovery formulas show that the vectors $\partial/\partial z_j$ and $\partial/\partial \bar z_j$ determine, and are determined by, the complexified real coordinate frame. Thus these vector fields give the local frames for the splitting $T_\mathbb C X=T^{1,0}X\oplus T^{0,1}X$. [/example] This decomposition is not merely notation; it should recover holomorphicity itself. After seeing the coordinate frames, the natural question is whether a smooth map is holomorphic exactly when its differential respects the complex tangent directions. This motivates the following [differential criterion for holomorphic maps](/theorems/7007). [quotetheorem:7007] [citeproof:7007] The smoothness hypothesis ensures that the differential exists as a bundle map and that the Cauchy-Riemann equations can be interpreted pointwise. The condition is also genuinely restrictive: complex conjugation $F:\mathbb C \to \mathbb C$, $F(z)=\bar z$, is smooth, but its differential anticommutes with the standard complex structure rather than commuting with it, so it is not holomorphic. Like the coordinate definition, the criterion is local; it checks holomorphicity chart by chart and does not by itself encode global properties such as injectivity, properness, or being a biholomorphism. The value of the criterion is that it separates two layers of structure. First, a complex manifold gives a smooth manifold with an endomorphism $J$ of its tangent bundle. Second, holomorphic maps are exactly the smooth maps whose differentials are complex-linear for these endomorphisms. The next question is therefore intrinsic: if a smooth manifold is given such an endomorphism in advance, what extra condition guarantees that it came from holomorphic coordinate charts rather than only from pointwise linear algebra? This criterion explains why tangent-bundle language is efficient: holomorphicity becomes a statement about compatibility with $J$. It also suggests a broader question: if a smooth manifold has such an endomorphism $J$, when does it come from holomorphic charts? That question leads from complex manifolds to almost complex manifolds. [definition: Almost Complex Manifold] An almost complex manifold is a smooth manifold $M$ equipped with a bundle endomorphism \begin{align*} J:TM \to TM \end{align*} satisfying $J^2=-\operatorname{id}_{TM}$. [/definition] Every complex manifold gives an almost complex manifold, but the converse requires an integrability condition. The obstruction must involve derivatives of $J$, because the pointwise condition $J^2=-\operatorname{id}_{TM}$ only describes linear algebra on each tangent space. The tensor measuring this obstruction is the Nijenhuis tensor. [definition: Nijenhuis Tensor] Let $(M,J)$ be an almost complex manifold. The Nijenhuis tensor of $J$ is the tensor \begin{align*} N_J \in \Gamma(\Lambda^2T^*M\otimes TM) \end{align*} whose value on vector fields $X,Y \in \mathfrak X(M)$ is defined by \begin{align*} N_J(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]. \end{align*} [/definition] Thus $N_J$ may be read either as a tensor field or as the corresponding alternating $C^\infty(M)$-bilinear map $\mathfrak X(M)\times \mathfrak X(M)\to \mathfrak X(M)$. For a complex manifold, this tensor vanishes because the local holomorphic coordinate vector fields have bracket relations compatible with the splitting into $T^{1,0}$ and $T^{0,1}$. The central issue is whether this vanishing condition is also sufficient for holomorphic coordinates to exist. This motivates the following integrability theorem. [quotetheorem:7003] The theorem is used here as the analytic integrability criterion for almost complex structures. Its content is that the tensorial condition $N_J=0$, which is stated only in terms of Lie brackets and $J$, is exactly the condition needed for local holomorphic coordinates inducing $J$ to exist. The condition is not automatic. On a small neighbourhood of $0$ in $\mathbb C^2$ with coordinates $(z_1,z_2)$, define a complex rank-two distribution by \begin{align*} L_1&=\frac{\partial}{\partial \bar z_1},& L_2&=\frac{\partial}{\partial \bar z_2}+\bar z_1\frac{\partial}{\partial z_1}. \end{align*} Together with its complex conjugate, this distribution gives an almost complex structure near $0$ by declaring it to be the $(-i)$-eigenspace. However \begin{align*} [L_1,L_2]=\frac{\partial}{\partial z_1}, \end{align*} which is a $(1,0)$ direction rather than a section of the chosen $(0,1)$ distribution. Thus the bracket-closure condition fails, so the corresponding Nijenhuis tensor is nonzero and no holomorphic coordinate system induces this almost complex structure. The proof of Newlander-Nirenberg is much deeper than this tensorial statement suggests. It solves a nonlinear first-order analytic problem for local coordinates, so the theorem functions here as an existence result rather than as an explicit recipe for constructing charts from $J$. [remark: Role Of Integrability] The tangent-bundle splitting $T_\mathbb C M=T^{1,0}M\oplus T^{0,1}M$ exists for every almost complex manifold. Integrability says that this splitting is compatible with Lie brackets in the way needed for local holomorphic coordinates to exist. Thus the first chapter ends with the bridge between coordinate complex geometry and the intrinsic tensor language used in Hermitian geometry. [/remark] Once the complex tangent bundle has been split into its holomorphic and antiholomorphic parts, the natural differential calculus must respect that decomposition. The next chapter develops forms of type $(p,q)$ so that $ar\partial$ can be treated as the operator that measures compatibility with the complex structure. # 2. Differential Forms of Type (p,q) This chapter develops the calculus of differential forms adapted to the complex structure. It assumes the previous chapter's construction of the complexified tangent bundle, the splitting into $(1,0)$ and $(0,1)$ directions, and the basic calculus of smooth differential forms from real differential geometry. With the notation $T^{1,0}X$ and $T^{0,1}X$ fixed there, we now pass from vector fields to exterior forms and obtain the bigrading that underlies Dolbeault theory. The main questions are how the exterior derivative interacts with this splitting, how holomorphic maps preserve type, and why the local equation $\bar{\partial}u=\beta$ has solutions on polydiscs. ## Complex-Valued Forms and Type Decomposition A real differential form is not flexible enough to record holomorphic and anti-holomorphic directions separately. Since the complexified cotangent bundle decomposes into dual pieces, the natural first step is to allow complex coefficients and then ask how many factors come from each side. [definition: Complex-Valued Differential Form] Let $X$ be a smooth manifold. A complex-valued differential $k$-form on $X$ is a smooth section of $\Lambda^k T^*X \otimes_{\mathbb R} \mathbb C$. The space of complex-valued differential $k$-forms is denoted $A^k(X)$. [/definition] Complex-valued forms are obtained from real forms by allowing coefficients in $\mathbb C$. Thus every $\alpha \in A^k(X)$ can be written uniquely as $\alpha = \alpha_1 + i\alpha_2$ with $\alpha_1,\alpha_2$ real $k$-forms, and the exterior derivative extends by complex linearity. [example: Complex Differential of a Coordinate Function] On $\mathbb C$ with coordinate $z=x+iy$, complex-valued one-forms are smooth complex-linear combinations of $dx$ and $dy$. The two forms \begin{align*} dz=dx+i\,dy, \qquad d\bar z=dx-i\,dy \end{align*} are complex-linearly independent: if $a\,dz+b\,d\bar z=0$, then \begin{align*} a(dx+i\,dy)+b(dx-i\,dy)=(a+b)\,dx+i(a-b)\,dy=0. \end{align*} Since $dx$ and $dy$ are independent, $a+b=0$ and $i(a-b)=0$, hence $a=b$ and $a=-b$, so $a=b=0$. Adding and subtracting the two defining identities gives \begin{align*} dz+d\bar z=(dx+i\,dy)+(dx-i\,dy)=2\,dx. \end{align*} Therefore \begin{align*} dx=\frac{1}{2}(dz+d\bar z). \end{align*} Similarly, \begin{align*} dz-d\bar z=(dx+i\,dy)-(dx-i\,dy)=2i\,dy. \end{align*} Dividing by $2i$ gives \begin{align*} dy=\frac{1}{2i}(dz-d\bar z). \end{align*} Thus $dz$ and $d\bar z$ span the same complexified cotangent directions as $dx$ and $dy$, but reorganised into holomorphic and anti-holomorphic directions. [/example] The example separates the two basic one-form directions, but a higher-degree form may contain several holomorphic factors and several anti-holomorphic factors at once. To make that count intrinsic, we need a definition that records the number of factors of each kind in holomorphic coordinates. [definition: Form of Type P Q] Let $X$ be a complex manifold of complex dimension $n$. A complex-valued differential form $\alpha$ has type $(p,q)$ if, in every holomorphic coordinate chart, it can be written as a finite sum \begin{align*} \alpha = \sum_{|I|=p,\,|J|=q} \alpha_{I\bar J}\, dz_I \wedge d\bar z_J, \end{align*} where $\alpha_{I\bar J}$ are smooth complex-valued functions, $dz_I=dz_{i_1}\wedge \cdots \wedge dz_{i_p}$, and $d\bar z_J=d\bar z_{j_1}\wedge \cdots \wedge d\bar z_{j_q}$. [/definition] The previous definition gives the local shape of a single pure-type form. To use these forms systematically, we need to collect all smooth forms of a fixed bidegree into one space; otherwise statements such as direct-sum decompositions or operators changing bidegree would have no fixed domain and codomain. The notation below records that bookkeeping globally on the manifold. [definition: Space of P Q Forms] Let $X$ be a complex manifold. The space of smooth differential forms of type $(p,q)$ on $X$ is denoted $A^{p,q}(X)$. [/definition] The type notation refines the usual degree. A form in $A^{p,q}(X)$ has total degree $p+q$, and no such forms exist when $p<0$, $q<0$, $p>n$, or $q>n$. The next issue is whether every complex-valued form has a unique sum of such type pieces, since later operators will be defined by projecting onto those pieces. [quotetheorem:7008] [citeproof:7008] The decomposition allows us to write $\alpha^{p,q}$ for the $(p,q)$ component of a form, and it gives a coordinate-free meaning to separating holomorphic and anti-holomorphic covectors. The hypothesis that coordinate changes are holomorphic is essential: if a smooth coordinate change involved both $z$ and $\bar z$, then the pullback of a new holomorphic differential could contain both $dz$ and $d\bar z$ terms, so the local type projections would not glue. On a non-integrable almost complex manifold, there is still a pointwise decomposition of the complexified cotangent space, but the local holomorphic coordinate argument is unavailable and exterior differentiation can produce extra torsion-type components. The theorem also does not say that $d$ preserves a single bidegree; it only decomposes each form into pieces before any differentiation is applied. The next example makes the projection operation concrete in ordinary coordinates before we study how $d$ interacts with it. [example: Decomposing a One-Form on C Two] Let $\alpha=dx_1+i\,dy_2$ on $\mathbb C^2$, where $z_j=x_j+iy_j$. From $dz_j=dx_j+i\,dy_j$ and $d\bar z_j=dx_j-i\,dy_j$, adding gives $dz_j+d\bar z_j=2\,dx_j$, so \begin{align*} dx_j=\frac{1}{2}(dz_j+d\bar z_j). \end{align*} Subtracting gives $dz_j-d\bar z_j=2i\,dy_j$, so \begin{align*} dy_j=\frac{1}{2i}(dz_j-d\bar z_j). \end{align*} Applying these identities with $j=1$ and $j=2$ gives \begin{align*} \alpha=\frac{1}{2}(dz_1+d\bar z_1)+i\cdot \frac{1}{2i}(dz_2-d\bar z_2). \end{align*} Since $i/(2i)=1/2$, this becomes \begin{align*} \alpha=\frac{1}{2}dz_1+\frac{1}{2}d\bar z_1+\frac{1}{2}dz_2-\frac{1}{2}d\bar z_2. \end{align*} Grouping the $dz$ terms and the $d\bar z$ terms, \begin{align*} \alpha=\frac{1}{2}(dz_1+dz_2)+\frac{1}{2}(d\bar z_1-d\bar z_2). \end{align*} The first summand has type $(1,0)$ and the second has type $(0,1)$, so \begin{align*} \alpha^{1,0}=\frac{1}{2}(dz_1+dz_2), \qquad \alpha^{0,1}=\frac{1}{2}(d\bar z_1-d\bar z_2). \end{align*} This separates the same real one-form into its holomorphic and anti-holomorphic components. [/example] Conjugation exchanges the two halves of the decomposition. If $\alpha \in A^{p,q}(X)$, then $\bar\alpha \in A^{q,p}(X)$, since complex conjugation sends $dz_i$ to $d\bar z_i$ and reverses the roles of holomorphic and anti-holomorphic coefficients. ## The Operators Partial and Dbar The exterior derivative raises total degree by one. Once forms have a type decomposition, the next question is whether $d$ raises the holomorphic degree, the anti-holomorphic degree, or both. [definition: Dolbeault Operators] Let $X$ be a complex manifold. For $\alpha \in A^{p,q}(X)$, define \begin{align*} \partial \alpha = (d\alpha)^{p+1,q}, \qquad \bar\partial \alpha = (d\alpha)^{p,q+1}. \end{align*} The maps $\partial:A^{p,q}(X)\to A^{p+1,q}(X)$ and $\bar\partial:A^{p,q}(X)\to A^{p,q+1}(X)$ are called the Dolbeault operators. [/definition] The previous definition names the two projected pieces of $d$, but a priori the exterior derivative of a pure-type form could also have components in other bidegrees. The key question is whether integrability of the complex structure rules out those extra terms. The result below is the structural fact that makes $\partial$ and $\bar\partial$ a genuine splitting of $d$ on complex manifolds. [quotetheorem:7004] [citeproof:7004] Thus the de Rham differential splits into two pieces on a complex manifold. The integrability hypothesis is doing real work: for a general almost complex structure, $d$ can have components of type $(p+2,q-1)$ and $(p-1,q+2)$, so the formula $d=\partial+\bar\partial$ is no longer the full story. Even in the integrable case, the theorem does not say that $\partial$ or $\bar\partial$ differentiates coefficients while leaving all algebraic signs unchanged; the displayed formulae still include the wedge order and hence the usual exterior-sign bookkeeping. The point of the result is that all new terms fall into exactly two adjacent bidegrees, which makes it possible to isolate holomorphic and anti-holomorphic derivatives. The following computation shows this split for functions, where $\bar\partial f$ records precisely the anti-holomorphic variation. [example: Computing Dbar of Functions on C N] Let $U\subset \mathbb C^n$ be open and let $f\in C^\infty(U)$. Since a function has type $(0,0)$, its exterior derivative decomposes as \begin{align*} df=\sum_{j=1}^n \frac{\partial f}{\partial z_j}\,dz_j+\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\,d\bar z_j. \end{align*} The $\bar\partial$ operator keeps the $(0,1)$ part of $df$, so \begin{align*} \bar\partial f=\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\,d\bar z_j. \end{align*} Now take $f(z)=|z_1|^2+z_2=z_1\bar z_1+z_2$ on $\mathbb C^n$ with $n\ge 2$. For the first anti-holomorphic derivative, \begin{align*} \frac{\partial f}{\partial \bar z_1}=\frac{\partial}{\partial \bar z_1}(z_1\bar z_1)+\frac{\partial z_2}{\partial \bar z_1}=z_1+0=z_1. \end{align*} For $j=2$, \begin{align*} \frac{\partial f}{\partial \bar z_2}=\frac{\partial}{\partial \bar z_2}(z_1\bar z_1)+\frac{\partial z_2}{\partial \bar z_2}=0+0=0. \end{align*} For $j\ge 3$, neither $z_1\bar z_1$ nor $z_2$ depends on $\bar z_j$, hence \begin{align*} \frac{\partial f}{\partial \bar z_j}=0. \end{align*} Substituting these derivatives into the formula for $\bar\partial f$ gives \begin{align*} \bar\partial f=z_1\,d\bar z_1. \end{align*} Thus the anti-holomorphic variation comes entirely from the factor $\bar z_1$ in $|z_1|^2$, while the holomorphic summand $z_2$ contributes no $\bar\partial$ term. [/example] The sign rules are inherited from the graded Leibniz rule for $d$. Since $\partial$ and $\bar\partial$ each raise total degree by one, the same sign appears for both operators. [quotetheorem:7009] [citeproof:7009] These identities make the sign convention compatible with the product structure on forms. The total degree $p+q$ is necessary in the exponent: for instance, if $\alpha=d\bar z$ and $\beta=f$ on $\mathbb C$, then $\bar\partial(d\bar z\, f)$ carries the sign from moving a degree-one form past the derivative of $f$. The theorem does not assert commutativity of the wedge product; exchanging two forms still introduces the sign determined by their total degrees, not by the two bidegrees separately. Its role is to let later calculations with $\partial$ and $\bar\partial$ use the same graded algebra discipline as de Rham theory. The next relations come from the single identity $d^2=0$ after separating bidegrees. [quotetheorem:7010] [citeproof:7010] The identity $\bar\partial^2=0$ is the starting point of Dolbeault cohomology. The complex-manifold hypothesis is again essential: on a non-integrable almost complex manifold the corresponding anti-holomorphic operator need not square to zero, and the failure measures the obstruction to integrability. These equations do not imply that $\partial$ and $\bar\partial$ commute; the middle identity says they anti-commute as odd operators. They also do not force every $\bar\partial$-closed form to be globally $\bar\partial$-exact, because exactness is a separate local or global question. What the theorem gives is a cochain complex for each fixed $p$: \begin{align*} A^{p,0}(X)\xrightarrow{\bar\partial}A^{p,1}(X)\xrightarrow{\bar\partial}A^{p,2}(X)\xrightarrow{\bar\partial}\cdots \end{align*} is a cochain complex for each fixed $p$. ## Pullbacks and Coordinate Computations Differential forms are useful because they can be pulled back along smooth maps. In complex geometry, the relevant question is whether holomorphic maps respect the type splitting. A general smooth map need not do so: complex conjugation $C:\mathbb C\to\mathbb C$, $C(z)=\bar z$, satisfies $C^*(dw)=d\bar z$, so it sends a $(1,0)$-form on the target to a $(0,1)$-form on the source. This failure identifies holomorphicity as the exact condition that prevents type mixing. [quotetheorem:7011] [citeproof:7011] This theorem isolates a functorial feature of holomorphic geometry: holomorphic maps carry the holomorphic cotangent directions to holomorphic cotangent directions and the anti-holomorphic directions to anti-holomorphic ones. The hypothesis cannot be weakened to smoothness, as complex conjugation on $\mathbb C$ sends $dw$ to $d\bar z$ and reverses type. The theorem does not say that pullback preserves positivity, metrics, or harmonicity; it only concerns the algebraic bidegree of the resulting form. It also does not require $F$ to be a local biholomorphism, so some pulled-back forms may vanish when $dF$ has kernel that is not zero. The next question is stronger: once type is preserved, does pullback also commute with the two pieces of the exterior derivative? [quotetheorem:7012] [citeproof:7012] The preceding statement is frequently used in charts because it makes $\partial$ and $\bar\partial$ natural under holomorphic coordinate changes. Holomorphicity is necessary here for the same reason as before: if $F(z)=\bar z$, then $F^*(\partial w)=F^*(dw)=d\bar z$, while $\partial(F^*w)=\partial\bar z=0$, so the first identity fails. The theorem does not say that arbitrary smooth pullback commutes with the projected operators; smooth pullback commutes with $d$, but projection to type is not natural unless type is preserved. This result is what lets Dolbeault computations be made in one holomorphic chart and then transported to another. The next example uses the transition map on projective space as a basic model. [example: Forms on Projective One-Space in Affine Charts] Let $\mathbb{CP}^1$ have affine charts $U_0=\{[Z_0:Z_1]:Z_0\ne 0\}$ with coordinate $z=Z_1/Z_0$ and $U_1=\{[Z_0:Z_1]:Z_1\ne 0\}$ with coordinate $w=Z_0/Z_1$. On $U_0\cap U_1$ both $Z_0$ and $Z_1$ are nonzero, so \begin{align*} w=\frac{Z_0}{Z_1}=\frac{1}{Z_1/Z_0}=\frac{1}{z}=z^{-1}. \end{align*} Differentiating $w=z^{-1}$ gives \begin{align*} dw=d(z^{-1})=-z^{-2}\,dz. \end{align*} Conjugating $w=z^{-1}$ gives $\bar w=\bar z^{-1}$, and differentiating this identity gives \begin{align*} d\bar w=d(\bar z^{-1})=-\bar z^{-2}\,d\bar z. \end{align*} A local $(1,0)$-form on $U_1$ of the form $f(w)\,dw$ is rewritten in the $z$-chart by substituting $w=1/z$ and $dw=-z^{-2}\,dz$: \begin{align*} f(w)\,dw=f(1/z)(-z^{-2}\,dz)=-f(1/z)z^{-2}\,dz. \end{align*} This is still a smooth coefficient times $dz$, so it has type $(1,0)$ on the overlap. Similarly, a local $(0,1)$-form $g(w)\,d\bar w$ becomes \begin{align*} g(w)\,d\bar w=g(1/z)(-\bar z^{-2}\,d\bar z)=-g(1/z)\bar z^{-2}\,d\bar z. \end{align*} This is a smooth coefficient times $d\bar z$, so it has type $(0,1)$. Thus the projective transition map does not mix $dz$ with $d\bar z$; it only multiplies the coefficient by the corresponding holomorphic or anti-holomorphic Jacobian factor. [/example] Coordinate formulae also clarify how wedge signs interact with bidegree. If $\alpha\in A^{p,q}(X)$ and $\beta\in A^{r,s}(X)$, then $\alpha\wedge\beta\in A^{p+r,q+s}(X)$, and exchanging the order introduces the sign $(-1)^{(p+q)(r+s)}$. [example: A Sign in Bidegree Two] Let $\alpha=f\,dz_1\wedge d\bar z_2\in A^{1,1}(\mathbb C^2)$ and $\beta=g\,d\bar z_1\in A^{0,1}(\mathbb C^2)$. Using bilinearity of the wedge product and multiplication of coefficient functions, we get \begin{align*} \alpha\wedge\beta=(f\,dz_1\wedge d\bar z_2)\wedge(g\,d\bar z_1)=fg\,dz_1\wedge d\bar z_2\wedge d\bar z_1. \end{align*} The one-forms $d\bar z_2$ and $d\bar z_1$ anticommute, so \begin{align*} d\bar z_2\wedge d\bar z_1=-\,d\bar z_1\wedge d\bar z_2. \end{align*} Substituting this into the previous expression gives \begin{align*} \alpha\wedge\beta=fg\,dz_1\wedge(-\,d\bar z_1\wedge d\bar z_2)=-fg\,dz_1\wedge d\bar z_1\wedge d\bar z_2. \end{align*} The final form contains one holomorphic factor, $dz_1$, and two anti-holomorphic factors, $d\bar z_1$ and $d\bar z_2$, so it has type $(1,2)$. The minus sign is exactly the sign produced by moving the degree-one form $d\bar z_1$ past the degree-one form $d\bar z_2$. [/example] ## The Local Dbar Equation The identity $\bar\partial^2=0$ gives a necessary condition for solving $\bar\partial u=\beta$: the form $\beta$ must be $\bar\partial$-closed. The local theory asks whether this condition is also sufficient in simple coordinate domains. [definition: Dbar-Closed Form] Let $U$ be a complex manifold and let $\beta\in A^{p,q}(U)$. The form $\beta$ is $\bar\partial$-closed if $\bar\partial\beta=0$. [/definition] Being $\bar\partial$-closed is the complex analogue of being closed in de Rham theory. The natural local problem is to decide whether a closed form of positive anti-holomorphic degree has a local primitive. On polydiscs this is the complex counterpart of the Poincare lemma, with analytic input from the [Cauchy transform](/page/Cauchy%20Transform). In the quoted statement, $\mathcal{E}^{p,q}(D)$ denotes the smooth $(p,q)$-forms on $D$, the same objects written above as $A^{p,q}(D)$, and $\mathcal{E}^{p,\bullet}_D$ denotes the corresponding Dolbeault complex in fixed holomorphic degree $p$. [quotetheorem:3410] [citeproof:3410] The lemma says that the complex $A^{p,\bullet}$ has no local cohomology in positive anti-holomorphic degree on polydiscs. The assumption $q\ge 1$ is necessary because solving $\bar\partial\gamma=\beta$ lowers the anti-holomorphic degree; for $q=0$ there is no space $A^{p,-1}$, and the equation would instead be a statement that a function is the zero derivative of something of negative degree. The smaller polydisc $P'\Subset P$ is used because the Cauchy-transform construction relies on cut-offs supported away from the boundary, avoiding [boundary regularity](/theorems/99) and compatibility conditions. The theorem is local and does not say that a closed form on an arbitrary domain has a global primitive; annuli, compact complex manifolds, and domains with covering data that do not glue can carry Dolbeault cohomology that is not zero. This local-versus-global distinction is the bridge to Chapter 3, where sheaf cohomology records how holomorphic data are glued from local solutions. [example: Local Primitive for a Dbar-Closed (0,1)-Form] Let $P\subset\mathbb C$ be a disc and let $\beta=f(z)\,d\bar z$ with $f\in C^\infty(P)$. Since $d\bar z\wedge d\bar z=0$, every $(0,1)$-form on a one-dimensional complex domain is $\bar\partial$-closed. On a smaller disc $P'\Subset P$, choose $\chi\in C_c^\infty(P)$ with $\chi=1$ on a neighbourhood of $P'$, and set \begin{align*} g(\zeta)=\chi(\zeta)f(\zeta). \end{align*} Define the Cauchy transform \begin{align*} u(z)=\frac{1}{\pi}\int_{\mathbb C}\frac{g(\zeta)}{z-\zeta}\,d\mathcal L^2(\zeta). \end{align*} By the one-variable *Cauchy transform identity* $\partial_{\bar z}\bigl(\pi^{-1}(z-\zeta)^{-1}\bigr)=\delta_\zeta$ in the distribution sense, differentiating under the distribution pairing gives \begin{align*} \frac{\partial u}{\partial\bar z}(z)=g(z). \end{align*} Because $\chi=1$ on a neighbourhood of $P'$, for every $z\in P'$ we have \begin{align*} g(z)=\chi(z)f(z)=f(z). \end{align*} Therefore, on $P'$, \begin{align*} \bar\partial u=\frac{\partial u}{\partial\bar z}\,d\bar z=f(z)\,d\bar z=\beta|_{P'}. \end{align*} Thus the cut-off Cauchy transform produces a local primitive for $\beta$, and this is the one-variable model used in the polydisc argument. [/example] The example solves an inhomogeneous $\bar\partial$ equation, while the homogeneous equation $\bar\partial f=0$ should detect the absence of antiholomorphic variation. This needs verification because holomorphicity was first defined by coordinate functions and Cauchy-Riemann equations, whereas $\bar\partial$ was introduced through the type decomposition of forms. The compatibility result below identifies these two viewpoints for smooth functions. [quotetheorem:7013] [citeproof:7013] This characterisation is the bridge to later chapters. The smoothness hypothesis matters in the stated form because the proof uses classical partial derivatives; weaker distributional versions require additional regularity theorems rather than the elementary Cauchy-Riemann argument. The theorem does not say that every $\bar\partial$-closed higher-degree form is holomorphic data of the same kind as a function; for higher bidegrees, closedness defines cohomology classes rather than functions. It places holomorphic functions at degree zero of the Dolbeault complex, parallel to the way locally constant functions sit at degree zero of the de Rham complex. Holomorphic vector bundles, Hermitian connections, curvature, positivity, and later sheaf-theoretic interpretations all use the same split between $(1,0)$ and $(0,1)$ directions, with $\bar\partial$ encoding the holomorphic structure. The $(p,q)$-decomposition turns holomorphicity into a differential condition, and that is what makes cohomological methods possible. With $ar\partial$ in place, the next chapter studies how local holomorphic data fail to glue globally and how that failure is recorded by Dolbeault cohomology. # 3. Dolbeault Cohomology and Holomorphic Functions Dolbeault cohomology is the bridge between the differential form language of the previous chapter and holomorphic objects. The operator $\bar\partial$ detects holomorphicity, but it also forms complexes whose cohomology measures the failure of local $\bar\partial$-solutions to patch globally. This chapter introduces those complexes, relates them to the sheaf of holomorphic forms, and records the first computations on domains where the local analytic theory has no global obstruction. ## The Dolbeault Complex The first question is how much of de Rham cohomology survives after we remember the type decomposition of complex-valued forms. Since Chapter 2 proved $d = \partial + \bar\partial$ and $d^2 = 0$, the identities there imply $\bar\partial^2 = 0$, so each fixed holomorphic degree $p$ gives a cochain complex in the antiholomorphic degree. [definition: Dolbeault Complex] Let $X$ be a complex manifold. For each $p \ge 0$, the Dolbeault complex of $(p,*)$-forms is \begin{align*} 0 \longrightarrow A^{p,0}(X) \xrightarrow{\bar\partial} A^{p,1}(X) \xrightarrow{\bar\partial} A^{p,2}(X) \xrightarrow{\bar\partial} \cdots, \end{align*} where $A^{p,q}(X)$ denotes the space of smooth complex-valued $(p,q)$-forms on $X$. [/definition] This complex packages a family of differential equations. To extract an invariant from it, we need to quotient closed forms by forms that already arise as $\bar\partial$ of something one degree lower. [definition: Dolbeault Cohomology] Let $X$ be a complex manifold. The Dolbeault cohomology group of bidegree $(p,q)$ is \begin{align*} H^{p,q}_{\bar\partial}(X) := \frac{\ker(\bar\partial: A^{p,q}(X) \to A^{p,q+1}(X))}{\operatorname{im}(\bar\partial: A^{p,q-1}(X) \to A^{p,q}(X))}, \end{align*} with the convention that $A^{p,-1}(X)=0$. [/definition] The group $H^{p,q}_{\bar\partial}(X)$ records global obstructions to solving $\bar\partial\beta=\alpha$. The first case to test is $q=0$, where there are no incoming coboundaries and the cohomology is just the kernel of $\bar\partial$. [example: Holomorphic Functions As Degree Zero Dolbeault Classes] For $p=q=0$, the space $A^{0,0}(X)$ is the space $C^\infty(X,\mathbb C)$ of smooth complex-valued functions. Since $A^{0,-1}(X)=0$, the image term in degree zero is zero, so the definition of Dolbeault cohomology gives \begin{align*} H^{0,0}_{\bar\partial}(X)=\ker(\bar\partial:A^{0,0}(X)\to A^{0,1}(X)). \end{align*} In a holomorphic coordinate chart $(z_1,\dots,z_n)$, a smooth function $f$ satisfies \begin{align*} \bar\partial f=\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\,d\bar z_j. \end{align*} Because the forms $d\bar z_1,\dots,d\bar z_n$ are pointwise linearly independent, $\bar\partial f=0$ is equivalent to \begin{align*} \frac{\partial f}{\partial \bar z_1}=\cdots=\frac{\partial f}{\partial \bar z_n}=0. \end{align*} These are exactly the Cauchy-Riemann equations in holomorphic coordinates, so the kernel consists precisely of the holomorphic functions on $X$. Therefore \begin{align*} H^{0,0}_{\bar\partial}(X)=\mathcal O(X). \end{align*} Thus the first Dolbeault group recovers a holomorphic object from the smooth differential-form complex. [/example] This example shows that the degree-zero Dolbeault condition is holomorphicity. For $(p,0)$-forms the same obstruction is coefficientwise: a smooth expression with only $dz$-factors is not holomorphic unless its local coefficient functions have no antiholomorphic variation. The following terminology records exactly those $(p,0)$-forms killed by $\bar\partial$. [definition: Holomorphic P-Form] Let $X$ be a complex manifold. A holomorphic $p$-form on $X$ is a smooth form $\alpha \in A^{p,0}(X)$ such that $\bar\partial\alpha = 0$. [/definition] This definition is intrinsic because the operator $\bar\partial$ was defined using the type decomposition, which is invariant under holomorphic coordinate changes. Once holomorphic $p$-forms have been characterized as $\bar\partial$-closed $(p,0)$-forms, the remaining bookkeeping question is whether this agrees with the standard notation $\Omega_X^p(X)$ for global sections of the sheaf of holomorphic $p$-forms. The identification below ties the Dolbeault complex to that sheaf-theoretic language. [quotetheorem:3389] [citeproof:3389] The identification is natural in $X$, not just a coincidence of vector spaces. Its limitation is also important: it uses $q=0$, where there are no incoming coboundaries and the condition is only $\bar\partial$-closedness. For $q>0$, a Dolbeault class is no longer a single holomorphic object; it is a closed smooth form modulo choices of lower-degree primitives, so higher-degree groups measure obstruction rather than spaces of holomorphic forms. The complex-structure hypothesis is equally concrete: on a merely smooth manifold there is no intrinsic decomposition of complex-valued forms into $(p,q)$-types and no intrinsic sheaf $\Omega_X^p$ of holomorphic $p$-forms. A merely smooth map between complex manifolds need not preserve the decomposition into $(p,q)$-types, so there is no reason for it to induce maps on Dolbeault cohomology. For example, an antiholomorphic change of coordinate sends $dz$-directions to $d\bar z$-directions and therefore does not respect the fixed bidegree complex. This is why the next result assumes holomorphicity, and why Dolbeault cohomology is functorial in the holomorphic category rather than in the smooth category. [quotetheorem:7014] Functoriality lets Dolbeault classes be restricted, pulled back to charts, and compared across maps. The hypothesis that $F$ is holomorphic cannot be dropped: if $F$ reverses the complex structure, the pullback of a $(1,0)$-form may become a $(0,1)$-form, so it does not define an endomorphism of a fixed Dolbeault complex. The result is therefore a tool for comparing holomorphic local models, not arbitrary smooth parametrisations. The most basic case is the inclusion of an open subset, which will be used repeatedly when local and global arguments are compared. [example: Restriction To An Open Complex Submanifold] Let $U\subset X$ be open, and let $j:U\hookrightarrow X$ be the inclusion. Since $U$ has the induced complex structure, $j$ is holomorphic, and pullback of forms is ordinary restriction: \begin{align*} j^*\alpha=\alpha|_U. \end{align*} If $\alpha\in A^{p,q}(X)$ satisfies $\bar\partial\alpha=0$, then holomorphic pullback commutes with $\bar\partial$, so \begin{align*} \bar\partial(j^*\alpha)=j^*(\bar\partial\alpha)=j^*(0)=0. \end{align*} Thus restriction sends cocycles on $X$ to cocycles on $U$. The induced map on cohomology is well-defined because cohomologous representatives remain cohomologous after restriction. If $\alpha'=\alpha+\bar\partial\beta$ for some $\beta\in A^{p,q-1}(X)$, then \begin{align*} j^*\alpha'=j^*\alpha+j^*(\bar\partial\beta)=j^*\alpha+\bar\partial(j^*\beta). \end{align*} Hence $j^*\alpha'$ and $j^*\alpha$ define the same class in $H^{p,q}_{\bar\partial}(U)$, giving a restriction map \begin{align*} j^*:H^{p,q}_{\bar\partial}(X)\longrightarrow H^{p,q}_{\bar\partial}(U). \end{align*} When $q=0$, this is exactly the usual restriction of holomorphic $p$-forms from $X$ to $U$; when $q>0$, it means that a global Dolbeault obstruction class can be examined after passing to smaller open subsets. [/example] ## The Smooth Resolution Of Holomorphic Forms The preceding section defined Dolbeault cohomology using global smooth forms. The central problem now is to explain why this analytic-looking construction computes sheaf cohomology of holomorphic objects. The mechanism is a resolution: holomorphic forms sit inside smooth forms as the first kernel of a locally exact complex. [definition: Sheaf Of Holomorphic Functions] Let $X$ be a complex manifold. The sheaf $\mathcal O_X$ is the contravariant functor from the category of open subsets of $X$ with inclusions to the category of rings defined as follows: \begin{align*} \mathcal O_X(U) := \{f:U\to\mathbb C \mid f \text{ is holomorphic}\} \end{align*} for each open set $U\subset X$, and for each inclusion $V\subset U$ the restriction map is \begin{align*} \rho_{UV}:\mathcal O_X(U)\to\mathcal O_X(V),\qquad \rho_{UV}(f)=f|_V. \end{align*} [/definition] The locality and gluing axioms say that if $(U_i)_{i\in I}$ covers $U$, sections agreeing after restriction to every $U_i$ are equal, and compatible sections $f_i\in\mathcal O_X(U_i)$ glue uniquely to a section $f\in\mathcal O_X(U)$. Thus $\mathcal O_X$ keeps track not only of global holomorphic functions but also of their local restrictions and gluing. To compare with $(p,*)$ Dolbeault complexes, we need the same sheaf language for holomorphic forms of fixed degree $p$. [definition: Sheaf Of Holomorphic P-Forms] Let $X$ be a complex manifold and let $p\ge 0$. The sheaf $\Omega_X^p$ is the contravariant functor from the category of open subsets of $X$ with inclusions to the category of complex vector spaces defined by \begin{align*} \Omega_X^p(U):=\{\alpha\in A^{p,0}(U):\bar\partial\alpha=0\} \end{align*} for each open set $U\subset X$, with restriction maps \begin{align*} \rho_{UV}:\Omega_X^p(U)\to\Omega_X^p(V),\qquad \rho_{UV}(\alpha)=\alpha|_V \end{align*} for each inclusion $V\subset U$. [/definition] The sheaf axioms for $\Omega_X^p$ express the same locality and gluing property, now for holomorphic forms rather than functions. This sheaf is the object whose cohomology Dolbeault theory will compute. A typical obstruction problem starts with local equations $\bar\partial\beta_i=\alpha$ on open sets $U_i$: on overlaps, the differences $\beta_i-\beta_j$ are $\bar\partial$-closed, and these differences may prevent the local primitives from patching to one global primitive. Before studying that global failure, we need to know that no obstruction is already present on sufficiently small coordinate neighbourhoods. This is the local Dolbeault-Poincare input for $\bar\partial$: after shrinking to coordinate polydiscs, positive antiholomorphic degree is locally exact, while degree zero is controlled by holomorphicity. Local exactness gives a candidate resolution, and global cohomology records the remaining gluing obstruction. The following theorem packages this into the smooth $\bar\partial$ resolution. [quotetheorem:7015] [citeproof:7015] The resolution replaces a holomorphic sheaf by sheaves of smooth forms, but the theorem is not a vanishing theorem for global Dolbeault cohomology. Exactness is a stalkwise assertion, so it does not say that every global $\bar\partial$-closed form on $X$ is globally $\bar\partial$-exact. For example, on a compact Riemann surface of genus $g>0$, nonzero classes in $H^{0,1}_{\bar\partial}(X)$ remain even though the smooth $\bar\partial$ resolution is exact as a sequence of sheaves. The smoothness hypothesis is also essential for the resolution method: smooth partitions of unity make the sheaves $\mathcal A_X^{p,q}$ acyclic, while holomorphic partitions of unity would force holomorphic functions to be too flexible and generally do not exist on compact connected manifolds. Thus the resolution turns a hard holomorphic gluing problem into the cohomology of a global differential complex, but only after using the acyclicity special to smooth sheaves. [quotetheorem:3389] [citeproof:3389] Dolbeault's theorem is the main structural result of the chapter. Its hypotheses do several separate jobs. The complex structure is needed to define the $(p,q)$ decomposition and the sheaf $\Omega_X^p$; replacing $X$ by a merely smooth manifold leaves no intrinsic $\bar\partial$ complex to compare with holomorphic sheaf cohomology. Smooth forms are needed because the acyclic resolution uses partitions of unity; using holomorphic forms in every degree would fail on compact connected manifolds, where holomorphic functions cannot be used to localise arbitrary data. The conclusion also does not say that Dolbeault groups always vanish: on a compact Riemann surface of genus $g>0$, $H^{0,1}_{\bar\partial}(X)\cong H^1(X,\mathcal O_X)$ has dimension $g$. Rather, the theorem says that nonzero Dolbeault classes are exactly the sheaf-theoretic obstructions to gluing holomorphic $p$-forms. On spaces with enough local acyclic sets these groups can often be computed from Cech covers, while on compact manifolds they are often computed analytically using [harmonic representatives](/theorems/2747). [remark: What Dolbeault Theorem Changes] Dolbeault cohomology is defined by differential equations on smooth forms, while $H^q(X,\Omega_X^p)$ is defined by gluing local holomorphic data. The theorem identifies these two languages. In later chapters, Hermitian metrics and elliptic theory will give analytic representatives of the same classes. [/remark] ## Local Holomorphic Functions On Polydiscs The next problem is to understand the simplest spaces where global obstructions disappear. Polydiscs are the local coordinate domains of complex geometry, so they are the testing ground for holomorphic functions, local exactness, and vanishing of Dolbeault cohomology. [definition: Polydisc] For $r_1,\dots,r_n>0$, the polydisc of polyradius $(r_1,\dots,r_n)$ centred at $a=(a_1,\dots,a_n)\in\mathbb C^n$ is \begin{align*} P(a;r_1,\dots,r_n):=\{z\in\mathbb C^n: |z_i-a_i|<r_i \text{ for } i=1,\dots,n\}. \end{align*} [/definition] Polydiscs are preferred over general small coordinate balls in several arguments because products of one-variable Cauchy formulae apply directly. The first local consequence is the power-series control of holomorphic functions on slightly smaller polydiscs. [quotetheorem:3419] [citeproof:3419] This local expansion theorem is analytic rather than cohomological, but it explains why holomorphic functions form a rigid sheaf. The smaller-polydisc hypothesis matters: the [power series](/page/Power%20Series) is guaranteed on $P'\Subset P$, while boundary behaviour on all of $P$ may be uncontrolled. For instance, $f(z)=1/(1-z)$ is holomorphic on the unit disc but its Taylor series at $0$ cannot converge on any disc crossing the boundary point $z=1$; the radius gap is what keeps the [Cauchy estimates](/theorems/2571) uniform. The product shape of the polydisc also matters for this proof because the distinguished boundary torus and the iterated kernels separate the variables. The same product structure that gives power-series estimates also makes the $\bar\partial$ equation solvable on a polydisc in positive antiholomorphic degree. This is the first place where local analytic control becomes a cohomological computation: if every $\bar\partial$-closed $(p,q)$-form with $q>0$ has a global primitive on the polydisc, then the local pieces used in the smooth resolution are not merely stalkwise exact but acyclic on this coordinate model. The next theorem records that vanishing statement. [quotetheorem:3388] [citeproof:3388] The vanishing theorem gives the full Dolbeault cohomology of the polydisc once degree zero is included. Each hypothesis is doing work. The product geometry lets the proof solve one complex variable at a time without leaving the domain; for arbitrary domains, this particular homotopy construction has no reason to preserve the domain. The Stein local nature is also essential for vanishing: polydiscs are the first instance of the broader principle that Stein spaces have enough holomorphic functions and convexity to force higher coherent sheaf cohomology to vanish. Annuli and other Stein domains still have good holomorphic function theory, while compact complex manifolds need not have positive-degree Dolbeault vanishing at all. For instance, compact Riemann surfaces of positive genus have nonzero $H^{0,1}_{\bar\partial}$, so global topology can preserve classes that disappear on coordinate polydiscs. Thus polydiscs are the local acyclic pieces used behind the sheaf-theoretic comparison, not a model for all global behaviour; later vanishing theorems require stronger global hypotheses or additional positivity. [example: Dolbeault Cohomology Of A Polydisc] Let $P\subset\mathbb C^n$ be a polydisc with holomorphic coordinates $z_1,\dots,z_n$. For $q>0$, *Dolbeault Vanishing On Polydiscs* says that every $\bar\partial$-closed $(p,q)$-form on $P$ is $\bar\partial$ of a $(p,q-1)$-form, so \begin{align*} H^{p,q}_{\bar\partial}(P)=0. \end{align*} In degree $q=0$, the convention $A^{p,-1}(P)=0$ gives \begin{align*} H^{p,0}_{\bar\partial}(P)=\ker(\bar\partial:A^{p,0}(P)\to A^{p,1}(P)). \end{align*} If $0\le p\le n$, write a smooth $(p,0)$-form as \begin{align*} \alpha=\sum_{|I|=p} f_I(z)\,dz_I, \end{align*} where $I=(i_1<\cdots<i_p)$ and $dz_I=dz_{i_1}\wedge\cdots\wedge dz_{i_p}$. Since $\bar\partial(dz_I)=0$ and $\bar\partial$ differentiates the coefficient functions in the antiholomorphic directions, \begin{align*} \bar\partial\alpha=(-1)^p\sum_{|I|=p}\sum_{j=1}^n \frac{\partial f_I}{\partial\bar z_j}\,dz_I\wedge d\bar z_j. \end{align*} The forms $dz_I\wedge d\bar z_j$ form the coordinate basis for $(p,1)$-forms, so $\bar\partial\alpha=0$ is equivalent to \begin{align*} \frac{\partial f_I}{\partial\bar z_j}=0\quad\text{for every }I\text{ and every }j. \end{align*} Thus each coefficient $f_I$ is holomorphic, and \begin{align*} H^{p,0}_{\bar\partial}(P)\cong H^0(P,\Omega_P^p). \end{align*} For $p=0$, this is $\mathcal O(P)$. For $p>0$, it is the space of holomorphic sums $\sum_{|I|=p} f_I(z)\,dz_I$; if $p>n$, there are no such multiindices and the group is $0$. Hence the only nonzero Dolbeault groups of a polydisc occur in antiholomorphic degree $0$, where they are exactly the holomorphic $p$-forms. [/example] The computation also matches Dolbeault's theorem. Since polydiscs have no higher coherent sheaf cohomology in this local setting, the sheaf cohomology side vanishes in positive degree as well. ## A First Global Computation On The Riemann Sphere Local vanishing does not mean global vanishing. The next question is how topology and compactness create Dolbeault classes that cannot be removed by solving a global $\bar\partial$ equation. The Riemann sphere $\mathbb{CP}^1$ is the first compact example where Dolbeault cohomology reflects holomorphic line bundles. [example: First Cohomology Of The Riemann Sphere] Let $X=\mathbb{CP}^1$. Applying *Dolbeault Theorem For Holomorphic P-Forms* with $p=0$ and $q=1$ gives \begin{align*} H^{0,1}_{\bar\partial}(\mathbb{CP}^1)\cong H^1(\mathbb{CP}^1,\Omega^0_{\mathbb{CP}^1}). \end{align*} By definition, $\Omega^0_{\mathbb{CP}^1}$ is the sheaf of holomorphic $0$-forms, hence the structure sheaf $\mathcal O_{\mathbb{CP}^1}$. Therefore \begin{align*} H^1(\mathbb{CP}^1,\Omega^0_{\mathbb{CP}^1})=H^1(\mathbb{CP}^1,\mathcal O_{\mathbb{CP}^1}). \end{align*} The standard notation for holomorphic line bundles on $\mathbb{CP}^1$ has $\mathcal O(0)=\mathcal O_{\mathbb{CP}^1}$. The line-bundle cohomology table on $\mathbb{CP}^1$ says \begin{align*} H^1(\mathbb{CP}^1,\mathcal O(k))=0\quad\text{for }k\ge -1. \end{align*} Substituting $k=0$, and using $0\ge -1$, gives \begin{align*} H^1(\mathbb{CP}^1,\mathcal O_{\mathbb{CP}^1})=H^1(\mathbb{CP}^1,\mathcal O(0))=0. \end{align*} Combining the displayed identifications yields \begin{align*} H^{0,1}_{\bar\partial}(\mathbb{CP}^1)=0. \end{align*} Thus this Dolbeault group vanishes because the corresponding structure-sheaf cohomology group vanishes; no Hodge-theoretic argument is being used here. [/example] This vanishing is a useful warning about dimension one. The first possible antiholomorphic cohomology group does not automatically detect compactness; it depends on the holomorphic sheaf being used. For the structure sheaf on $\mathbb{CP}^1$, the comparison with line bundles removes the obstruction. [remark: Where Nonzero Classes First Appear] On a compact Riemann surface of genus $g$, Dolbeault's theorem identifies $H^{0,1}_{\bar\partial}(X)$ with $H^1(X,\mathcal O_X)$, whose dimension is $g$. Thus $\mathbb{CP}^1$ is the genus-zero case. Later, Hodge theory will explain this group by harmonic $(0,1)$-forms once a Hermitian metric is chosen. [/remark] The chapter's main point is that $\bar\partial$ is both a differential operator and a cohomological resolution. Locally, the $\bar\partial$-Poincare lemma makes the smooth complex exact in positive antiholomorphic degree. Globally, the remaining cohomology is exactly sheaf cohomology of holomorphic forms, so analytic equations and holomorphic gluing data become two descriptions of the same invariant. Dolbeault cohomology shows that analytic solvability and holomorphic gluing are two faces of the same problem. Having identified that bridge, the course now moves from scalar forms to holomorphic vector bundles, where the same ideas are expressed with bundle-valued sections. # 4. Holomorphic Vector Bundles This chapter moves from the tangent bundle of a complex manifold to vector bundles whose fibres vary holomorphically. It uses the preceding material on complex manifolds, holomorphic coordinate changes, tangent bundles, and projective space, together with the basic language of smooth complex vector bundles. The guiding question is how much of the local theory of holomorphic functions survives when the values lie in a varying complex vector space rather than in a fixed copy of $\mathbb C^r$. We also introduce the standard line bundles on projective space, since they are the basic examples behind the divisor construction in Chapter 11 and the curvature computations in Chapters 8 through 10. ## Holomorphic Frames and Transition Functions A smooth complex vector bundle over a complex manifold has local frames, but complex geometry asks for frames compatible with the holomorphic coordinate changes on the base. The first problem is to identify the extra structure needed so that a section can be called holomorphic in local coordinates independently of the chosen local product chart. [definition: Holomorphic Vector Bundle] Let $X$ be a complex manifold. A holomorphic vector bundle of rank $r$ over $X$ is a smooth complex vector bundle $\pi:E\to X$ together with a complex-manifold structure on $E$, a holomorphic projection $\pi$, an open cover $(U_i)_{i\in I}$, and biholomorphisms \begin{align*} \Phi_i: \pi^{-1}(U_i) \to U_i\times \mathbb C^r \end{align*} which are fibrewise complex-linear and whose change-of-frame functions \begin{align*} g_{ij}:U_i\cap U_j\to GL(r,\mathbb C) \end{align*} are holomorphic. [/definition] The condition is imposed on the transition maps because a vector bundle is assembled by gluing local products. If $v_i$ and $v_j$ denote the coordinate column vectors of the same point of $E$ over $U_i\cap U_j$, then $v_i=g_{ij}v_j$, so holomorphicity of $g_{ij}$ is exactly the compatibility condition between the two local descriptions. This motivates isolating the local bases in which sections are measured. [definition: Holomorphic Frame] Let $E\to X$ be a holomorphic vector bundle of rank $r$. A holomorphic frame on an open set $U\subset X$ is an ordered $r$-tuple of holomorphic maps $e_a:U\to E|_U$ for $1\le a\le r$, with $\pi\circ e_a=\operatorname{id}_U$, such that $(e_1(x),\dots,e_r(x))$ is a basis of $E_x$ for every $x\in U$. [/definition] A frame turns a section $s$ over $U$ into a vector of component functions $s=\sum_{a=1}^r s_a e_a$. The point of a holomorphic frame is that holomorphicity of $s$ can then be tested by the ordinary holomorphicity of the component functions $s_a:U\to\mathbb C$. For this test to be independent of the frame, the transition matrices between local frames must be holomorphic. This hypothesis is doing real work: the same smooth bundle can carry frames whose change-of-frame functions are only smooth, and then holomorphicity of component functions is not preserved under a change of frame. For instance, on the product smooth line bundle over an open set in $\mathbb C$, replacing the standard frame $e$ by $e'=e^{\bar z}e$ gives a non-holomorphic transition factor, so a section with holomorphic coefficient in one frame need not have holomorphic coefficient in the other. The practical construction problem is therefore to start with holomorphic transition matrices satisfying the cocycle identities and glue the local products from that data. [quotetheorem:7016] [citeproof:7016] Cocycles turn bundle construction into a computable overlap problem. The identity $g_{ij}g_{jk}=g_{ik}$ is necessary because a vector transported from the $k$-th local product chart to the $i$-th one must give the same result whether it passes directly or through the $j$-th local product chart. Without this identity, the proposed gluing can assign two different values to the same vector on a triple overlap: for instance, if $U_1\cap U_2\cap U_3\ne\varnothing$ and line-bundle data satisfy $g_{12}=g_{23}=1$ but $g_{13}=2$, then the point $(x,v)_3$ would be identified with $(x,v)_1$ through $U_2$ and with $(x,2v)_1$ directly. For $v\ne0$ this is inconsistent, so the quotient would not have well-defined fibres over the triple overlap. The classification statement also has precise limits. It classifies bundles after a cover and rank have been fixed, and comparisons across different covers require passing to a common refinement; it is not a canonical list of bundles independent of all choices of cover. It also classifies holomorphic bundles, not smooth complex bundles with extra notation: the changes of local product chart $h_i$ must be holomorphic, and allowing merely smooth $h_i$ would identify bundles that are smoothly isomorphic but carry different holomorphic structures. Finally, the theorem records isomorphism classes of bundles, not preferred global frames or preferred transition functions. A bundle can be represented by many cocycles, and a nontrivial line bundle may still be locally described by the scalar transition functions of this theorem. This motivates looking at a simple rank-one situation, where the matrices reduce to nonzero holomorphic functions and the obstruction to a global frame is visible in the gluing functions. [example: Mobius-Type Gluing Over A Complex Torus] Let $X=\mathbb C/\Lambda$ be a complex torus, fix $p\in X$, and choose a cover $X=U_0\cup U_1$ where $U_0$ is a coordinate disc around $p$ with coordinate $t$ and $t(p)=0$, while $U_1=X\setminus\{p\}$. On $U_0\cap U_1$ one has $t\ne0$, so \begin{align*} g_{01}=t:U_0\cap U_1\to\mathbb C^* \end{align*} is a nowhere-vanishing holomorphic transition function. Since this is a two-set cover, the cocycle condition has no nontrivial triple-overlap check, and the data define the divisor line bundle $\mathcal O_X(p)$. We show that $\mathcal O_X(p)$ has no single global nowhere-vanishing holomorphic frame. Suppose such a frame existed. In the local frames over $U_0$ and $U_1$, it would be represented by nowhere-vanishing holomorphic functions $h_0$ on $U_0$ and $h_1$ on $U_1$. The convention $v_i=g_{ij}v_j$ gives, on $U_0\cap U_1$, \begin{align*} h_0=g_{01}h_1=t h_1. \end{align*} Because $t\ne0$ on the overlap, this is equivalent there to \begin{align*} h_1=\frac{h_0}{t}. \end{align*} Since $h_0$ is nowhere zero on $U_0$, in the coordinate $t$ it has an expansion \begin{align*} h_0(t)=a_0+a_1t+a_2t^2+\cdots \end{align*} with $a_0=h_0(p)\ne0$. Therefore near $p$, \begin{align*} \frac{h_0(t)}{t}=\frac{a_0}{t}+a_1+a_2t+\cdots, \end{align*} so $h_1$ extends across $p$ as a [meromorphic function](/page/Meromorphic%20Function) with a pole of order exactly $1$. This meromorphic function has no zeros: on $U_1$ it equals the nowhere-vanishing function $h_1$, and on $U_0\setminus\{p\}$ it equals $h_0/t$, where both $h_0$ and $t$ are nonzero. Thus its divisor is \begin{align*} \operatorname{div}(h_1)=-p, \end{align*} which has degree $-1$. This contradicts *Degree Zero of Principal Divisors*, the standard fact that every principal divisor on a compact Riemann surface has degree $0$. Hence $\mathcal O_X(p)$ is locally trivial but cannot be presented by one global nowhere-vanishing holomorphic frame. [/example] The example shows that bundles have meaningful global geometry even when all local pieces are products. To compare such objects, we need maps that preserve the fibres and remain holomorphic in local frames, which motivates the notion of holomorphic bundle morphism. [definition: Holomorphic Bundle Morphism] Let $\pi_E:E\to X$ and $\pi_F:F\to X$ be holomorphic vector bundles. A holomorphic bundle morphism $\varphi:E\to F$ over $X$ is a map satisfying $\pi_F\circ\varphi=\pi_E$ whose restriction $\varphi_x:E_x\to F_x$ is complex-linear for every $x\in X$, and such that, in any holomorphic frames for $E$ and $F$, its representing matrix has holomorphic entries. [/definition] This definition is independent of frames because changing frames multiplies the representing matrix on the left and right by holomorphic invertible matrices. Thus the category of holomorphic vector bundles has objects given by holomorphic gluing data and arrows given locally by holomorphic matrices. ## Line Bundles and Projective Examples Rank-one bundles are the first global objects where holomorphic gluing has geometric content. Their transition functions take values in $\mathbb C^*$, so they can record how local defining equations for hypersurfaces transform on overlaps. [definition: Holomorphic Line Bundle] A holomorphic line bundle over a complex manifold $X$ is a holomorphic vector bundle of rank $1$ over $X$. [/definition] For a line bundle, each transition function is a nowhere-vanishing holomorphic function. This makes line bundles close to divisors: if local equations for a hypersurface differ by multiplication by units, those units are precisely transition functions for a line bundle. This motivates the divisor construction as the first geometric source of line bundles. [example: Divisor Transition Functions] Let $D\subset X$ be a smooth hypersurface, and choose open sets $U_i$ with holomorphic defining functions $f_i\in\mathcal O_X(U_i)$ such that $D\cap U_i=\{f_i=0\}$. On an overlap $U_i\cap U_j$, two defining functions for the same smooth hypersurface differ by a holomorphic unit, so there is a nowhere-vanishing holomorphic function $u_{ij}$ with \begin{align*} f_i=u_{ij}f_j. \end{align*} On a triple overlap $U_i\cap U_j\cap U_k$, the two ways of comparing $f_i$ with $f_k$ give \begin{align*} f_i=u_{ij}f_j=u_{ij}u_{jk}f_k. \end{align*} The direct comparison also gives \begin{align*} f_i=u_{ik}f_k. \end{align*} Hence $(u_{ij}u_{jk}-u_{ik})f_k=0$. Away from $D$, the function $f_k$ is nonzero, so $u_{ij}u_{jk}=u_{ik}$ there; since both sides are holomorphic, the equality extends across the hypersurface. Also $u_{ii}=1$, because $f_i=1\cdot f_i$. Thus the functions $g_{ij}=u_{ij}$ are a $\mathbb C^*$-valued holomorphic cocycle, and with the convention $v_i=g_{ij}v_j$ they define the line bundle denoted $\mathcal O_X(D)$. The local functions $f_i$ are compatible with this same convention, because on $U_i\cap U_j$ one has \begin{align*} f_i=u_{ij}f_j=g_{ij}f_j. \end{align*} Therefore the $f_i$ are the local representatives of a global holomorphic section of $\mathcal O_X(D)$. Its zero set is locally $\{f_i=0\}=D\cap U_i$, so its zero divisor is $D$ with the chosen reduced defining equations. If one instead used the opposite coordinate convention, the transition functions would be $u_{ij}^{-1}$, producing the dual line bundle usually denoted $\mathcal O_X(-D)$. [/example] Divisors explain why line bundles matter, while projective space supplies the examples used for computation. The next construction is motivated by the geometry of a point of projective space, which is itself a line through the origin. [definition: Tautological Line Bundle On Projective Space] The tautological line bundle $\mathcal O_{\mathbb{CP}^n}(-1)$ over $\mathbb{CP}^n$ is the holomorphic line bundle whose fibre over a point $[z]\in\mathbb{CP}^n$ is the complex line \begin{align*} \mathcal O_{\mathbb{CP}^n}(-1)_{[z]}=\{\lambda z:\lambda\in\mathbb C\}\subset\mathbb C^{n+1}. \end{align*} [/definition] On the affine chart $U_i=\{[z_0:\cdots:z_n]:z_i\ne 0\}$, the vector \begin{align*} \frac{z}{z_i} \end{align*} gives a local holomorphic frame for $\mathcal O_{\mathbb{CP}^n}(-1)$. With the convention that coordinate vectors satisfy $v_i=g_{ij}v_j$, comparing these frames gives transition functions \begin{align*} g_{ij}=\frac{z_i}{z_j} \end{align*} on $U_i\cap U_j$, and this motivates a concrete chart computation in the smallest projective case. [example: The Tautological Bundle On $\mathbb{CP}^1$] On $\mathbb{CP}^1$, use the standard charts \begin{align*} U_0=\{[z_0:z_1]:z_0\ne0\}, \qquad U_1=\{[z_0:z_1]:z_1\ne0\}, \end{align*} with affine coordinates \begin{align*} w=\frac{z_1}{z_0}, \qquad \zeta=\frac{z_0}{z_1}. \end{align*} On the overlap $U_0\cap U_1$, both $z_0$ and $z_1$ are nonzero, so \begin{align*} w\zeta=\frac{z_1}{z_0}\frac{z_0}{z_1}=1. \end{align*} Hence $\zeta=1/w$ on $U_0\cap U_1$. The tautological line over $[z_0:z_1]$ is the line $\mathbb C(z_0,z_1)\subset\mathbb C^2$. On $U_0$, the vector \begin{align*} e_0=\frac{(z_0,z_1)}{z_0}=(1,w) \end{align*} spans this line holomorphically. On $U_1$, the vector \begin{align*} e_1=\frac{(z_0,z_1)}{z_1}=(\zeta,1) \end{align*} spans the same line holomorphically. On $U_0\cap U_1$, \begin{align*} \zeta e_0=\zeta(1,w)=(\zeta,\zeta w). \end{align*} Since $\zeta w=1$, this becomes \begin{align*} \zeta e_0=(\zeta,1)=e_1. \end{align*} Now write the same vector in the two local frames: \begin{align*} v_0e_0=v_1e_1. \end{align*} Using $e_1=\zeta e_0$, the right-hand side is \begin{align*} v_1e_1=v_1(\zeta e_0)=(\zeta v_1)e_0. \end{align*} Because $e_0$ is a frame vector and is therefore nonzero in each fibre, \begin{align*} v_0=\zeta v_1. \end{align*} With the convention $v_i=g_{ij}v_j$, this gives \begin{align*} g_{01}=\zeta=\frac{1}{w}. \end{align*} Thus the transition function for $\mathcal O_{\mathbb{CP}^1}(-1)$ has a simple pole in the $w$-coordinate at $w=0$, which is the local sign of the negative-degree convention. [/example] The tautological bundle records lines inside a fixed vector space. Its dual records linear functionals on those lines; with the transition convention fixed above, this motivates the hyperplane bundle and its tensor powers. [definition: Hyperplane Bundle And Twisting Sheaves] The hyperplane bundle on $\mathbb{CP}^n$ is \begin{align*} \mathcal O_{\mathbb{CP}^n}(1):=\mathcal O_{\mathbb{CP}^n}(-1)^*. \end{align*} For $k\in\mathbb Z$, define $\mathcal O_{\mathbb{CP}^n}(k)$ by $\mathcal O_{\mathbb{CP}^n}(1)^{\otimes k}$ when $k\ge0$, and by $\mathcal O_{\mathbb{CP}^n}(-1)^{\otimes(-k)}$ when $k<0$. [/definition] The transition functions of $\mathcal O(k)$ are the $k$-th powers of the transition functions of $\mathcal O(1)$. This compact notation is used constantly: twisting a bundle by $\mathcal O(k)$ changes its projective growth behaviour. It also motivates the basic section computation in terms of homogeneous polynomials. [example: Sections Of $\mathcal O(k)$] Let $k\ge0$, and let $P(z_0,\dots,z_n)$ be homogeneous of degree $k$. On the chart $U_i=\{z_i\ne0\}$, write affine coordinates \begin{align*} u_a=\frac{z_a}{z_i}\quad\text{for }a\ne i. \end{align*} By homogeneity, \begin{align*} P(z_0,\dots,z_n)=z_i^k P\left(\frac{z_0}{z_i},\dots,\frac{z_{i-1}}{z_i},1,\frac{z_{i+1}}{z_i},\dots,\frac{z_n}{z_i}\right). \end{align*} Therefore the local representative \begin{align*} s_i=\frac{P(z)}{z_i^k} \end{align*} equals a polynomial in the affine coordinates on $U_i$, so $s_i$ is holomorphic. On $U_i\cap U_j$, both $z_i$ and $z_j$ are nonzero. The representative on $U_j$ is \begin{align*} s_j=\frac{P(z)}{z_j^k}. \end{align*} Multiplying by the transition function of $\mathcal O_{\mathbb{CP}^n}(k)$ gives \begin{align*} \left(\frac{z_j}{z_i}\right)^k s_j=\left(\frac{z_j}{z_i}\right)^k\frac{P(z)}{z_j^k}. \end{align*} Since $\left(z_j/z_i\right)^k/z_j^k=1/z_i^k$, this becomes \begin{align*} \left(\frac{z_j}{z_i}\right)^k s_j=\frac{P(z)}{z_i^k}=s_i. \end{align*} Thus the local holomorphic functions $s_i$ satisfy the same overlap rule as sections of $\mathcal O_{\mathbb{CP}^n}(k)$, and hence glue to a global holomorphic section. Homogeneous degree $k$ is exactly what makes the quotient $P(z)/z_i^k$ independent of the chosen representative of the projective point. [/example] ## Operations on Holomorphic Vector Bundles Once holomorphic bundles and morphisms are in place, the next question is whether the usual linear-algebra constructions on fibres remain holomorphic globally. The answer is affirmative because sums, duals, tensor products, determinants, and exterior powers can all be performed on transition matrices by holomorphic matrix operations. [definition: Holomorphic Section] Let $E\to X$ be a holomorphic vector bundle. A holomorphic section of $E$ over an open set $U\subset X$ is a holomorphic map $s:U\to E$ such that $\pi\circ s=\operatorname{id}_U$. [/definition] In a holomorphic frame, a section is holomorphic exactly when its component functions are holomorphic. The sheaf of holomorphic sections is denoted $\mathcal O(E)$ or $\mathcal O_X(E)$ when the base should be emphasized. Since many geometric constructions compare sections and quotient bundles, this motivates the fibrewise language of exact sequences. [definition: Exact Sequence Of Holomorphic Vector Bundles] A sequence of holomorphic vector bundles and holomorphic bundle morphisms over $X$ \begin{align*} 0\to E'\xrightarrow{\alpha}E\xrightarrow{\beta}E''\to0 \end{align*} is exact if for every $x\in X$ the induced sequence of complex vector spaces \begin{align*} 0\to E'_x\xrightarrow{\alpha_x}E_x\xrightarrow{\beta_x}E''_x\to0 \end{align*} is exact. [/definition] Exactness is a fibrewise condition, but holomorphic splitting is stronger than smooth splitting. A short exact sequence of smooth vector bundles always splits after choosing a Hermitian metric; a holomorphic splitting may be obstructed by complex geometry. This motivates recording the standard fibrewise operations that can be applied to exact sequences and their terms. [definition: Dual Tensor And Exterior Bundles] Let $E\to X$ and $F\to X$ be holomorphic vector bundles. The dual bundle $E^*$, [tensor product](/page/Tensor%20Product) $E\otimes F$, exterior power $\Lambda^q E$, and determinant line bundle $\det E$ are the holomorphic bundles obtained by applying the corresponding linear-algebra construction to each fibre: \begin{align*} (E^*)_x=(E_x)^*, \qquad (E\otimes F)_x=E_x\otimes F_x, \end{align*} \begin{align*} (\Lambda^q E)_x=\Lambda^q(E_x), \qquad \det E=\Lambda^r E\quad\text{if }\operatorname{rank}E=r. \end{align*} [/definition] The transition functions are obtained functorially: $E^*$ uses $(g_{ij}^{-1})^\top$, $E\otimes F$ uses $g_{ij}\otimes h_{ij}$, and $\Lambda^qE$ uses $\Lambda^q g_{ij}$. These formulas are holomorphic because they are polynomial or rational expressions in the entries of invertible holomorphic matrices, with determinant nonzero on the overlap. Among these constructions, the top exterior power is especially important, which motivates making the determinant cocycle explicit. [quotetheorem:7017] [citeproof:7017] The rank and invertibility hypotheses are essential here. Without constant rank there is no top exterior power line bundle: for example, the coherent sheaf on $\mathbb C$ whose fibre is $\mathbb C$ away from $0$ and $\mathbb C^2$ at $0$ has no locally constant fibre dimension near $0$, so there is no single integer $r$ for which $\Lambda^r$ is fibrewise one-dimensional. Without invertible transition matrices the determinant may vanish, so it cannot define a $\mathbb C^*$-valued cocycle. The determinant reduces rank-$r$ information to a line bundle, but it forgets most of the vector bundle; for example, $\mathcal O_{\mathbb{CP}^1}\oplus\mathcal O_{\mathbb{CP}^1}$ and $\mathcal O_{\mathbb{CP}^1}(1)\oplus\mathcal O_{\mathbb{CP}^1}(-1)$ have the same determinant but are not isomorphic as holomorphic bundles. The equality of determinants says only that the sums of their degrees agree. Their splitting types differ: by the Birkhoff--Grothendieck theorem, a rank-two bundle on $\mathbb{CP}^1$ decomposes as $\mathcal O(a)\oplus\mathcal O(b)$ with $a\ge b$, and the ordered pair $(a,b)$ is an isomorphism invariant. Thus $(0,0)$ and $(1,-1)$ define different bundles even though both have determinant $\mathcal O_{\mathbb{CP}^1}$. This also avoids a tempting but false shortcut: both bundles have two global holomorphic sections, since $h^0(\mathcal O)=1$, $h^0(\mathcal O(1))=2$, and $h^0(\mathcal O(-1))=0$. What the determinant preserves is the top-order gluing factor, which is exactly the part seen by the top exterior power. This is the first place where characteristic classes will enter later. For example, positivity of a vector bundle is often detected first through its determinant. This motivates a direct calculation for split bundles. [example: Determinant Of A Direct Sum] Let $L_1,\dots,L_r$ be holomorphic line bundles over $X$, and choose local frames $e_i^{(a)}$ for $L_a$ on $U_i$. Write their transition functions with the convention that coordinate functions satisfy $v_i^{(a)}=g_{ij}^{(a)}v_j^{(a)}$. Then the ordered frame \begin{align*} (e_i^{(1)},\dots,e_i^{(r)}) \end{align*} for $E=L_1\oplus\cdots\oplus L_r$ has transition matrix \begin{align*} G_{ij}=\operatorname{diag}\left(g_{ij}^{(1)},\dots,g_{ij}^{(r)}\right). \end{align*} Its determinant is computed from the determinant formula: \begin{align*} \det(G_{ij})=\sum_{\sigma\in S_r}\operatorname{sgn}(\sigma)\prod_{a=1}^r (G_{ij})_{a,\sigma(a)}. \end{align*} Since $(G_{ij})_{a,b}=0$ when $a\ne b$, every term with $\sigma\ne\operatorname{id}$ vanishes. The only surviving term is the identity permutation, so \begin{align*} \det(G_{ij})=\prod_{a=1}^r (G_{ij})_{a,a}=g_{ij}^{(1)}\cdots g_{ij}^{(r)}. \end{align*} On the tensor product $L_1\otimes\cdots\otimes L_r$, the local tensor frame is \begin{align*} e_i^{(1)}\otimes\cdots\otimes e_i^{(r)}. \end{align*} On $U_i\cap U_j$, the frame transformation gives \begin{align*} e_j^{(1)}\otimes\cdots\otimes e_j^{(r)}=\left(g_{ij}^{(1)}e_i^{(1)}\right)\otimes\cdots\otimes\left(g_{ij}^{(r)}e_i^{(r)}\right). \end{align*} By multilinearity of the tensor product, \begin{align*} e_j^{(1)}\otimes\cdots\otimes e_j^{(r)}=\left(g_{ij}^{(1)}\cdots g_{ij}^{(r)}\right)\left(e_i^{(1)}\otimes\cdots\otimes e_i^{(r)}\right). \end{align*} Thus the tensor product line bundle has transition function $g_{ij}^{(1)}\cdots g_{ij}^{(r)}$, which is the same transition function as $\det E$. Therefore \begin{align*} \det E\cong L_1\otimes\cdots\otimes L_r. \end{align*} This is why, after a bundle is decomposed into line-bundle summands, its determinant is obtained by multiplying the corresponding line-bundle factors. [/example] ## The Euler Sequence On Projective Space The tangent bundle of projective space is not introduced by writing down vector fields in every chart separately. A more invariant construction relates it to the quotient of the product bundle $\mathbb{CP}^n\times\mathbb C^{n+1}$ by the tautological line in each fibre. [quotetheorem:7018] [citeproof:7018] This sequence is the standard computational tool for projective space. Its existence relies on a special feature of $\mathbb{CP}^n$: each point is itself a line in the fixed vector space $\mathbb C^{n+1}$, so the tautological line subbundle sits canonically inside the product bundle $\mathbb{CP}^n\times\mathbb C^{n+1}$. The quotient by this line records infinitesimal holomorphic motion of the line inside the ambient vector space, which is why it recovers the holomorphic tangent bundle after twisting. The precise hypothesis-dependence is that the construction needs both a fixed ambient vector space and a canonical line subbundle inside its product bundle; without those, there is no quotient bundle $Q=(X\times V)/\mathcal O(-1)$ and no natural identification $T^{1,0}X\cong\operatorname{Hom}(\mathcal O(-1),Q)$. On a general complex manifold there is no canonical ambient product bundle and no tautological subline to quotient out, so the Euler sequence has no direct analogue without extra structure such as an embedding into projective space. A concrete failure occurs for a compact complex torus $X=\mathbb C^n/\Lambda$: its holomorphic tangent bundle is globally framed by translation-invariant vector fields, but $X$ has no preferred presentation as a space of lines in a fixed vector space, hence no canonical line subbundle whose quotient produces $T^{1,0}X$ in the projective-space manner. Even when a manifold is embedded in projective space, restricting the Euler sequence gives information about the ambient holomorphic tangent bundle; recovering $T^{1,0}X$ also requires the normal sequence. Thus the projective Euler sequence expresses the holomorphic tangent bundle using line bundles whose sections and transition functions are well understood, but that expression depends on the ambient linear geometry of projective space. [example: Determinant Of The Tangent Bundle Of Projective Space] Let $q_{ij}=z_j/z_i$ be the transition function of $\mathcal O_{\mathbb{CP}^n}(1)$ on $U_i\cap U_j$. In the direct-sum frame of $\mathcal O_{\mathbb{CP}^n}(1)^{\oplus(n+1)}$, the transition matrix is \begin{align*} Q_{ij}=\operatorname{diag}(q_{ij},q_{ij},\dots,q_{ij}). \end{align*} Using the determinant formula for a diagonal matrix, \begin{align*} \det(Q_{ij})=q_{ij}\cdot q_{ij}\cdots q_{ij}=q_{ij}^{n+1}. \end{align*} By the definition of twisting powers, $\mathcal O_{\mathbb{CP}^n}(n+1)$ has transition function $q_{ij}^{n+1}$. Now use the Euler sequence \begin{align*} 0\to\mathcal O_{\mathbb{CP}^n}\to\mathcal O_{\mathbb{CP}^n}(1)^{\oplus(n+1)}\to T^{1,0}\mathbb{CP}^n\to0. \end{align*} The line bundle $\mathcal O_{\mathbb{CP}^n}$ has transition function $1$, so its determinant has transition function $1$, and its inverse determinant has transition function $1^{-1}=1$. For a short exact sequence, determinants multiply as \begin{align*} \det E\cong \det E'\otimes \det E''. \end{align*} Applying this identity to the Euler sequence gives \begin{align*} \det\left(\mathcal O_{\mathbb{CP}^n}(1)^{\oplus(n+1)}\right)\cong \det(\mathcal O_{\mathbb{CP}^n})\otimes \det T^{1,0}\mathbb{CP}^n. \end{align*} Tensoring both sides by $\det(\mathcal O_{\mathbb{CP}^n})^{-1}$ gives \begin{align*} \det T^{1,0}\mathbb{CP}^n\cong \det\left(\mathcal O_{\mathbb{CP}^n}(1)^{\oplus(n+1)}\right)\otimes \det(\mathcal O_{\mathbb{CP}^n})^{-1}. \end{align*} The transition function on the right is \begin{align*} q_{ij}^{n+1}\cdot 1^{-1}=q_{ij}^{n+1}. \end{align*} These are exactly the transition functions of $\mathcal O_{\mathbb{CP}^n}(n+1)$, hence \begin{align*} \det T^{1,0}\mathbb{CP}^n\cong\mathcal O_{\mathbb{CP}^n}(n+1). \end{align*} Since the canonical bundle is the dual of the determinant of the holomorphic tangent bundle, this calculation gives $K_{\mathbb{CP}^n}\cong\mathcal O_{\mathbb{CP}^n}(-n-1)$. [/example] Holomorphic vector bundles therefore combine local analytic data with global gluing. The next part of the course adds Hermitian metrics, allowing these holomorphic objects to be studied using connections, curvature, and positivity. Holomorphic vector bundles show that the Dolbeault viewpoint is not limited to functions or forms on the manifold itself. The next chapter equips these bundles with Hermitian metrics so that their holomorphic structure can be studied through connections, curvature, and positivity. # 5. The dbar Operator on Bundles Chapters 2 and 3 built the scalar Dolbeault complex on a complex manifold and used the splitting of complex-valued forms into types. In geometry, however, the objects we differentiate are often sections of vector bundles rather than ordinary functions. This chapter explains how a holomorphic vector bundle carries its own Dolbeault operator, how such operators can be recognised intrinsically on a smooth complex vector bundle, and how short exact sequences of bundles produce exact sequences in Dolbeault cohomology. ## Bundle-Valued Forms and the Dolbeault Operator The scalar operator $\bar\partial$ differentiates functions and forms in antiholomorphic directions. If $E \to X$ is a vector bundle, a section $s \in \Gamma(X,E)$ has values in fibres of $E$, so differentiating its coefficient functions depends on a choice of local frame. The first problem is to identify the extra structure that makes the antiholomorphic derivative of $E$-valued forms coordinate-independent. [definition: Bundle-Valued Form] Let $X$ be a complex manifold and let $E \to X$ be a smooth complex vector bundle. The space of smooth $E$-valued $(p,q)$-forms is \begin{align*} A^{p,q}(X,E) := \Gamma\bigl(X,\Lambda^{p,q}T^*X \otimes E\bigr). \end{align*} [/definition] Thus an element of $A^{p,q}(X,E)$ is locally a finite sum of ordinary $(p,q)$-forms multiplied by local smooth sections of $E$. To differentiate such objects, we need a rule that differentiates the form part by the scalar $\bar\partial$ and also records how the bundle component changes. [definition: Dolbeault Operator on a Holomorphic Bundle] Let $E \to X$ be a holomorphic vector bundle. The Dolbeault operator of $E$ is the first-order differential operator \begin{align*} \bar\partial_E : A^{p,q}(X,E) \to A^{p,q+1}(X,E) \end{align*} which, in a local holomorphic frame $e_1,\dots,e_r$, is given by \begin{align*} \bar\partial_E\left(\sum_{i=1}^r \alpha_i \otimes e_i\right) = \sum_{i=1}^r \bar\partial \alpha_i \otimes e_i. \end{align*} [/definition] The definition uses holomorphic frames, so the next problem is whether it depends on the chosen frame. A theorem is needed because the same section has different coefficient functions in different frames, and only holomorphic transition matrices should make the coefficientwise formula compatible on overlaps. [quotetheorem:7019] [citeproof:7019] This theorem says that holomorphicity of the bundle is precisely what lets the scalar Dolbeault operator act on coefficients without an added correction term. The holomorphic-frame hypothesis is essential: if a smooth frame changes by a matrix $g$ with $\bar\partial g\ne 0$, then applying $\bar\partial$ to the transformed coefficients produces an extra term involving $g^{-1}\bar\partial g$, so the coefficientwise formula does not patch. The theorem does not assert that every smooth frame is compatible with this formula, nor does it construct holomorphic frames for an arbitrary smooth bundle. It only proves that once a holomorphic bundle structure is already present, the local coefficientwise rule defines a global operator. The next question is whether the resulting sequence is a cochain complex, since cohomology can be formed only when the square of the differential vanishes. [quotetheorem:7020] [citeproof:7020] The square-zero identity turns bundle-valued forms into a complex. Its proof uses more than the formal Leibniz rule: locally the bundle must admit frames in which the connection matrix has no $(0,1)$ contribution, or more generally the associated $(0,1)$-connection must be flat. If a smooth bundle is equipped with a local operator $D''=\bar\partial+A$ with $\bar\partial A+A\wedge A\ne 0$, then $(D'')^2$ is curvature rather than zero, so closedness and exactness would not define cohomology. The theorem does not say that every first-order antiholomorphic operator squares to zero; it says this for the operator coming from a holomorphic vector bundle. This motivates defining the cohomology groups that measure the failure of local $\bar\partial_E$-closed data to be globally $\bar\partial_E$-exact. [definition: Bundle-Valued Dolbeault Cohomology] Let $E \to X$ be a holomorphic vector bundle. The Dolbeault cohomology of $X$ with coefficients in $E$ is \begin{align*} H^{p,q}_{\bar\partial}(X,E) := \frac{\ker(\bar\partial_E:A^{p,q}(X,E)\to A^{p,q+1}(X,E))}{\operatorname{im}(\bar\partial_E:A^{p,q-1}(X,E)\to A^{p,q}(X,E))}. \end{align*} [/definition] This definition recovers scalar Dolbeault cohomology when $E$ is the product holomorphic line bundle. In degree $(0,0)$, it recovers holomorphic sections. [example: Computing in a Holomorphic Frame] Let $X\subset \mathbb C^n$ be open and let $E=X\times \mathbb C^r$ have its standard holomorphic frame $e_1,\dots,e_r$. If $s=\sum_{i=1}^r f_i e_i$ is a smooth section, then, identifying $f_i e_i$ with $f_i\otimes e_i$, the defining coefficientwise formula in this holomorphic frame gives \begin{align*} \bar\partial_E s=\bar\partial_E\left(\sum_{i=1}^r f_i\otimes e_i\right)=\sum_{i=1}^r \bar\partial f_i\otimes e_i. \end{align*} Since $e_1,\dots,e_r$ is a frame, an $E$-valued $(0,1)$-form is zero exactly when each coefficient of $e_i$ is zero. Therefore $\bar\partial_Es=0$ exactly when \begin{align*} \bar\partial f_i=0 \quad \text{for every } i=1,\dots,r. \end{align*} By the Cauchy-Riemann equations, this is equivalent to each coefficient function $f_i$ being holomorphic on $X$. More generally, for $\alpha=\sum_{i=1}^r \alpha_i\otimes e_i\in A^{p,q}(X,E)$, the same holomorphic-frame formula gives \begin{align*} \bar\partial_E\alpha=\sum_{i=1}^r \bar\partial\alpha_i\otimes e_i. \end{align*} Thus, for the product holomorphic bundle, the bundle-valued Dolbeault complex is computed by applying the scalar $\bar\partial$ separately to each component. [/example] The example shows why holomorphic frames are computationally privileged. If the frame is merely smooth, the operator acquires a matrix of $(0,1)$-forms, and this observation leads to the intrinsic definition of a holomorphic structure. ## Holomorphic Structures as Flat Antiholomorphic Connections A smooth complex vector bundle does not come with holomorphic transition functions. The next question is whether holomorphicity can be encoded by a differential operator alone, without first choosing a holomorphic atlas for the bundle. The answer is that a holomorphic structure is the same kind of data as a flat connection in the $(0,1)$ directions. [definition: Antiholomorphic Connection] Let $E \to X$ be a smooth complex vector bundle. A $(0,1)$-connection on $E$ is a $\mathbb C$-linear first-order operator \begin{align*} D'' : A^{0,0}(X,E) \to A^{0,1}(X,E) \end{align*} satisfying \begin{align*} D''(fs)=\bar\partial f\otimes s + fD''s \end{align*} for all $f\in C^\infty(X;\mathbb C)$ and $s\in A^{0,0}(X,E)$. [/definition] The Leibniz rule determines how $D''$ extends to all bundle-valued forms: differentiate the scalar form part and apply $D''$ to the bundle part with the usual sign. To produce a Dolbeault-type cohomology theory, we need the extended operator to have square zero. [definition: Flat Antiholomorphic Connection] Let $E \to X$ be a smooth complex vector bundle and let $D''$ be a $(0,1)$-connection. Its extension is the sequence of operators \begin{align*} D'' : A^{0,q}(X,E) \to A^{0,q+1}(X,E), \qquad q\ge 0, \end{align*} determined by the graded Leibniz rule. The connection is flat if the extended operator satisfies \begin{align*} (D'')^2=0. \end{align*} [/definition] Flatness is the integrability condition. In a smooth frame, it appears as a Maurer-Cartan equation for the matrix of connection forms. [example: Local Matrix of a Holomorphic Structure] Let $e_1,\dots,e_r$ be a smooth local frame on $U$, and write \begin{align*} D''e_j=\sum_{i=1}^r A_{ij}\otimes e_i,\qquad A_{ij}\in A^{0,1}(U). \end{align*} For a section $s=\sum_{j=1}^r f_j e_j$, the Leibniz rule for a $(0,1)$-connection gives \begin{align*} D''s=\sum_{j=1}^r D''(f_j e_j)=\sum_{j=1}^r \bar\partial f_j\otimes e_j+\sum_{j=1}^r f_jD''e_j. \end{align*} Substituting the expression for $D''e_j$ gives \begin{align*} D''s=\sum_{j=1}^r \bar\partial f_j\otimes e_j+\sum_{j=1}^r\sum_{i=1}^r f_jA_{ij}\otimes e_i. \end{align*} Reindexing the first sum by the frame vector $e_i$ and grouping the coefficients of each $e_i$ gives \begin{align*} D''s=\sum_{i=1}^r\left(\bar\partial f_i+\sum_{j=1}^r A_{ij}f_j\right)\otimes e_i. \end{align*} Now apply $D''$ to the frame vector $e_j$ twice. Since each $A_{ij}$ has degree $1$, the graded Leibniz rule gives \begin{align*} (D'')^2e_j=\sum_{i=1}^r \bar\partial A_{ij}\otimes e_i-\sum_{i=1}^r A_{ij}\wedge D''e_i. \end{align*} Substituting $D''e_i=\sum_{k=1}^r A_{ki}\otimes e_k$ gives \begin{align*} (D'')^2e_j=\sum_{i=1}^r \bar\partial A_{ij}\otimes e_i-\sum_{i=1}^r\sum_{k=1}^r A_{ij}\wedge A_{ki}\otimes e_k. \end{align*} The coefficient of $e_k$ is \begin{align*} \bar\partial A_{kj}-\sum_{i=1}^r A_{ij}\wedge A_{ki}. \end{align*} Because $(0,1)$-forms anticommute, $-A_{ij}\wedge A_{ki}=A_{ki}\wedge A_{ij}$, so this coefficient is \begin{align*} \bar\partial A_{kj}+\sum_{i=1}^r A_{ki}\wedge A_{ij}. \end{align*} Thus $(D'')^2=0$ is equivalent, entry by entry, to \begin{align*} \bar\partial A+A\wedge A=0, \end{align*} where $(A\wedge A)_{kj}=\sum_i A_{ki}\wedge A_{ij}$. In a holomorphic frame one has $D''e_j=0$ for every $j$, so $A=0$; the matrix $A$ records exactly the correction term introduced by using a general smooth frame. [/example] The local formula suggests that solving $D''e'_j=0$ should produce holomorphic frames, but this is a system of $\bar\partial$-equations for a whole frame, not just a formal change of notation. The obstruction is the square of the operator: if $(D'')^2$ is nonzero, applying it to a supposedly flat local frame would immediately contradict $D''e'_j=0$. The theorem below says that the vanishing of this obstruction is also sufficient locally, so flat $(0,1)$-connections are exactly holomorphic structures on the fixed smooth bundle. [quotetheorem:7021] Flatness is the essential hypothesis: in a local smooth frame with $D''=\bar\partial+A$, the obstruction to solving for a frame annihilated by $D''$ is the curvature term $\bar\partial A+A\wedge A$. For instance, if this matrix of $(0,2)$-forms is nonzero on an open set, then applying $(D'')^2$ to a local section detects a nonzero obstruction, so no holomorphic frame compatible with $D''$ can exist there. The theorem does not say that the equivalence is a purely formal consequence of the Leibniz rule; the construction of the annihilated frames uses elliptic or several complex variables methods for local $\bar\partial$-systems. It also does not classify holomorphic bundles up to isomorphism, but identifies which operators on a fixed smooth bundle define holomorphic structures. [remark: Integrability and Geometry] The theorem is the vector bundle analogue of the Newlander-Nirenberg philosophy: an algebraic square-zero condition on antiholomorphic differentiation is equivalent to the existence of holomorphic coordinates, now in the fibres of the bundle. It allows us to define holomorphic structures on a fixed smooth bundle by writing down operators rather than transition functions. [/remark] Once this equivalence is accepted, bundle-valued Dolbeault cohomology can be defined from a flat $(0,1)$-connection alone. This viewpoint is especially useful when deforming holomorphic structures, because small changes of $D''$ are represented by $(0,1)$-forms with values in endomorphism bundles. ## Exact Sequences of Dolbeault Complexes Many holomorphic bundles arise as subbundles, quotient bundles, or extensions. The natural question is whether an exact sequence of holomorphic bundles remains exact after passing to bundle-valued forms, and what cohomological information is produced when global exactness fails. The answer is a long exact sequence in Dolbeault cohomology. [definition: Short Exact Sequence of Holomorphic Vector Bundles] A sequence of holomorphic vector bundles over $X$ \begin{align*} 0 \to E \xrightarrow{i} F \xrightarrow{p} G \to 0 \end{align*} is short exact if $i$ is injective, $p$ is surjective, and $\operatorname{im} i=\ker p$ as holomorphic subbundles. [/definition] Exactness at the bundle level gives exactness on smooth forms because smooth vector bundles admit local splittings and smooth partitions of unity. The next theorem is needed because the Dolbeault differentials must also commute with the induced maps, otherwise exactness would not be exactness of complexes. [quotetheorem:7022] [citeproof:7022] The holomorphicity of the bundle maps is essential here, because it is exactly what makes $i_*$ and $p_*$ commute with the corresponding Dolbeault operators. If a smooth bundle map has a local matrix $M$ with $\bar\partial M\ne 0$, then $\bar\partial(M\alpha)$ contains the extra term $(\bar\partial M)\wedge\alpha$, so the induced map is not a cochain map. The theorem also does not assert that the original sequence splits holomorphically; surjectivity on smooth forms uses smooth local lifts and partitions of unity, which are unavailable in the holomorphic category. Passing from this short exact sequence of complexes to cohomology is then a general homological algebra construction. In the Dolbeault setting it produces connecting maps that record the obstruction to lifting a closed quotient-valued form to a closed middle-valued form. [quotetheorem:3471] [citeproof:3471] The connecting map is often the most important part of the sequence. Exactness of the bundle sequence is needed because the construction first lifts to the middle bundle and then identifies the derivative of that lift with a form in the subbundle. If the original maps did not form a short exact sequence of complexes, this derivative might not land in the subbundle or might depend on choices in a non-exact way. The long exact sequence does not compute the groups by itself; it relates them and becomes effective only when enough neighbouring Dolbeault cohomology groups are already known. In geometric applications, it packages extension data into a cohomology class. [example: Extension Classes in Dolbeault Cohomology] Consider a short exact sequence of holomorphic bundles \begin{align*} 0 \to F \to V \to E \to 0. \end{align*} Choose a smooth splitting, so that every smooth section of $V$ is written uniquely as a pair $(f,e)$ with $f\in A^{0,0}(X,F)$ and $e\in A^{0,0}(X,E)$. Since the inclusion $F\to V$ and projection $V\to E$ are holomorphic, $\bar\partial_V(f,0)=(\bar\partial_F f,0)$ and the $E$-component of $\bar\partial_V(f,e)$ is $\bar\partial_E e$. Hence the remaining part of $\bar\partial_V(0,e)$ is an $F$-valued $(0,1)$-form depending $C^\infty$-linearly on $e$; call it $\beta(e)$, with \begin{align*} \beta\in A^{0,1}(X,\operatorname{Hom}(E,F)). \end{align*} Thus, in the smooth splitting $V\cong F\oplus E$, \begin{align*} \bar\partial_V(f,e)=(\bar\partial_F f+\beta(e),\bar\partial_E e). \end{align*} Apply $\bar\partial_V$ twice to the section $(0,e)$. First, \begin{align*} \bar\partial_V(0,e)=(\beta(e),\bar\partial_E e). \end{align*} Applying $\bar\partial_V$ again gives \begin{align*} \bar\partial_V^2(0,e)=(\bar\partial_F(\beta(e))+\beta(\bar\partial_E e),\bar\partial_E^2e). \end{align*} The second component is zero because $\bar\partial_E^2=0$, and $\bar\partial_V^2=0$ forces \begin{align*} \bar\partial_F(\beta(e))+\beta(\bar\partial_E e)=0. \end{align*} By the definition of the Dolbeault operator on $\operatorname{Hom}(E,F)$, the left-hand side is $(\bar\partial_{\operatorname{Hom}(E,F)}\beta)(e)$, so \begin{align*} \bar\partial_{\operatorname{Hom}(E,F)}\beta=0. \end{align*} Now change the smooth splitting by a smooth bundle map $\varphi\in A^{0,0}(X,\operatorname{Hom}(E,F))$, so that the new lift of $e$ is $(\varphi(e),e)$. In the old splitting, \begin{align*} \bar\partial_V(\varphi(e),e)=(\bar\partial_F(\varphi(e))+\beta(e),\bar\partial_E e). \end{align*} To express this in the new splitting, subtract the new lift of the $E$-component, namely $(\varphi(\bar\partial_E e),\bar\partial_E e)$. The new $F$-component is therefore \begin{align*} \bar\partial_F(\varphi(e))+\beta(e)-\varphi(\bar\partial_E e). \end{align*} By the defining formula for $\bar\partial_{\operatorname{Hom}(E,F)}\varphi$, this equals \begin{align*} \beta(e)+(\bar\partial_{\operatorname{Hom}(E,F)}\varphi)(e). \end{align*} Thus the new upper-right entry is \begin{align*} \beta'=\beta+\bar\partial_{\operatorname{Hom}(E,F)}\varphi. \end{align*} Therefore the Dolbeault cohomology class \begin{align*} [\beta]\in H^{0,1}_{\bar\partial}(X,\operatorname{Hom}(E,F)) \end{align*} is independent of the chosen smooth splitting. It vanishes exactly when some choice of splitting has $\beta=0$, and in that case $\bar\partial_V=\bar\partial_F\oplus\bar\partial_E$ in the splitting, so the short exact sequence splits holomorphically. [/example] This example is the concrete form of the connecting homomorphism in degree zero. A holomorphic section of the quotient bundle lifts smoothly to the middle bundle, and applying $\bar\partial$ to the lift produces the obstruction class in the subbundle. ## Projective Space and the Euler Sequence The formalism becomes useful when applied to standard exact sequences on complex projective space. Returning to the Euler sequence from Chapter 4, it describes the holomorphic tangent bundle of $\mathbb{CP}^n$ in terms of line bundles, so it converts questions about tangent-valued forms into questions about forms with coefficients in $\mathcal O$ and $\mathcal O(1)$. [quotetheorem:7023] For this course, the Euler sequence is used as a standard structural input rather than proved from the tautological line bundle. The hypotheses are specific to projective space and its standard line bundle $\mathcal O(1)$; replacing $\mathbb{CP}^n$ by an arbitrary complex manifold gives no comparable sequence without extra geometry. Exactness is also essential: without the injection of $\mathcal O$ and the quotient description of $T^{1,0}\mathbb{CP}^n$, the associated Dolbeault complexes would not produce the stated long exact sequence. The Euler sequence does not by itself compute tangent-valued Dolbeault cohomology; it reduces such computations to line-bundle-valued groups and the connecting maps between them. Its importance here is that it can be fed directly into the Dolbeault machinery developed above. [example: Euler Sequence as a Dolbeault Exact Sequence] By the *[Euler Sequence on Projective Space](/theorems/7023)*, there is a short exact sequence of holomorphic vector bundles \begin{align*} 0 \to \mathcal O \to \mathcal O(1)^{\oplus(n+1)} \to T^{1,0}\mathbb{CP}^n \to 0. \end{align*} For fixed $p$, applying the functor $A^{p,q}(\mathbb{CP}^n,-)$ in each degree $q$ gives \begin{align*} 0 \to A^{p,q}(\mathbb{CP}^n,\mathcal O) \to A^{p,q}(\mathbb{CP}^n,\mathcal O(1)^{\oplus(n+1)}) \to A^{p,q}(\mathbb{CP}^n,T^{1,0}\mathbb{CP}^n) \to 0. \end{align*} Since forms with values in a direct sum are direct sums of forms with values in each summand, \begin{align*} A^{p,q}(\mathbb{CP}^n,\mathcal O(1)^{\oplus(n+1)}) = A^{p,q}(\mathbb{CP}^n,\mathcal O(1))^{\oplus(n+1)}. \end{align*} Thus, degree by degree, the sequence becomes \begin{align*} 0 \to A^{p,q}(\mathbb{CP}^n,\mathcal O) \to A^{p,q}(\mathbb{CP}^n,\mathcal O(1))^{\oplus(n+1)} \to A^{p,q}(\mathbb{CP}^n,T^{1,0}\mathbb{CP}^n) \to 0. \end{align*} The maps in the Euler sequence are holomorphic, so they commute with the corresponding bundle Dolbeault operators by the *Short Exact Sequence of Dolbeault Complexes*. Therefore these degreewise exact sequences assemble into the short exact sequence of cochain complexes \begin{align*} 0 \to A^{p,\bullet}(\mathbb{CP}^n,\mathcal O) \to A^{p,\bullet}(\mathbb{CP}^n,\mathcal O(1))^{\oplus(n+1)} \to A^{p,\bullet}(\mathbb{CP}^n,T^{1,0}\mathbb{CP}^n) \to 0. \end{align*} Applying the *Long Exact Sequence in Bundle Dolbeault Cohomology* then gives, for each $q$, \begin{align*} \cdots \to H^{p,q}_{\bar\partial}(\mathbb{CP}^n,\mathcal O) \to H^{p,q}_{\bar\partial}(\mathbb{CP}^n,\mathcal O(1))^{\oplus(n+1)} \to H^{p,q}_{\bar\partial}(\mathbb{CP}^n,T^{1,0}\mathbb{CP}^n) \xrightarrow{\delta} H^{p,q+1}_{\bar\partial}(\mathbb{CP}^n,\mathcal O) \to \cdots . \end{align*} Thus tangent-bundle-valued Dolbeault cohomology on projective space is related to the line-bundle-valued groups for $\mathcal O$ and $\mathcal O(1)$, and in deformation problems $H^{0,1}_{\bar\partial}(\mathbb{CP}^n,T^{1,0}\mathbb{CP}^n)$ is the group where infinitesimal changes of complex structure live. [/example] The chapter has now moved from local coefficientwise differentiation to global cohomological consequences. The key point is that the equation $\bar\partial_E^2=0$ is both an analytic integrability condition and the algebraic reason that bundle-valued Dolbeault cohomology exists. Hermitian metrics add a way to measure lengths and angles, but they also prepare the ground for differentiating bundle-valued objects. The next chapter introduces connections on complex vector bundles, the device that compares fibres and turns local trivializations into geometric information. # 6. Hermitian Metrics on Complex Manifolds and Bundles Hermitian geometry begins when the holomorphic objects from the previous chapters are equipped with smoothly varying positive definite inner products. The main question is how to measure vectors, forms, and bundle-valued tensors in a way compatible with the complex structure. This chapter introduces Hermitian metrics on complex vector bundles, then specialises to the holomorphic tangent bundle of a complex manifold, where the same data determines both a real Riemannian metric and a distinguished $(1,1)$-form. ## Hermitian Metrics on Complex Vector Bundles A complex vector bundle has complex-linear transition functions, but the transition functions alone do not choose lengths or angles in the fibres. The first problem is to add such a choice without destroying the smooth or holomorphic structure of the bundle. [definition: Hermitian Metric On A Complex Vector Bundle] Let $E \to X$ be a smooth complex vector bundle over a smooth manifold $X$. A Hermitian metric on $E$ is a smooth assignment \begin{align*} h_x : E_x \times E_x \to \mathbb C, \qquad x \in X, \end{align*} such that for every $x \in X$, the map $h_x$ is linear in the first variable, conjugate-linear in the second variable, satisfies $h_x(u,v)=\overline{h_x(v,u)}$, and satisfies $h_x(u,u)>0$ for every nonzero $u \in E_x$. [/definition] The associated fibre norm is $|u|_h=(h_x(u,u))^{1/2}$ for $u \in E_x$. Smoothness means that if $s,t$ are smooth local sections of $E$, then the function $x \mapsto h_x(s(x),t(x))$ is smooth. [example: Standard Metric On A Trivial Bundle] Let $E=X\times \mathbb C^r$, and write two local sections as $s=(s_1,\dots,s_r)$ and $t=(t_1,\dots,t_r)$. The standard Hermitian metric is \begin{align*} h(s,t)=\sum_{\alpha=1}^{r}s_\alpha\overline{t_\alpha}. \end{align*} It is linear in the first variable because, for $a,b\in\mathbb C$ and sections $s,q,t$, \begin{align*} h(as+bq,t)=\sum_{\alpha=1}^{r}(as_\alpha+bq_\alpha)\overline{t_\alpha}=a\sum_{\alpha=1}^{r}s_\alpha\overline{t_\alpha}+b\sum_{\alpha=1}^{r}q_\alpha\overline{t_\alpha}=ah(s,t)+bh(q,t). \end{align*} It is conjugate-linear in the second variable because \begin{align*} h(s,at+bq)=\sum_{\alpha=1}^{r}s_\alpha\overline{at_\alpha+bq_\alpha}=\bar a\sum_{\alpha=1}^{r}s_\alpha\overline{t_\alpha}+\bar b\sum_{\alpha=1}^{r}s_\alpha\overline{q_\alpha}=\bar a h(s,t)+\bar b h(s,q). \end{align*} Hermitian symmetry follows from \begin{align*} \overline{h(t,s)}=\overline{\sum_{\alpha=1}^{r}t_\alpha\overline{s_\alpha}}=\sum_{\alpha=1}^{r}\overline{t_\alpha}s_\alpha=\sum_{\alpha=1}^{r}s_\alpha\overline{t_\alpha}=h(s,t), \end{align*} and positivity follows from \begin{align*} h(s,s)=\sum_{\alpha=1}^{r}s_\alpha\overline{s_\alpha}=\sum_{\alpha=1}^{r}|s_\alpha|^2, \end{align*} which is positive exactly when at least one component $s_\alpha$ is nonzero. For the standard frame $e_1,\dots,e_r$, the section $e_\alpha$ has component $1$ in the $\alpha$-th slot and component $0$ in every other slot. Hence \begin{align*} h(e_\alpha,e_\beta)=\sum_{\gamma=1}^{r}(e_\alpha)_\gamma\overline{(e_\beta)_\gamma}=\delta_{\alpha\beta}. \end{align*} Thus the local matrix $H=(h_{\alpha\bar\beta})$ of the standard metric is the identity matrix. In an arbitrary trivialisation, the same rule is replaced by a positive definite Hermitian matrix of smooth functions, so the standard metric is the coordinate model for all local Hermitian metric matrices. [/example] The standard metric is local, while vector bundles are assembled from local pieces. A first obstruction appears if one tries to glue the standard matrices directly: on overlaps the transition functions need not be unitary, so the identity matrix in one trivialisation becomes $A^*A$ in another. Without a smoothing mechanism, these incompatible local choices would not define a global fibre inner product. The next result explains why paracompactness removes this obstruction for smooth complex vector bundles. [quotetheorem:7024] [citeproof:7024] The hypothesis that $X$ is paracompact is used through the smooth [partition of unity](/page/Partition%20of%20Unity); on spaces without such partitions, local fibre metrics need not glue by this argument. The theorem asserts smooth existence, not uniqueness, compatibility with a prescribed holomorphic structure, or curvature control. For instance, the standard metric in two different holomorphic trivialisations of a nontrivial line bundle will not usually agree on overlaps, so the constructed metric depends on the partition of unity rather than on holomorphic transition functions alone. This existence result gives the background metric needed to discuss duals, tensor products, determinant bundles, and eventually Chern connections. A metric on $E$ itself is only the starting point: later calculations also evaluate covectors on vectors, so the metric must provide a compatible way to measure elements of $E^*$. The failure to choose this induced metric is not cosmetic; in a non-orthonormal frame, using the same matrix $H$ on covectors gives the wrong size for the functional dual to a long vector. The inverse matrix must appear, which is why the dual metric needs its own definition. [definition: Induced Dual Metric] Let $(E,h)$ be a Hermitian vector bundle over $X$. The induced Hermitian metric $h^*$ on $E^*$ is defined fibrewise by \begin{align*} h_x^* : E_x^* \times E_x^* \to \mathbb C, \qquad x \in X, \end{align*} where $h_x^*$ is the unique Hermitian form whose norm satisfies \begin{align*} |\lambda|_{h^*}=\sup\{ |\lambda(u)| : u \in E_x,\ |u|_h=1\}, \qquad \lambda \in E_x^*. \end{align*} [/definition] With respect to an $h$-orthonormal frame $e_1,\dots,e_r$ and dual frame $e^1,\dots,e^r$, the dual frame is $h^*$-orthonormal. If the local matrix of $h$ in a general frame is $H=(h_{\alpha\bar\beta})$, then the local matrix of $h^*$ in the dual frame is $H^{-1}$ with the corresponding conjugate index convention. This inverse-matrix behaviour is forced by the pairing $E^*\times E\to \mathbb C$; otherwise the estimate $|\lambda(u)|\le |\lambda|_{h^*}|u|_h$ would fail in rescaled frames. To measure pairings and bundle-valued tensors rather than single covectors, the same compatibility requirement leads to the tensor product metric. [definition: Tensor Product Metric] Let $(E,h_E)$ and $(F,h_F)$ be Hermitian vector bundles over $X$. The tensor product metric $h_E\otimes h_F$ on $E\otimes F$ is determined on decomposable tensors by \begin{align*} (h_E\otimes h_F)_x : (E_x\otimes F_x)\times (E_x\otimes F_x)\to \mathbb C, \end{align*} with \begin{align*} (h_E\otimes h_F)(u_1\otimes v_1,u_2\otimes v_2)=h_E(u_1,u_2)h_F(v_1,v_2). \end{align*} [/definition] This formula extends sesquilinearly to arbitrary finite sums. It also gives induced metrics on $E^{\otimes a}\otimes (E^*)^{\otimes b}$, so tensors with bundle indices can be measured once $E$ has a Hermitian metric. Without this product rule, a tensor such as a bundle-valued one-form would have no coordinate-independent pointwise norm: rescaling the bundle frame and rescaling the cotangent coframe would not cancel correctly. The remaining functorial construction needed constantly in complex geometry is the top exterior power, where the metric should measure fibrewise volume. Here a product of individual lengths is insufficient, since it gives the same value to an [orthonormal basis](/page/Orthonormal%20Basis) and to a nearly dependent collection of unit vectors. The determinant metric corrects this failure by measuring the full Gram determinant. [definition: Determinant Metric] Let $(E,h)$ be a Hermitian vector bundle of rank $r$. The determinant metric on $\det E=\Lambda^r E$ is defined by \begin{align*} (\det h)_x : (\det E_x)\times(\det E_x)\to \mathbb C, \qquad x\in X, \end{align*} where on decomposable top exterior vectors its norm satisfies \begin{align*} |u_1\wedge \cdots \wedge u_r|^2_{\det h}=\det(h(u_\alpha,u_\beta))_{\alpha,\beta=1}^{r}. \end{align*} [/definition] The determinant metric records the Hermitian volume of a parallelepiped in a fibre. The determinant is necessary because multiplying the lengths $|u_1|_h\cdots |u_r|_h$ would ignore angles; two nearly dependent vectors can all have length one while spanning very small volume. This metric will be used in Chapter 9 to define curvature forms of determinant line bundles and the Chern-Ricci form. [example: Metrics On The Line Bundles O(k)] On $\mathbb{CP}^n$, let $O(-1)$ denote the tautological line subbundle of the trivial bundle $\mathbb{CP}^n\times \mathbb C^{n+1}$, equipped with the metric induced by the standard Hermitian metric on $\mathbb C^{n+1}$. Its dual $O(1)=O(-1)^*$ then carries the dual metric, and tensor powers give the metrics on $O(k)$. On the affine chart $Z_0\ne 0$, write $z_j=Z_j/Z_0$. The tautological local frame of $O(-1)$ is \begin{align*} s(z)=(1,z_1,\dots,z_n)\in \mathbb C^{n+1}. \end{align*} Using the standard Hermitian metric on $\mathbb C^{n+1}$, \begin{align*} |s(z)|^2=\langle s(z),s(z)\rangle=\sum_{\alpha=0}^{n}s_\alpha(z)\overline{s_\alpha(z)}=1+\sum_{j=1}^{n}z_j\overline{z_j}=1+|z_1|^2+\cdots+|z_n|^2. \end{align*} Let $s^*$ be the dual local frame of $O(1)$, so $s^*(s)=1$. Since $O(1)$ has the dual metric, the defining dual norm gives \begin{align*} |s^*(z)|^2=\frac{1}{|s(z)|^2}. \end{align*} Substituting the expression for $|s(z)|^2$ gives \begin{align*} |s^*(z)|^2=(1+|z_1|^2+\cdots+|z_n|^2)^{-1}. \end{align*} For tensor powers, the frame $(s^*)^{\otimes k}$ of $O(k)$ has squared norm $(1+|z_1|^2+\cdots+|z_n|^2)^{-k}$, while $s^{\otimes k}$ in $O(-k)$ has squared norm $(1+|z_1|^2+\cdots+|z_n|^2)^k$. Thus the local weight of the metric records the projective factor $1+|z|^2$, which is the basic way Hermitian line bundle metrics encode projective geometry. [/example] ## Hermitian Metrics On The Holomorphic Tangent Bundle On a complex manifold $X$, the most important vector bundle is the holomorphic tangent bundle $T^{1,0}X$. An arbitrary real Riemannian metric on $TX$ need not respect the complex structure: it may assign different lengths to $u$ and $Ju$, so multiplication by $i$ would not be an isometry on complex tangent directions. Hermitian geometry rules out this failure by measuring the complex tangent bundle directly. A Hermitian metric on this bundle measures complex tangent vectors, but it also contains real differential-geometric information because $T^{1,0}X$ is built from the real tangent bundle and the complex structure. [definition: Hermitian Metric On A Complex Manifold] Let $X$ be a complex manifold. A Hermitian metric on $X$ is a Hermitian metric $h$ on the holomorphic tangent bundle $T^{1,0}X$. [/definition] Thus a Hermitian manifold is a pair $(X,h)$ consisting of a complex manifold and such a metric. If $z_1,\dots,z_n$ are local holomorphic coordinates, the frame $\partial_{z_1},\dots,\partial_{z_n}$ gives local coefficients \begin{align*} h_{i\bar j}=h(\partial_{z_i},\partial_{z_j}). \end{align*} The next problem is to recover real lengths and two-dimensional oriented areas from these complex coefficients. We fix the convention \begin{align*} u^{1,0}=\frac{1}{2}(u-iJu)\in T_x^{1,0}X \end{align*} for the image of a real tangent vector $u\in T_xX$ in the $(1,0)$ tangent space. With this normalisation, the associated real Riemannian metric and fundamental form are defined as follows. [definition: Associated Riemannian Metric And Fundamental Form] Let $(X,h)$ be a Hermitian manifold with complex structure $J$ on $TX$. For real tangent vectors $u,v \in T_xX$, define \begin{align*} g(u,v)=2\operatorname{Re} h(u^{1,0},v^{1,0}), \end{align*} and \begin{align*} \omega(u,v)=g(Ju,v), \end{align*} where $u^{1,0}=\frac{1}{2}(u-iJu)$. [/definition] Different normalisations of the map $TX\to T^{1,0}X$ move factors of $2$ between $h$, $g$, and $\omega$; the convention above is fixed for the rest of this chapter. The formulas are useful only if they genuinely produce Riemannian and differential-form data compatible with $J$. If $J$ is not an isometry for the real metric, then $\omega(u,v)=g(Ju,v)$ need not be skew-symmetric, so it would not be a differential two-form. The next theorem verifies that Hermitian metrics are exactly the real metrics for which this obstruction disappears. [quotetheorem:7025] [citeproof:7025] This theorem is the reason Hermitian geometry sits between complex geometry and Riemannian geometry. The $J$-invariance hypothesis in the converse is necessary: on $\mathbb C$ with coordinates $(x,y)$, the real metric $dx^2+2dy^2$ is positive definite but does not satisfy $g(J\partial_x,J\partial_x)=g(\partial_x,\partial_x)$, and the associated expression $g(Ju,v)$ is not skew-symmetric. The theorem does not say that every Riemannian metric on a complex manifold is Hermitian, nor does it impose the stronger Kahler condition $d\omega=0$. What it does provide is the dictionary needed later to move between metric contractions on $T^{1,0}X$, real volume forms, adjoints of differential operators, and the differential form $\omega$. [example: Flat Metric On A Complex Torus] Let $\pi:\mathbb C^n\to X=\mathbb C^n/\Lambda$ be the quotient map. For $\lambda\in \Lambda$, translation $T_\lambda(z)=z+\lambda$ has differential $dT_\lambda=\operatorname{id}_{\mathbb C^n}$, so for tangent vectors $u,v\in T_z\mathbb C^n\cong \mathbb C^n$, \begin{align*} h_0(dT_\lambda u,dT_\lambda v)=h_0(u,v)=\sum_{j=1}^{n}u_j\overline{v_j}. \end{align*} Thus $h_0$ has the same value on any two representatives of the same tangent vectors on the quotient, and it defines a Hermitian metric on $X$. In the induced holomorphic coordinates, the coordinate frame satisfies \begin{align*} h_0(\partial_{z_i},\partial_{z_j})=\delta_{ij}. \end{align*} Hence the local Hermitian matrix is the identity matrix, and the associated fundamental form is \begin{align*} \omega=i\sum_{i,j=1}^{n}\delta_{ij}\,dz_i\wedge d\bar z_j=i\sum_{j=1}^{n}dz_j\wedge d\bar z_j. \end{align*} Each coefficient $\delta_{ij}$ is constant, so every first derivative of the metric coefficients is zero in these coordinates. Consequently any local curvature term built from derivatives of the metric coefficients vanishes, making the complex torus with this metric the reference flat model for Hermitian manifolds. [/example] ## Local Expressions And Unitary Frames After the global definitions, the next task is computational: express Hermitian geometry in local coordinates and choose frames that simplify the metric at a point or on a small open set. These local descriptions are the ones used in curvature calculations later in the course. [definition: Local Hermitian Matrix] Let $(E,h)$ be a Hermitian vector bundle and let $e_1,\dots,e_r$ be a smooth local frame over an open set $U\subset X$. The local Hermitian matrix of $h$ in this frame is \begin{align*} H=(h_{\alpha\bar\beta})_{\alpha,\beta=1}^{r}, \qquad h_{\alpha\bar\beta}=h(e_\alpha,e_\beta). \end{align*} [/definition] The matrix $H(x)$ is positive definite Hermitian for each $x\in U$. If the frame changes by $e'=eA$, where $A:U\to GL(r,\mathbb C)$, then the matrix changes by \begin{align*} H'=A^*HA, \end{align*} where $A^*$ is the conjugate transpose. If one computes contractions in an arbitrary frame and forgets this transformation law, even the squared length of a section can appear to change after a rescaling of the frame. This is the basic local computational failure that unitary frames are designed to avoid. The most convenient frames for metric computations are therefore frames in which $H$ is the identity matrix, which motivates the definition of a unitary frame. [definition: Unitary Frame] Let $(E,h)$ be a Hermitian vector bundle. A local frame $e_1,\dots,e_r$ over $U\subset X$ is unitary if \begin{align*} h(e_\alpha,e_\beta)=\delta_{\alpha\beta} \end{align*} for all $\alpha,\beta\in\{1,\dots,r\}$ on $U$. [/definition] Unitary frames make metric contractions look like the standard Hermitian inner product. They rarely preserve holomorphicity unless the metric is special, but the computational value of such frames depends on whether they exist locally. A pointwise orthonormal basis is not enough for geometry, since differentiating it would make no sense unless the basis varies smoothly. The next theorem supplies this local smooth existence result. [quotetheorem:6247] [citeproof:6247] The smoothness hypothesis is essential: a merely continuous Hermitian metric can produce continuous orthonormal frames, but differentiating those frames in connection formulas may fail. The theorem also does not say that a holomorphic vector bundle admits holomorphic unitary frames; for a holomorphic line frame $e$ with $h(e,e)=e^{-\varphi}$, the unitary frame $e^{\varphi/2}e$ is holomorphic only when $\varphi$ is locally the real part of a holomorphic function in the required way. Thus the result solves the smooth metric-normalisation problem while leaving the holomorphic structure visible in non-unitary holomorphic frames. This tension between holomorphic frames and unitary frames is what produces the Chern connection in the next stage of the course. [example: A Nonconstant Metric In A Holomorphic Line Frame] Let $L\to U$ be a holomorphic line bundle over a coordinate domain, and let $e$ be a nowhere-vanishing holomorphic frame. Since every local section has the form $s=f e$ for a smooth complex-valued function $f$, a Hermitian metric is fixed in this frame by the positive smooth function \begin{align*} a=h(e,e). \end{align*} Indeed, if $s=f e$ and $t=g e$, then sesquilinearity of $h$ gives \begin{align*} h(s,t)=h(f e,g e)=f\overline{g}\,h(e,e)=f\overline{g}\,a. \end{align*} Because $a>0$, the function $\varphi=-\log a$ is smooth and real-valued, and $a=e^{-\varphi}$. Define \begin{align*} u=e^{\varphi/2}e. \end{align*} Then $u$ is a smooth local frame, and its squared norm is \begin{align*} h(u,u)=h(e^{\varphi/2}e,e^{\varphi/2}e)=e^{\varphi/2}\overline{e^{\varphi/2}}\,h(e,e)=e^\varphi e^{-\varphi}=1. \end{align*} Thus $e^{\varphi/2}e$ is unitary as a smooth frame. The original holomorphic frame records the holomorphic trivialisation, while the real weight $\varphi$ records exactly how the metric differs from the constant unit metric in that frame. [/example] ## The Fundamental Form And Volume Form A Hermitian metric on $T^{1,0}X$ gives a real two-form of type $(1,1)$. The guiding problem is to understand how this form encodes the metric and how its top exterior power gives the natural volume density. The convention $u^{1,0}=\frac{1}{2}(u-iJu)$ and $g(u,v)=2\operatorname{Re}h(u^{1,0},v^{1,0})$ fixes the constants in the coordinate formulas below. [quotetheorem:7026] [citeproof:7026] This local formula is the computational heart of Hermitian geometry. The holomorphic-coordinate hypothesis is necessary: in a general real coordinate system the decomposition into $dz_i$ and $d\bar z_j$ is not available, and a two-form need not display type $(1,1)$ in this matrix form. The formula does not assert that $\omega$ is closed; for example, a conformal change $h=e^f h_0$ on a complex torus gives $\omega=e^f\omega_0$, and $d\omega$ is nonzero when $df\wedge\omega_0$ is nonzero. It says that the metric matrix and the fundamental form contain the same information, written once as fibrewise linear algebra and once as a differential form. Once the metric is encoded by $\omega$, the natural volume density should be the top exterior power of this two-form, which motivates the Hermitian volume form. [definition: Hermitian Volume Form] Let $(X,h)$ be a Hermitian manifold of complex dimension $n$ with fundamental form $\omega$. The Hermitian volume form is \begin{align*} dV_h=\frac{\omega^n}{n!}. \end{align*} [/definition] In local holomorphic coordinates, this volume form is proportional to $\det(h_{i\bar j})$ times the standard real coordinate volume. A formula using only the diagonal entries would fail after a non-unitary change of holomorphic coordinates, because off-diagonal terms record the angles between coordinate vector fields. The determinant is exactly the determinant metric applied to the top exterior power of the holomorphic cotangent bundle. [quotetheorem:7027] [citeproof:7027] The determinant in the volume formula is the same determinant that appears in change-of-variables and in the metric on $\det T^{1,0}X$. Positivity of $H$ is necessary here: a degenerate semipositive matrix would have zero determinant at some point and would not define a Riemannian volume form. The theorem does not make the coordinate volume globally constant; the factor $\det(h_{i\bar j})$ changes from chart to chart exactly so that the top-degree form is globally defined. This is the first sign that curvature of determinant bundles will control canonical volume data. [example: Fubini-Study Metric On Projective Space] On $\mathbb{CP}^n$, use the affine chart $Z_0\ne 0$ with coordinates $z_j=Z_j/Z_0$, and put \begin{align*} \rho=1+|z_1|^2+\cdots+|z_n|^2=1+\sum_{\ell=1}^{n}z_\ell\overline{z_\ell}. \end{align*} The Fubini-Study fundamental form is defined on this chart by \begin{align*} \omega_{FS}=i\partial\bar\partial\log\rho. \end{align*} Since \begin{align*} \frac{\partial \rho}{\partial z_i}=\overline{z_i} \end{align*} and \begin{align*} \frac{\partial \rho}{\partial \overline{z_j}}=z_j, \end{align*} the first derivative of the potential is \begin{align*} \frac{\partial}{\partial z_i}\log\rho=\frac{1}{\rho}\frac{\partial\rho}{\partial z_i}=\frac{\overline{z_i}}{\rho}. \end{align*} Differentiating this coefficient with respect to $\overline{z_j}$ gives \begin{align*} \frac{\partial}{\partial \overline{z_j}}\left(\frac{\overline{z_i}}{\rho}\right)=\frac{\delta_{ij}\rho-\overline{z_i}z_j}{\rho^2}. \end{align*} Thus \begin{align*} \omega_{FS}=i\sum_{i,j=1}^{n}\frac{\delta_{ij}\rho-\overline{z_i}z_j}{\rho^2}\,dz_i\wedge d\overline{z_j}. \end{align*} Equivalently, the local Hermitian matrix is \begin{align*} h_{i\bar j}=\frac{\delta_{ij}\rho-\overline{z_i}z_j}{\rho^2}. \end{align*} To see positivity, let $\xi=(\xi_1,\dots,\xi_n)\in\mathbb C^n$. Then \begin{align*} \sum_{i,j=1}^{n}h_{i\bar j}\xi_i\overline{\xi_j}=\frac{1}{\rho}\sum_{i=1}^{n}|\xi_i|^2-\frac{1}{\rho^2}\left(\sum_{i=1}^{n}\overline{z_i}\xi_i\right)\left(\sum_{j=1}^{n}z_j\overline{\xi_j}\right). \end{align*} The second product is \begin{align*} \left(\sum_{i=1}^{n}\overline{z_i}\xi_i\right)\left(\sum_{j=1}^{n}z_j\overline{\xi_j}\right)=\left|\sum_{i=1}^{n}\overline{z_i}\xi_i\right|^2. \end{align*} By Cauchy-Schwarz for the standard Hermitian inner product on $\mathbb C^n$, \begin{align*} \left|\sum_{i=1}^{n}\overline{z_i}\xi_i\right|^2\le \left(\sum_{i=1}^{n}|z_i|^2\right)\left(\sum_{i=1}^{n}|\xi_i|^2\right). \end{align*} Therefore \begin{align*} \sum_{i,j=1}^{n}h_{i\bar j}\xi_i\overline{\xi_j}\ge \frac{1}{\rho}\sum_{i=1}^{n}|\xi_i|^2-\frac{\rho-1}{\rho^2}\sum_{i=1}^{n}|\xi_i|^2=\frac{1}{\rho^2}\sum_{i=1}^{n}|\xi_i|^2. \end{align*} This is positive for every nonzero $\xi$, so the displayed matrix is positive definite. On an overlap with the chart $Z_k\ne 0$, write $w_0=Z_0/Z_k=1/z_k$ and $w_j=Z_j/Z_k=z_j/z_k$ for $j\ne k$. Then \begin{align*} 1+\sum_{j=1}^{n}|z_j|^2=|z_k|^2\left(1+|w_0|^2+\sum_{j\ne k}|w_j|^2\right). \end{align*} Taking logarithms gives \begin{align*} \log\left(1+\sum_{j=1}^{n}|z_j|^2\right)=\log|z_k|^2+\log\left(1+|w_0|^2+\sum_{j\ne k}|w_j|^2\right). \end{align*} Since $z_k$ is holomorphic and nonvanishing on the overlap, locally \begin{align*} \log|z_k|^2=\log z_k+\log\overline{z_k}. \end{align*} The term $\log z_k$ is holomorphic, so $\bar\partial\log z_k=0$, and the term $\log\overline{z_k}$ is antiholomorphic, so $\partial\log\overline{z_k}=0$. Hence \begin{align*} \partial\bar\partial\log|z_k|^2=0. \end{align*} Thus the local formulas define the same global $(1,1)$-form on projective space. The Fubini-Study metric is therefore the Hermitian metric whose local coefficients are obtained from the projective potential $\log(1+|z|^2)$. [/example] The Fubini-Study metric is the standard compact positive model, while the complex torus is the flat model. Together they show how Hermitian metrics can reflect very different global geometries even though their local definitions are the same. [remark: Hermitian Versus Kahler] Every Kahler metric is Hermitian, but a Hermitian metric need not be Kahler. The extra Kahler condition is $d\omega=0$, which will be treated later. For now, the important point is that Hermitian metrics exist on every complex manifold admitting smooth partitions of unity, while Kahler metrics impose additional global and differential constraints. [/remark] Connections provide the language for transporting sections, but on a complex manifold they must interact correctly with the holomorphic splitting. The Chern connection is singled out in the next chapter because it is the unique connection that matches both the holomorphic structure and the Hermitian metric. # 7. Connections on Complex Vector Bundles This chapter moves from holomorphic structures and forms to the differential-geometric device that measures how sections of a complex vector bundle vary. The central questions are how to differentiate sections without choosing global coordinates, how this differentiation is represented by matrices of one-forms in a frame, and how curvature records the failure of second covariant derivatives to commute. The complex setting adds two further themes: compatibility with Hermitian metrics and the splitting of every connection into $(1,0)$ and $(0,1)$ parts. ## Smooth Connections and Curvature A vector bundle has no canonical way to compare vectors in different fibres. The first problem is therefore to define a derivative of a section which is local, first-order, and compatible with multiplication by functions in exactly the way ordinary differentiation is. [definition: Smooth Connection] Let $E \to X$ be a smooth complex vector bundle over a smooth manifold $X$. A smooth connection on $E$ is a $\mathbb C$-[linear map](/page/Linear%20Map) \begin{align*} \nabla: \Gamma(E) \longrightarrow \Omega^1(X;E) \end{align*} such that, for every $f \in C^\infty(X;\mathbb C)$ and every $s \in \Gamma(E)$, \begin{align*} \nabla(fs) = df \otimes s + f\nabla s. \end{align*} [/definition] The Leibniz rule is the whole point of the definition: it says that the connection differentiates the scalar coefficient and the section separately. Once a vector field $V \in \mathfrak X(X)$ is inserted into the one-form part, we write $\nabla_V s \in \Gamma(E)$ for the covariant derivative of $s$ along $V$. [example: Product Bundle With A Matrix One-Form] Let $E=X\times \mathbb C^r$, let $e_1,\dots,e_r$ be the standard constant frame, and write a section as a column of smooth functions $s=(s_1,\dots,s_r)^\top$. Given $A\in \Omega^1(X;\operatorname{End}(\mathbb C^r))$, define \begin{align*} \nabla s = ds+As. \end{align*} For $f\in C^\infty(X;\mathbb C)$, \begin{align*} \nabla(fs)=d(fs)+A(fs)=df\otimes s+f\,ds+f\,As=df\otimes s+f\nabla s. \end{align*} Thus $s\mapsto ds+As$ satisfies the Leibniz rule and is a smooth connection. Conversely, let $\nabla$ be any smooth connection on $X\times \mathbb C^r$. Since each $\nabla e_j$ is an $E$-valued one-form, there are unique scalar one-forms $A_{ij}\in\Omega^1(X;\mathbb C)$ such that \begin{align*} \nabla e_j=\sum_{i=1}^r A_{ij}\otimes e_i. \end{align*} For $s=\sum_{j=1}^r s_j e_j$, the Leibniz rule gives \begin{align*} \nabla s=\sum_{j=1}^r \nabla(s_j e_j)=\sum_{j=1}^r ds_j\otimes e_j+\sum_{j=1}^r s_j\nabla e_j. \end{align*} Substituting the displayed formula for $\nabla e_j$ gives \begin{align*} \nabla s=\sum_{i=1}^r ds_i\otimes e_i+\sum_{i=1}^r\left(\sum_{j=1}^r A_{ij}s_j\right)\otimes e_i. \end{align*} Therefore the coefficient of $e_i$ in $\nabla s$ is $ds_i+\sum_j A_{ij}s_j$, which is exactly the $i$th entry of $ds+As$. So every connection on the product bundle is represented in the standard frame by a matrix-valued one-form $A$. [/example] The example is local rather than special: every vector bundle becomes a product bundle after choosing a local frame. To compare computations made in different product presentations, we need a precise local object attached to a frame. [definition: Connection One-Form] Let $E \to X$ be a smooth complex vector bundle of rank $r$, and let $e = (e_1,\dots,e_r)$ be a local frame over an open set $U \subset X$. The connection one-form of $\nabla$ in the frame $e$ is the matrix $A = (A_{ij}) \in \Omega^1(U;\mathfrak{gl}(r,\mathbb C))$ determined by \begin{align*} \nabla e_j = \sum_{i=1}^r A_{ij} \otimes e_i. \end{align*} [/definition] The connection one-form turns $\nabla$ into matrix calculus on $U$, but it depends on the chosen frame. The next question is to determine the exact transformation law under $e' = eg$, because that law tells us which later expressions are intrinsic and which are only gauge-dependent. [quotetheorem:6235] [citeproof:6235] The extra term $g^{-1}dg$ is the inhomogeneous part of the transformation law, and it is exactly where the hypotheses matter. The new frame must be related to the old one by a smooth map $g:U\to GL(r,\mathbb C)$: if $g$ is not invertible, then $e'=eg$ is not a frame, so coefficient vectors cannot be converted back by $g^{-1}$; if $g$ is not smooth, the term $dg$ is not a smooth one-form and does not define a smooth connection matrix. The theorem also does not say that $A$ itself is a global form on $X$; it says that the local representatives of one fixed connection are glued by this affine transformation law. For instance, even the zero connection matrix in one frame need not remain zero after a nonconstant change of frame, because $A'=g^{-1}dg$. This failure of tensorial transformation is the reason curvature is introduced next. Since a connection form contains frame artefacts, the next problem is to build from $\nabla$ a tensorial object that captures intrinsic information about the bundle. [definition: Curvature] Let $\nabla$ be a smooth connection on a complex vector bundle $E \to X$. For each $k \ge 0$, extend $\nabla$ to a map \begin{align*} \nabla: \Omega^k(X;E) \longrightarrow \Omega^{k+1}(X;E) \end{align*} by \begin{align*} \nabla(\alpha \otimes s) = d\alpha \otimes s + (-1)^k \alpha \wedge \nabla s, \qquad \alpha \in \Omega^k(X;\mathbb C),\ s \in \Gamma(E). \end{align*} The curvature of $\nabla$ is the operator \begin{align*} F_\nabla = \nabla\circ\nabla: \Gamma(E) \longrightarrow \Omega^2(X;E). \end{align*} [/definition] Although the connection has a first-order part, its square is tensorial: $F_\nabla(fs)=fF_\nabla(s)$. This means $F_\nabla$ may be viewed as a two-form with values in $\operatorname{End}(E)$, and it remains to compute that two-form from the local matrix $A$. [quotetheorem:1540] [citeproof:1540] The term $A\wedge A$ is where noncommutativity enters: for line bundles it vanishes as a matrix commutator issue, while for higher rank bundles it usually contributes. The formula is local because it uses a chosen frame; changing the frame changes the matrix $A$, so $dA+A\wedge A$ should not be read as an absolute two-form with fixed matrix entries. Nor does the existence of such a formula imply that $E$ is globally a product bundle, since every vector bundle admits local frames even when no global frame exists. The necessity of the quadratic term is already visible for rank two: if two matrix coefficients fail to commute, replacing $F_A$ by $dA$ would give an expression with the wrong gauge behaviour. Before studying the general gauge behaviour of curvature, it helps to see the rank-one case where the formula reduces to an ordinary exterior derivative. [example: Curvature Of A Line Bundle Connection] Let $L \to X$ be a complex line bundle, let $e$ be a local nonvanishing section over $U$, and write \begin{align*} \nabla e=A\otimes e \end{align*} with $A\in \Omega^1(U;\mathbb C)$. For a local section $s=fe$, the Leibniz rule gives \begin{align*} \nabla(fe)=df\otimes e+fA\otimes e=(df+fA)\otimes e. \end{align*} Applying the extended connection to the $L$-valued one-form $(df+fA)\otimes e$ gives \begin{align*} \nabla\circ\nabla(fe)=d(df+fA)\otimes e-(df+fA)\wedge A\otimes e. \end{align*} Now $d(df)=0$ and $d(fA)=df\wedge A+f\,dA$, so \begin{align*} d(df+fA)=df\wedge A+f\,dA. \end{align*} Also \begin{align*} (df+fA)\wedge A=df\wedge A+f\,A\wedge A. \end{align*} Since $A$ is a scalar one-form, $A\wedge A=-A\wedge A$, hence $2A\wedge A=0$ and $A\wedge A=0$ over $\mathbb C$. Therefore \begin{align*} \nabla\circ\nabla(fe)=f\,dA\otimes e. \end{align*} Thus the curvature is represented in the frame $e$ by the scalar two-form $dA$. If $e'=eg$ for a smooth function $g:U\to \mathbb C^\times$, then \begin{align*} \nabla e'=\nabla(ge)=dg\otimes e+gA\otimes e=(g^{-1}dg+A)\otimes e', \end{align*} so $A'=A+g^{-1}dg$. Taking exterior derivatives, \begin{align*} dA'=dA+d(g^{-1}dg). \end{align*} Since $d(g^{-1})=-g^{-2}dg$ and $d^2g=0$, \begin{align*} d(g^{-1}dg)=d(g^{-1})\wedge dg+g^{-1}d^2g=-g^{-2}dg\wedge dg=0. \end{align*} Hence $dA'=dA$, so in rank one the curvature two-form is independent of the chosen local frame. [/example] The line bundle computation suggests that curvature is more intrinsic than the connection form. In higher rank the correct invariant statement cannot be equality of matrices, so the question is exactly how the curvature matrix changes under a gauge transformation. [quotetheorem:3838] [citeproof:3838] The same computation can be understood without coordinates: curvature is an $\operatorname{End}(E)$-valued two-form, and changing a frame changes the matrix of an endomorphism by conjugation. The invertibility and smoothness of $g$ are again essential, because conjugation by $g^{-1}$ is meaningful only for an actual smooth frame change. The theorem does not make the matrix entries of $F_A$ individually invariant; only conjugation-invariant expressions, such as traces of products of curvature matrices, have a chance of descending directly to global forms. A concrete failure case is to choose a nonconstant gauge on a rank-two product bundle: $F_A$ and $F_{A'}$ usually have different entries, even though they represent the same intrinsic curvature. This tensorial transformation law is the first sign that curvature has global content beyond the connection matrix. The next structural identity says that curvature is not arbitrary; every curvature form satisfies a differential constraint. [quotetheorem:1541] [citeproof:1541] The Bianchi identity is the differential shadow of the definition $F_\nabla = \nabla\circ\nabla$, but the operator in the identity must be the connection-induced exterior derivative $d_\nabla$, not the ordinary exterior derivative applied entrywise without regard to gauge. If one writes $dF_A=0$ as a replacement, the statement is not stable under changing frames: the extra terms produced by differentiating $g^{-1}F_Ag$ are exactly cancelled by $A\wedge F_A-F_A\wedge A$. The identity also does not say that the curvature vanishes; it is a constraint on how nonzero curvature may vary. Later, on Hermitian and Kähler manifolds, this is one of the inputs behind Chern-Weil theory, where traces of curvature expressions become closed forms, and behind identities for Chern connections. ## Hermitian Metrics and Unitary Connections A complex vector bundle in Hermitian geometry comes with a fibrewise Hermitian inner product. The next problem is to choose connections that differentiate this inner product compatibly, so that parallel transport preserves lengths and angles. [definition: Hermitian Metric On A Complex Vector Bundle] Let $E \to X$ be a smooth complex vector bundle. A Hermitian metric on $E$ is a smooth map \begin{align*} h: E \times_X E \longrightarrow \mathbb C \end{align*} such that, for each $p \in X$, the restriction $h_p:E_p\times E_p\to \mathbb C$ is a Hermitian inner product, linear in the first argument and conjugate-linear in the second. [/definition] The metric produces smooth functions $h(s,t)$ from pairs of smooth sections, so differentiating sections should interact with differentiating these functions. This motivates the compatibility condition: the connection should satisfy the product rule for the Hermitian pairing, not merely for scalar multiplication. [definition: Metric-Compatible Connection] Let $(E,h) \to X$ be a Hermitian vector bundle. A smooth connection $\nabla$ on $E$ is metric-compatible, or unitary, if for every vector field $V \in \mathfrak X(X)$ and every $s,t \in \Gamma(E)$, \begin{align*} V(h(s,t)) = h(\nabla_Vs,t) + h(s,\nabla_Vt). \end{align*} [/definition] This condition is the bundle version of the product rule for an inner product. To use it in computations we need its local matrix form, especially in frames adapted to the Hermitian metric. [quotetheorem:7028] [citeproof:7028] Thus a unitary connection has local connection form valued in the Lie algebra $\mathfrak u(r)$ in a unitary frame. The phrase "in a unitary frame" is essential: in a non-unitary frame the metric is represented by a positive Hermitian matrix $H$, and compatibility is expressed by \begin{align*} dH = A^\top H + H\overline{A}, \end{align*} where $\overline{A}$ denotes entrywise complex conjugation. This replaces $A^*+A=0$ outside unitary frames. The theorem therefore does not say that the connection matrix of a unitary connection is skew-Hermitian in every frame; it says that the metric can be used to choose frames where the condition has this simple form. This is the infinitesimal reason why its parallel transport maps are unitary transformations. [example: Flat Unitary Connection With Holonomy] Let $E=S^1\times \mathbb C$ with its standard frame $e$, and let $h$ be the standard Hermitian metric. Write a section as $s=ue$ for a smooth complex-valued function $u$. For a real constant $a$, define \begin{align*} \nabla(ue)=(du+ia\,u\,d\theta)\otimes e. \end{align*} The connection one-form is $A=ia\,d\theta$. Since $a$ is real and $d\theta$ is real-valued, \begin{align*} A^*=\overline{ia\,d\theta}=-ia\,d\theta=-A. \end{align*} Thus $A^*+A=0$, so the connection is unitary in the standard unitary frame. Its curvature in this frame is \begin{align*} F_A=dA+A\wedge A. \end{align*} The first term is \begin{align*} dA=d(ia\,d\theta)=ia\,d(d\theta)=0, \end{align*} because $a$ is constant and $d^2\theta=0$ locally. The second term is \begin{align*} A\wedge A=(ia\,d\theta)\wedge(ia\,d\theta)=(ia)^2\,d\theta\wedge d\theta=-a^2\,d\theta\wedge d\theta=0, \end{align*} because a one-form wedges with itself to zero. Hence $F_A=0$. Now parametrize the circle by $\theta=t$ for $0\le t\le 2\pi$. A section along this loop has the form $u(t)e$, and parallel transport means \begin{align*} 0=\nabla_{\partial_t}(u(t)e)=\left(\frac{du}{dt}+ia\,u(t)\right)e. \end{align*} Therefore $u$ satisfies \begin{align*} \frac{du}{dt}+ia\,u=0. \end{align*} Multiplying by $e^{iat}$ gives \begin{align*} \frac{d}{dt}\left(e^{iat}u(t)\right)=ia\,e^{iat}u(t)+e^{iat}\frac{du}{dt}=e^{iat}\left(\frac{du}{dt}+ia\,u(t)\right)=0. \end{align*} So $e^{iat}u(t)$ is constant, and \begin{align*} u(2\pi)=e^{-2\pi ia}u(0). \end{align*} Thus the connection is flat, but its holonomy around $S^1$ is multiplication by $e^{-2\pi ia}$, which is nontrivial whenever $a\notin \mathbb Z$. [/example] The example separates curvature from global holonomy. It also raises a basic existence question: after fixing a Hermitian metric, do compatible connections always exist, and how much freedom remains in choosing them? [quotetheorem:7029] This theorem says there is no uniqueness for metric-compatible connections alone. The skew-Hermitian condition on $B$ is necessary because the difference between two compatible product rules must contribute zero to the derivative of $h(s,t)$; if $B$ had a Hermitian part, adding it would change lengths under parallel transport. The affine description is also a limitation: it classifies compatible connections after one base connection has been chosen, rather than producing a canonical one from the metric alone. On a Hermitian line bundle, for example, the freedom is addition of an imaginary-valued one-form, so there are typically many unitary connections even on the same metric bundle. In Chapter 8, adding compatibility with a holomorphic structure will single out the Chern connection by fixing the $(0,1)$ part as well. ## Type Decomposition Of A Connection On a complex manifold, complex-valued one-forms split into $(1,0)$ and $(0,1)$ parts. The problem is to apply this splitting to the one-form component of a connection and understand what information is carried by each type. [definition: Type Decomposition Of A Connection] Let $X$ be a complex manifold and let $E \to X$ be a smooth complex vector bundle with connection $\nabla$. The type decomposition of $\nabla$ is \begin{align*} \nabla = \nabla^{1,0} + \nabla^{0,1}, \end{align*} where \begin{align*} \nabla^{1,0}: \Gamma(E) \to \Omega^{1,0}(X;E), \qquad \nabla^{0,1}: \Gamma(E) \to \Omega^{0,1}(X;E) \end{align*} are obtained by projecting the one-form part of $\nabla s$ onto types $(1,0)$ and $(0,1)$. [/definition] The two pieces are not symmetric in complex geometry: the $(0,1)$ part has the same direction as the Dolbeault operator, while the $(1,0)$ part differentiates in holomorphic directions. To make this precise, we must check that each component obeys the expected Leibniz rule with $\partial f$ and $\bar\partial f$. [quotetheorem:7030] [citeproof:7030] Thus $\nabla^{0,1}$ is the candidate for a holomorphic structure on $E$, but this statement depends on $X$ being a complex manifold so that complex one-forms have a canonical splitting into $(1,0)$ and $(0,1)$ parts. On a merely smooth manifold there is no intrinsic operator $\bar\partial f$ and hence no corresponding projection of $\nabla$ with this Leibniz rule. The theorem also does not say that either component is an ordinary smooth connection, since each component differentiates scalar functions only in its own type. The next issue is integrability: a Dolbeault-type operator defines holomorphic sections only when its square vanishes, so we need to locate that square inside the curvature of the original connection. [quotetheorem:7031] [citeproof:7031] The theorem explains why holomorphic vector bundles are usually encoded by operators of type $(0,1)$ whose square vanishes. The vanishing condition is a real obstruction, not a formal consequence of the Leibniz rule: a general operator $\bar\partial_E=\bar\partial+A^{0,1}$ may have $\bar\partial A^{0,1}+A^{0,1}\wedge A^{0,1}\ne 0$, and then local holomorphic frames cannot be consistently solved for. The result also does not determine the full curvature from the Dolbeault part, because the $(2,0)$ and $(1,1)$ components contain additional information from $\nabla^{1,0}$. In these notes, the connection viewpoint keeps the full curvature visible while isolating the component relevant to holomorphicity. [example: Type Decomposition In A Local Frame] Let $X$ have local holomorphic coordinates $(z_1,\dots,z_n)$, and let $E$ be the rank $r$ product bundle on this coordinate chart. Write a section as a column vector $s=(s_1,\dots,s_r)^\top$ and write the connection in the product frame as $\nabla s=ds+As$, where \begin{align*} A=\sum_{j=1}^n A_j\,dz_j+\sum_{j=1}^n A_{\bar j}\,d\bar z_j \end{align*} with each $A_j$ and $A_{\bar j}$ an $r\times r$ matrix of smooth functions. Since $ds=\partial s+\bar\partial s$, the connection expands as \begin{align*} \nabla s=\partial s+\bar\partial s+\sum_{j=1}^n A_j s\,dz_j+\sum_{j=1}^n A_{\bar j}s\,d\bar z_j. \end{align*} The terms involving $dz_j$ have type $(1,0)$, while the terms involving $d\bar z_j$ have type $(0,1)$. Therefore projection onto form types gives \begin{align*} \nabla^{1,0}s=\partial s+\sum_{j=1}^n A_js\,dz_j. \end{align*} Similarly, \begin{align*} \nabla^{0,1}s=\bar\partial s+\sum_{j=1}^n A_{\bar j}s\,d\bar z_j. \end{align*} Thus, as local operators on column-vector-valued functions, \begin{align*} \nabla^{1,0}=\partial+\sum_{j=1}^n A_j\,dz_j. \end{align*} Also, \begin{align*} \nabla^{0,1}=\bar\partial+\sum_{j=1}^n A_{\bar j}\,d\bar z_j. \end{align*} For the $(0,2)$ curvature term, only the $(0,1)$ part of $A$ can contribute. Set \begin{align*} A^{0,1}=\sum_{j=1}^n A_{\bar j}\,d\bar z_j. \end{align*} Then the $(0,2)$ part of $dA+A\wedge A$ is \begin{align*} F^{0,2}=\bar\partial A^{0,1}+A^{0,1}\wedge A^{0,1}. \end{align*} Expanding the first term gives \begin{align*} \bar\partial A^{0,1}=\sum_{k=1}^n\sum_{j=1}^n \frac{\partial A_{\bar j}}{\partial \bar z_k}\,d\bar z_k\wedge d\bar z_j. \end{align*} Expanding the second term by matrix multiplication gives \begin{align*} A^{0,1}\wedge A^{0,1}=\sum_{k=1}^n\sum_{j=1}^n A_{\bar k}A_{\bar j}\,d\bar z_k\wedge d\bar z_j. \end{align*} So the type decomposition is obtained by sorting the differential-form basis into $dz_j$ and $d\bar z_j$ components, not by diagonalising the matrices $A_j$ and $A_{\bar j}$; the $(0,2)$ curvature term is built exactly from the matrices $A_{\bar j}$, their $\bar z$-derivatives, and their wedge-matrix products. [/example] For a Hermitian bundle, a unitary connection ties the two type components together by conjugate transpose. This relation is the local algebra behind the Chern connection: the $(0,1)$ part is prescribed by the holomorphic structure, and the Hermitian metric then determines the $(1,0)$ part. [remark: Unitary Type Components] If $\nabla$ is unitary and $A$ is written in a local unitary frame, then $A^*+A=0$. Decomposing by type gives $(A^{1,0})^* = -A^{0,1}$ and $(A^{0,1})^*=-A^{1,0}$. Thus a unitary connection is determined locally by either type component together with the Hermitian metric. [/remark] The chapter leaves us with three complementary viewpoints on the same object. A connection is a covariant derivative on sections, a collection of local matrix one-forms obeying a gauge law, and a type-decomposed operator adapted to the complex structure. Curvature is the invariant two-form tying these viewpoints together, and metric compatibility restricts the local matrices to the unitary Lie algebra in unitary frames. The Chern connection is the point where holomorphic data and metric data meet in a canonical way. From there the course turns to curvature tensors, using the Chern connection as the geometric object from which those tensors are computed and interpreted. # 8. The Chern Connection Chapters 4 through 7 introduced the separate ingredients: holomorphic vector bundles determine the operator $\bar{\partial}_E$, Hermitian metrics measure lengths in each fibre, and smooth connections differentiate sections. This chapter explains how these two pieces of data determine a distinguished connection. The resulting Chern connection is the basic bridge between complex analysis and Hermitian differential geometry: it remembers holomorphic sections through its $(0,1)$-part and measures metric variation through its $(1,0)$-part. ## The Compatibility Problem A connection on a complex vector bundle is a smooth differential operator, but a holomorphic vector bundle already carries a first-order operator in the antiholomorphic directions. The first question is whether a connection can be forced to respect that holomorphic operator while also preserving a given Hermitian metric. The answer is yes, and the uniqueness is what makes the construction useful. [definition: Hermitian Compatibility] Let $E \to X$ be a complex vector bundle with Hermitian metric $h$, so each fibre has a Hermitian pairing $h_x:E_x\times E_x\to\mathbb C$ depending smoothly on $x\in X$. A connection $D:A^0(E)\to A^1(E)$ on $E$ is compatible with $h$ if, for all smooth sections $s,t \in A^0(E)$, \begin{align*} d(h(s,t)) = h(Ds,t) + h(s,Dt). \end{align*} [/definition] The metric compatibility condition is the complex analogue of preserving an inner product under parallel transport. It does not yet remember the holomorphic structure, so we also need a way to say that the connection agrees with the given Dolbeault operator in antiholomorphic directions. [definition: Connection of Type Compatible with a Holomorphic Structure] Let $E \to X$ be a holomorphic vector bundle with Dolbeault operator $\bar{\partial}_E:A^0(E)\to A^{0,1}(E)$. A connection $D:A^0(E)\to A^1(E)$ on the underlying smooth complex vector bundle has prescribed $(0,1)$-part if \begin{align*} D^{0,1} = \bar{\partial}_E. \end{align*} [/definition] This condition says that $D$ differentiates sections in antiholomorphic directions exactly as the holomorphic structure dictates. If this condition is omitted, a Hermitian bundle has many metric connections: on the product line bundle with constant metric, $D=d+i\alpha$ is metric-compatible for every real one-form $\alpha$. If metric compatibility is omitted, fixing $D^{0,1}=\bar{\partial}_E$ still leaves freedom to add arbitrary $(1,0)$-forms with values in $\operatorname{End}E$. The central problem is therefore whether imposing both conditions removes exactly these two freedoms and determines a single natural connection. [quotetheorem:3837] [citeproof:3837] The holomorphic hypothesis is essential: without an integrable Dolbeault operator there is no canonical condition $D^{0,1}=\bar{\partial}_E$ to impose. Positivity of $h$ is also part of the mechanism, since the local formula uses the inverse matrix $H^{-1}$; a degenerate fibre form would not determine the $(1,0)$-part. The theorem does not say that the connection matrix vanishes in every holomorphic frame, only that the operator is globally determined. In a non-holomorphic frame the same connection generally has a nonzero $(0,1)$ matrix contribution, so the simple formula belongs specifically to holomorphic frames and leads into the local computation below. [example: Flat Metric on a Product Holomorphic Bundle] Let $E=X\times\mathbb C^r$ with standard holomorphic frame $e_1,\dots,e_r$, and write the metric matrix in this frame as $H=I_r$. Since every entry of $I_r$ is constant, each entry has zero $\partial$-derivative, so \begin{align*} \partial H=\partial I_r=0. \end{align*} By the *[Chern Connection Matrix in a Holomorphic Frame](/theorems/7056)*, the connection matrix is \begin{align*} A=H^{-1}\partial H=I_r\cdot 0=0. \end{align*} Thus, for a section $s=\sum_{i=1}^r f_i e_i$ represented by the column vector $f=(f_1,\dots,f_r)^{\top}$, \begin{align*} Ds=df+Af=df+0\cdot f=df. \end{align*} So in this orthonormal holomorphic frame the Chern connection is exactly the ordinary exterior derivative on each component. [/example] ## The Local Formula in a Holomorphic Frame The abstract characterization is coordinate-free, but actual geometry is visible in the local matrix. The next task is to translate variation of the Hermitian metric into the connection one-form. This is where holomorphic frames are special: the entire connection matrix has type $(1,0)$. [quotetheorem:7056] [citeproof:7056] The holomorphic-frame assumption is necessary for the clean formula: if the frame is merely smooth, its own $\bar{\partial}$-variation contributes extra $(0,1)$ terms to the connection matrix. The invertibility of $H$ is also used directly, and it comes from the Hermitian metric being positive definite on each fibre. The theorem does not make $A$ tensorial under change of frame; connection matrices transform with an inhomogeneous term, so only the operator $D$ is invariant. What the formula does provide is the practical rule that Chern connection computations begin by writing the metric matrix in a holomorphic frame. [example: Line Bundle Connection Form] Let $L\to X$ be a holomorphic line bundle with local holomorphic frame $e$, and write $h(e,e)=h_e$ with $h_e:U\to\mathbb R_{+}$ smooth. Since the rank is one, the metric matrix in this frame is the $1\times 1$ matrix $H=(h_e)$, so $H^{-1}=(h_e^{-1})$ and \begin{align*} \partial H=(\partial h_e). \end{align*} By the *Chern Connection Matrix in a Holomorphic Frame*, the connection matrix is \begin{align*} A=H^{-1}\partial H=h_e^{-1}\partial h_e. \end{align*} Because $h_e$ is positive, $\log h_e$ is smooth, and the chain rule gives \begin{align*} \partial\log h_e=h_e^{-1}\partial h_e. \end{align*} Thus $A=\partial\log h_e$. For a local section $fe$, represented by the scalar function $f$, the formula $D=d+A$ gives \begin{align*} D(fe)=\left(df+Af\right)e=\left(df+f\,\partial\log h_e\right)e. \end{align*} So the connection one-form is $\partial\log h_e$, which is why line bundle Chern connection computations reduce to differentiating the logarithm of the metric weight. [/example] Rank greater than one introduces matrix multiplication, so the order of the factors matters. Even when the metric matrix is diagonal, the connection can encode different weights in different summands. [example: Rank Two Diagonal Metric] Let $E=U\times\mathbb C^2$ in a holomorphic frame over a domain $U\subset\mathbb C$, and let \begin{align*} H=\operatorname{diag}(e^{\varphi},e^{\psi}) \end{align*} for smooth real-valued functions $\varphi,\psi:U\to\mathbb R$. Since $e^\varphi$ and $e^\psi$ are positive, $H$ is invertible and \begin{align*} H^{-1}=\operatorname{diag}(e^{-\varphi},e^{-\psi}). \end{align*} The chain rule gives \begin{align*} \partial(e^\varphi)=e^\varphi\,\partial\varphi. \end{align*} Likewise, \begin{align*} \partial(e^\psi)=e^\psi\,\partial\psi. \end{align*} Therefore \begin{align*} \partial H=\operatorname{diag}(e^\varphi\,\partial\varphi,e^\psi\,\partial\psi). \end{align*} Using the holomorphic-frame formula for the Chern connection matrix, \begin{align*} A=H^{-1}\partial H. \end{align*} Multiplying the diagonal matrices entry by entry gives \begin{align*} A=\operatorname{diag}(e^{-\varphi}e^\varphi\,\partial\varphi,e^{-\psi}e^\psi\,\partial\psi). \end{align*} Since $e^{-\varphi}e^\varphi=1$ and $e^{-\psi}e^\psi=1$, \begin{align*} A=\operatorname{diag}(\partial\varphi,\partial\psi). \end{align*} Thus the two summands do not mix: the Chern connection is the direct sum of the two line bundle connections determined by the metric weights $\varphi$ and $\psi$. [/example] ## Curvature and Type Connections are compared through their curvature, and in complex geometry the type decomposition of curvature is as important as the curvature itself. For an arbitrary connection on a complex vector bundle, the curvature can have $(2,0)$, $(1,1)$, and $(0,2)$ components. The Chern connection is constrained enough that only the middle type remains. [definition: Curvature of a Connection] Let $D:A^0(E)\to A^1(E)$ be a connection on a complex vector bundle $E\to X$, extended to $E$-valued forms by the graded Leibniz rule. Its curvature is the induced $C^\infty(X)$-linear operator $F_D:A^0(E)\to A^2(E)$ defined by \begin{align*} F_D=D^2, \end{align*} [/definition] The curvature is tensorial: it is $C^\infty(X)$-linear in the section being acted on, so it corresponds to an endomorphism-valued two-form $F_D\in A^2(X;\operatorname{End}E)$. For a general connection there is no reason for a single type to survive: on the product line bundle over $\mathbb C^2$, the connection $D=d+z_1\,dz_2$ has curvature $dz_1\wedge dz_2$ of type $(2,0)$, while $D=d+\bar z_1\,d\bar z_2$ has curvature $d\bar z_1\wedge d\bar z_2$ of type $(0,2)$. The question specific to the Chern connection is which pieces of this two-form survive after the holomorphic and metric compatibility conditions are imposed. [quotetheorem:7032] [citeproof:7032] The holomorphic structure is necessary for the vanishing of the $(0,2)$ term, since a nonintegrable choice of $(0,1)$ operator would have $(D^{0,1})^2\ne 0$. Metric compatibility is what removes the $(2,0)$ freedom; the earlier connection $d+z_1\,dz_2$ on the product line bundle shows that a general connection can have nonzero $(2,0)$ curvature. The theorem does not say the curvature is zero, nor that every $(1,1)$ endomorphism-valued form can occur from the fixed metric. It narrows the computation to the remaining middle type, so we now keep the same holomorphic frame and evaluate that matrix directly. [quotetheorem:7033] The use of a holomorphic frame is again essential: in a smooth non-holomorphic frame the displayed expression acquires extra terms from the frame change. The theorem is also local in its presentation, so it does not claim that $H^{-1}\partial H$ is a globally defined matrix on its own; it is the curvature operator that patches invariantly. The formula explains why higher-rank curvature is sensitive to order of multiplication, a feature absent for line bundles. Since many geometric applications in this course concern line bundles, we now isolate the scalar case where the logarithm packages the connection form into a second derivative formula. [quotetheorem:3840] [citeproof:3840] The positivity of the metric weight is necessary because $\log h_e$ is used; a vanishing or sign-changing weight would not define a Hermitian metric and would break the formula. The holomorphic-frame condition is also necessary: replacing $e$ by a non-holomorphic local frame changes the connection form by non-holomorphic terms rather than by the harmless logarithm of a holomorphic transition function. The theorem does not say that $\log h_e$ is global, only that its $\partial\bar{\partial}$ transforms correctly between holomorphic trivialisations. This is precisely the mechanism behind the projective examples, where local potentials define global curvature forms. ## Projective Examples Projective space raises a concrete sign question: which natural line bundle has positive curvature, the tautological bundle or its dual? The answer cannot be read from the transition functions alone; it comes from differentiating the metric weight in affine charts. These examples connect the Chern connection to the Fubini-Study form and prepare the later language of positive Hermitian line bundles. [example: Curvature of the Tautological Bundle] Let $\mathcal O(-1)\to\mathbb{CP}^n$ be the tautological line bundle with the metric induced from the standard Hermitian metric on $\mathbb C^{n+1}$. On the affine chart $Z_0\neq 0$, write $w_i=Z_i/Z_0$ and use the holomorphic frame $e(w)=(1,w_1,\dots,w_n)$. The squared norm of this frame is \begin{align*} h_e(w)=h(e(w),e(w))=1+|w_1|^2+\cdots+|w_n|^2. \end{align*} Set $\rho=1+|w|^2=1+\sum_{i=1}^n w_i\bar w_i$. Then \begin{align*} \partial\rho=\sum_{i=1}^n \bar w_i\,dw_i. \end{align*} Also, \begin{align*} \bar{\partial}\rho=\sum_{j=1}^n w_j\,d\bar w_j. \end{align*} By the chain rule, \begin{align*} \partial\log\rho=\rho^{-1}\partial\rho=\frac{\sum_{i=1}^n \bar w_i\,dw_i}{1+|w|^2}. \end{align*} Using the *Line Bundle Curvature Formula*, \begin{align*} F_{\mathcal O(-1)}=-\partial\bar{\partial}\log\rho. \end{align*} Equivalently, \begin{align*} F_{\mathcal O(-1)}=-\partial\bar{\partial}\log(1+|w|^2). \end{align*} Thus the tautological bundle has curvature equal to the negative of the local Fubini-Study curvature form, up to the conventional global factor such as $i$ or $2\pi i$ used in Chern-Weil normalisations. [/example] The dual bundle reverses the metric weight and hence reverses the curvature. This sign reversal is why $\mathcal O(1)$ is the prototype of a positive holomorphic line bundle. [example: Curvature of the Hyperplane Bundle] Let $\mathcal O(1)=\mathcal O(-1)^*$ carry the dual metric. On the affine chart $Z_0\ne 0$, keep $\rho=1+|w|^2=1+\sum_{i=1}^n w_i\bar w_i$. In the dual frame $e^*$, the metric weight is the reciprocal of the tautological weight: \begin{align*} h_{e^*}(w)=\rho^{-1}=(1+|w|^2)^{-1}. \end{align*} Since $\rho>0$, the logarithm is defined, and \begin{align*} \log h_{e^*}=\log(\rho^{-1})=-\log\rho. \end{align*} Using the *Line Bundle Curvature Formula*, \begin{align*} F_{\mathcal O(1)}=-\partial\bar{\partial}\log h_{e^*}. \end{align*} Substituting $\log h_{e^*}=-\log\rho$ gives \begin{align*} F_{\mathcal O(1)}=-\partial\bar{\partial}(-\log\rho)=\partial\bar{\partial}\log\rho. \end{align*} Therefore \begin{align*} F_{\mathcal O(1)}=\partial\bar{\partial}\log(1+|w|^2). \end{align*} Thus passing from the tautological bundle to its dual reverses the sign of the curvature, which is the local differential-geometric origin of the positivity of the hyperplane bundle, up to the global normalisation factor such as $i$ or $2\pi i$ chosen for curvature forms. [/example] The preceding examples also show why local formulas must be interpreted invariantly. The function $\log(1+|w|^2)$ changes from chart to chart by the logarithm of the squared norm of a nonvanishing holomorphic transition function, and applying $\partial\bar{\partial}$ removes that ambiguity. ## Matrix Curvature Computations Line bundles hide noncommutativity. For higher rank bundles the expression $\bar{\partial}(H^{-1}\partial H)$ involves both differentiating the inverse metric matrix and multiplying matrix-valued forms in the correct order. A useful way to read the formula is that curvature measures the failure of the metric matrix to be holomorphically flattened to second order. [example: Rank Two Metric with Off-Diagonal Term] Let $a=1+|z|^2=1+z\bar z$ and write \begin{align*} \Delta=a^2-z\bar z=(1+z\bar z)^2-z\bar z=1+z\bar z+z^2\bar z^2. \end{align*} For $|z|$ sufficiently small, $\Delta>0$ and $a>0$, so $H$ is positive definite. The inverse of a $2\times 2$ matrix gives \begin{align*} (H^{-1})_{11}=(H^{-1})_{22}=\frac{a}{\Delta}, \qquad (H^{-1})_{12}=\frac{-z}{\Delta}, \qquad (H^{-1})_{21}=\frac{-\bar z}{\Delta}. \end{align*} Since $\partial z=dz$, $\partial\bar z=0$, and $\partial(z\bar z)=\bar z\,dz$, the nonzero entries of $\partial H$ are \begin{align*} (\partial H)_{11}=\bar z\,dz, \qquad (\partial H)_{12}=dz, \qquad (\partial H)_{22}=\bar z\,dz. \end{align*} By the *Chern Connection Matrix in a Holomorphic Frame*, $A=H^{-1}\partial H$. Multiplying the entries in order gives \begin{align*} A_{11}=\frac{a}{\Delta}\bar z\,dz+\frac{-z}{\Delta}\cdot 0=\frac{a\bar z}{\Delta}\,dz. \end{align*} Also, \begin{align*} A_{12}=\frac{a}{\Delta}dz+\frac{-z}{\Delta}\bar z\,dz=\frac{a-z\bar z}{\Delta}\,dz=\frac{1}{\Delta}\,dz. \end{align*} The lower-left entry is \begin{align*} A_{21}=\frac{-\bar z}{\Delta}\bar z\,dz+\frac{a}{\Delta}\cdot 0=-\frac{\bar z^2}{\Delta}\,dz. \end{align*} Finally, \begin{align*} A_{22}=\frac{-\bar z}{\Delta}dz+\frac{a}{\Delta}\bar z\,dz=\frac{\bar z(a-1)}{\Delta}\,dz=\frac{z\bar z^2}{\Delta}\,dz. \end{align*} By the *[Local Curvature Formula for the Chern Connection](/theorems/7033)*, $F_D=\bar{\partial}A$. Since \begin{align*} \bar{\partial}\Delta=\bar{\partial}(1+z\bar z+z^2\bar z^2)=z\,d\bar z+2z^2\bar z\,d\bar z=z(1+2z\bar z)\,d\bar z, \end{align*} the quotient rule already shows where the extra inverse-metric terms enter. For example, \begin{align*} \bar{\partial}\left(\frac{a\bar z}{\Delta}\right)=\frac{(1+2z\bar z)\Delta-a\bar z\cdot z(1+2z\bar z)}{\Delta^2}\,d\bar z=\frac{1+2z\bar z}{\Delta^2}\,d\bar z. \end{align*} Similarly, \begin{align*} \bar{\partial}\left(\frac{1}{\Delta}\right)=-\frac{z(1+2z\bar z)}{\Delta^2}\,d\bar z. \end{align*} For the lower-left entry, \begin{align*} \bar{\partial}\left(-\frac{\bar z^2}{\Delta}\right)=\frac{-2\bar z\,\Delta+\bar z^2z(1+2z\bar z)}{\Delta^2}\,d\bar z=-\frac{\bar z(2+z\bar z)}{\Delta^2}\,d\bar z. \end{align*} For the lower-right entry, \begin{align*} \bar{\partial}\left(\frac{z\bar z^2}{\Delta}\right)=\frac{2z\bar z\,\Delta-z\bar z^2\cdot z(1+2z\bar z)}{\Delta^2}\,d\bar z=\frac{z\bar z(2+z\bar z)}{\Delta^2}\,d\bar z. \end{align*} Therefore the curvature entries are these coefficient forms wedged with $dz$, because each entry of $A$ has the form $B\,dz$. This computation shows explicitly that in rank two the curvature is not obtained by differentiating only $\partial H$; differentiating the factor $H^{-1}$ contributes the quotient-rule terms involving $\bar{\partial}\Delta$. [/example] The Chern connection is therefore both canonical and computational. It is canonical because the holomorphic structure and Hermitian metric characterize it without choices; it is computational because a holomorphic frame converts it into the matrix $H^{-1}\partial H$. The rest of Hermitian geometry repeatedly studies what the resulting $(1,1)$ curvature says about positivity, Chern classes, and special metrics. Curvature has now been introduced as the central invariant that links local formulas to global geometry. The next chapter converts those curvature forms into topological classes via Chern-Weil theory, showing that the same local data determine global characteristic classes. # 9. Curvature Tensors in Hermitian Geometry Chapters 4 through 8 introduced holomorphic vector bundles, Hermitian metrics, and the Chern connection as the connection adapted to both the holomorphic structure and the metric. We now specialise that language to curvature tensors on Hermitian manifolds. The guiding questions are how the curvature of $T^{1,0}X$ is written in local indices, how its trace becomes the Chern-Ricci form, and how torsion changes the identities familiar from Riemannian geometry. Throughout this chapter $X$ is a complex manifold of complex dimension $n$, $h$ is a Hermitian metric on $T^{1,0}X$, and in a holomorphic coordinate chart $(U,z_1,\dots,z_n)$ we write \begin{align*} h_{i\bar j} = h\left(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_j}\right), \qquad (h^{i\bar j}) = (h_{i\bar j})^{-1}. \end{align*} The associated real $(1,1)$-form is \begin{align*} \omega = i\sum_{i,j} h_{i\bar j}\,dz_i\wedge d\bar z_j. \end{align*} ## Curvature of the Holomorphic Tangent Bundle The Chern connection gives a canonical way to differentiate holomorphic tangent vectors, but curvature is a second-order obstruction: it measures the failure of successive covariant derivatives to commute. For a Hermitian metric this obstruction has type $(1,1)$ and is best understood as the curvature of the holomorphic vector bundle $T^{1,0}X$. [definition: Chern Curvature Tensor] Let $\nabla^C$ be the Chern connection on $(T^{1,0}X,h)$, viewed as a map \begin{align*} \nabla^C: A^0(X;T^{1,0}X) \to A^1(X;T^{1,0}X) \end{align*} and extended to $T^{1,0}X$-valued forms by the usual graded rule. Its curvature is the $A^0(X;T^{1,0}X)$-linear operator \begin{align*} R^C=(\nabla^C)^2: A^0(X;T^{1,0}X) \to A^2(X;T^{1,0}X), \end{align*} with type decomposition \begin{align*} R^C \in A^{1,1}(X;\operatorname{End}(T^{1,0}X)). \end{align*} In a holomorphic frame $\partial_i = \partial/\partial z_i$, write \begin{align*} R^C(\partial_i) = \sum_{j,k,l} R^j{}_{i k\bar l}\,dz_k\wedge d\bar z_l\otimes \partial_j. \end{align*} The fully lowered components are \begin{align*} R_{i\bar j k\bar l} = \sum_p h_{p\bar j}R^p{}_{i k\bar l}. \end{align*} [/definition] This notation separates two roles: $i,\bar j$ refer to the bundle directions, while $k,\bar l$ refer to the differential-form directions. On a Kähler manifold these pairs acquire extra symmetries, but in the Hermitian setting the distinction matters. [example: Curvature Matrix for a Hermitian Line Bundle] Let $L\to X$ have local holomorphic frame $e$ and metric weight $h(e,e)=|e|_h^2=e^{-\varphi}$. Write the Chern connection in this frame as $\nabla e=\theta\otimes e$, where $\theta$ is a $(1,0)$-form. Metric compatibility gives \begin{align*} \partial h(e,e)=h(\nabla e,e)=\theta\,h(e,e). \end{align*} Since $h(e,e)=e^{-\varphi}$, we have \begin{align*} \partial h(e,e)=\partial(e^{-\varphi})=-e^{-\varphi}\partial\varphi. \end{align*} Substituting into $\partial h(e,e)=\theta h(e,e)$ and dividing by $e^{-\varphi}$ gives \begin{align*} \theta=-\partial\varphi. \end{align*} The curvature in this line frame is $\Theta=d\theta$, because $\theta\wedge\theta=0$ for scalar-valued one-forms. Therefore \begin{align*} \Theta=d(-\partial\varphi)=-\partial^2\varphi-\bar\partial\partial\varphi. \end{align*} Since $\partial^2=0$, this becomes \begin{align*} \Theta=-\bar\partial\partial\varphi. \end{align*} Using $d^2=0$ in bidegree $(1,1)$ gives $\partial\bar\partial\varphi+\bar\partial\partial\varphi=0$, so \begin{align*} \Theta=\partial\bar\partial\varphi. \end{align*} Thus, with the convention $\theta=h^{-1}\partial h$, the scalar curvature form of the line bundle is represented locally by $\partial\bar\partial\varphi$; reversing the curvature-matrix sign convention reverses this displayed sign. [/example] The tangent bundle case is the same calculation with a matrix-valued metric. The inverse metric appears because differentiating a frame requires converting covariant components back into vector components. [quotetheorem:7034] [citeproof:7034] The theorem is the basic computational tool for the rest of the chapter, but each hypothesis is doing work. The holomorphic frame is needed because the formula uses the equality $(\nabla^C)^{0,1}=\bar\partial$; in a non-holomorphic frame extra gauge terms enter the connection matrix. The Hermitian metric is needed because $h^{-1}\partial h$ is the metric-compatibility solution for the Chern connection; for a general connection on $T^{1,0}X$ there is no reason for its connection matrix to be determined by $h$ in this way. For example, even on the trivial bundle over a coordinate ball, replacing the Chern connection by $d+A$ with an arbitrary $(1,0)$ matrix-valued form $A$ gives curvature $\bar\partial A+A\wedge A$, not $\bar\partial(h^{-1}\partial h)$. The theorem does not say that the curvature has the full Riemannian or Kähler index symmetries. It computes the curvature of the Chern connection in a holomorphic frame; whether first derivatives of $h$ can be removed, or whether the two unbarred indices can be interchanged, is a further torsion question. [remark: Index Symmetries in the Hermitian Case] For a general Hermitian metric the components $R_{i\bar j k\bar l}$ are Hermitian in the endomorphism indices but need not satisfy the Kähler symmetries $R_{i\bar j k\bar l}=R_{k\bar j i\bar l}$ or $R_{i\bar j k\bar l}=R_{i\bar l k\bar j}$. The failure of these identities is controlled by torsion and its derivatives. [/remark] This warning is important because several traces of $R^C$ are available. The trace over the endomorphism indices gives the Chern-Ricci form, while traces involving the form indices lead to different scalar quantities in non-Kähler Hermitian geometry. ## The Chern-Ricci Form and the Canonical Bundle The Ricci tensor in Riemannian geometry is a trace of curvature. In complex geometry there is a more intrinsic route: take the determinant of the holomorphic tangent bundle, or equivalently the dual canonical bundle, and compute the induced curvature there. [definition: Chern-Ricci Form] The Chern-Ricci form of a Hermitian metric $h$ is the real $(1,1)$-form \begin{align*} \operatorname{Ric}^C(h) = i\sum_{k,l}\left(\sum_i R^i{}_{i k\bar l}\right)dz_k\wedge d\bar z_l. \end{align*} [/definition] This definition records the trace of curvature on $T^{1,0}X$, so it packages many tensor components into a single differential form. To understand why this trace is geometrically canonical rather than only a coordinate contraction, we need the line bundle whose local frame is the wedge of all holomorphic cotangent directions. [definition: Canonical Bundle] The canonical bundle of $X$ is the holomorphic line bundle \begin{align*} K_X = \Lambda^n (T^{1,0}X)^*. \end{align*} Its dual is \begin{align*} K_X^{-1}=\Lambda^n T^{1,0}X. \end{align*} [/definition] The canonical bundle packages top-degree holomorphic cotangent data, but curvature computations are rarely performed directly from that abstract exterior power. In a chart, a Hermitian metric on $T^{1,0}X$ appears as a matrix $(h_{i\bar j})$, and it is not yet clear which scalar expression in that matrix represents the trace of the Chern curvature. The obstruction is coordinate dependence: a useful formula must survive holomorphic changes of coordinates and still define a global form. The next result supplies exactly this determinant formula for the Chern-Ricci form. [quotetheorem:7035] [citeproof:7035] This formula is local, but the resulting form is global because changes of holomorphic coordinates alter $\log\det(h_{i\bar j})$ by the logarithm of the squared modulus of a nonvanishing holomorphic Jacobian determinant, whose $\partial\bar\partial$ vanishes. The Hermitian hypothesis is needed so that the matrix $(h_{i\bar j})$ is positive definite and defines a determinant metric; a non-Hermitian complex bilinear matrix need not give a real positive local weight and the logarithm in the formula is not the curvature of a Hermitian line bundle. The holomorphic coordinate hypothesis is also essential: under an arbitrary smooth change of frame, the determinant picks up non-holomorphic transition factors whose $\partial\bar\partial$ term need not vanish. A useful counterexample to overreading the theorem is a Hermitian metric with $\det(h_{i\bar j})$ constant but nonzero torsion or nonzero full Chern curvature. In complex dimension at least $2$, a matrix-valued Hermitian metric can vary inside $SL(n,\mathbb C)$-type directions while keeping determinant fixed; the Chern-Ricci form then vanishes although individual curvature components need not. Thus the formula computes only the trace curvature, not the whole Chern curvature tensor. The next result states the bundle-theoretic principle behind the coordinate invariance: determinant curvature is trace curvature for every Hermitian holomorphic vector bundle. [quotetheorem:7036] [citeproof:7036] Applying this theorem to $E=T^{1,0}X$ identifies the Chern-Ricci form with $i$ times the curvature of $K_X^{-1}$, and with the negative of $i$ times the curvature of $K_X$. The holomorphic bundle hypothesis is needed because the determinant line must inherit a Chern connection from holomorphic transition functions; for a merely smooth complex vector bundle with an arbitrary connection, determinant curvature is still the trace of curvature, but it is not a $(1,1)$ Chern curvature form attached to a holomorphic structure. The Hermitian hypothesis is what selects the Chern connection rather than an arbitrary compatible-looking connection. The theorem also has a sharp limitation: trace curvature forgets trace-free curvature. For instance, an $SU(r)$-connection has curvature with values in trace-free endomorphisms, so the determinant connection is flat even when the original connection has nonzero curvature. This is why the Chern-Ricci form detects the curvature of $K_X^{-1}$ and the first Chern class, but not the full curvature of $T^{1,0}X$. In cohomological terms, $[\operatorname{Ric}^C(h)/(2\pi)]$ represents $c_1(X)=c_1(K_X^{-1})$, so the local determinant calculation becomes a global bridge from curvature to holomorphic line bundles. [example: Fubini-Study Chern-Ricci Form] On the affine chart $[1:z_1:\cdots:z_n]$, set $\rho=1+|z|^2=1+\sum_{a=1}^n z_a\bar z_a$. The Fubini-Study potential is $\log\rho$, so the metric coefficients are \begin{align*} h_{i\bar j}=\partial_i\partial_{\bar j}\log\rho. \end{align*} First, \begin{align*} \partial_i\log\rho=\frac{\bar z_i}{\rho}. \end{align*} Differentiating this with respect to $\bar z_j$ gives \begin{align*} h_{i\bar j}=\partial_{\bar j}\left(\frac{\bar z_i}{\rho}\right)=\frac{\delta_{ij}\rho-\bar z_i z_j}{\rho^2}=\frac{\delta_{ij}}{\rho}-\frac{\bar z_i z_j}{\rho^2}. \end{align*} Thus, as a matrix, \begin{align*} (h_{i\bar j})=\rho^{-1}I-\rho^{-2}\bar z z^\top=\rho^{-1}\left(I-\rho^{-1}\bar z z^\top\right). \end{align*} Using the rank-one determinant identity $\det(I+uv^\top)=1+v^\topu$ with $u=-\rho^{-1}\bar z$ and $v=z$, we get \begin{align*} \det(h_{i\bar j})=\rho^{-n}\det\left(I-\rho^{-1}\bar z z^\top\right)=\rho^{-n}\left(1-\rho^{-1}\sum_{a=1}^n z_a\bar z_a\right). \end{align*} Since $\sum_{a=1}^n z_a\bar z_a=|z|^2$ and $\rho=1+|z|^2$, \begin{align*} 1-\rho^{-1}|z|^2=\frac{\rho-|z|^2}{\rho}=\frac{1}{\rho}. \end{align*} Therefore \begin{align*} \det(h_{i\bar j})=\rho^{-n}\rho^{-1}=\rho^{-(n+1)}=(1+|z|^2)^{-(n+1)}. \end{align*} By the *Chern-Ricci Formula*, \begin{align*} \operatorname{Ric}^C(\omega_{FS})=-i\partial\bar\partial\log\det(h_{i\bar j}). \end{align*} Substituting the determinant gives \begin{align*} \operatorname{Ric}^C(\omega_{FS})=-i\partial\bar\partial\left(-(n+1)\log\rho\right)=(n+1)i\partial\bar\partial\log\rho. \end{align*} Since $\omega_{FS}=i\partial\bar\partial\log\rho$, this becomes \begin{align*} \operatorname{Ric}^C(\omega_{FS})=(n+1)\omega_{FS}. \end{align*} Thus the Fubini-Study metric has positive Chern-Ricci curvature, and its Ricci form is exactly $(n+1)$ times the Fubini-Study form. [/example] The same formula also detects flatness in examples where the determinant metric is constant. Complex tori are the model case. [example: Flat Complex Torus] Let $X=\mathbb C^n/\Lambda$ carry the Hermitian metric induced by a constant positive Hermitian matrix $H=(H_{i\bar j})$. In the standard coordinates on $\mathbb C^n$, the lifted metric coefficients are \begin{align*} h_{i\bar j}=H_{i\bar j}. \end{align*} Each $H_{i\bar j}$ is constant, so $D=\det(H)>0$ is also constant. Hence \begin{align*} \partial\log\det(H)=\partial\log D=0. \end{align*} Applying $\bar\partial$ gives \begin{align*} \partial\bar\partial\log\det(H)=\partial(0)=0. \end{align*} By the *Chern-Ricci Formula*, \begin{align*} \operatorname{Ric}^C(H)=-i\partial\bar\partial\log\det(H)=-i\cdot 0=0. \end{align*} The full Chern curvature also vanishes. Since every coefficient $H_{i\bar j}$ is constant, \begin{align*} \partial H=0. \end{align*} Therefore the Chern connection matrix is \begin{align*} H^{-1}\partial H=H^{-1}\cdot 0=0. \end{align*} Using the local curvature formula $\Theta^C=\bar\partial(H^{-1}\partial H)$, we get \begin{align*} \Theta^C=\bar\partial(0)=0. \end{align*} Thus the flat complex torus has zero Chern-Ricci form and zero full Chern curvature; this is stronger than Ricci-flatness because every curvature component vanishes. [/example] The contrast between projective space and a torus motivates the cohomological role of $\operatorname{Ric}^C$. Locally it is built from a determinant, but globally it records the curvature of $K_X^{-1}$ and therefore the first Chern class. ## Torsion and Bianchi Identities for Hermitian Connections Curvature identities in Riemannian geometry are often stated without torsion because the Levi-Civita connection has none. The Chern connection on a Hermitian manifold is adapted to the complex structure, but it generally has torsion, so the curvature identities acquire additional terms. [definition: Chern Torsion] Let $\nabla^C$ be the Chern connection on $T^{1,0}X$, extended complex-linearly to complex vector fields of type $(1,0)$. Its torsion is the $C^\infty(X;\mathbb C)$-bilinear map \begin{align*} T^C: A^0(X;T^{1,0}X)\times A^0(X;T^{1,0}X)\to A^0(X;T^{1,0}X) \end{align*} defined by \begin{align*} T^C(u,v)=\nabla^C_u v-\nabla^C_v u-[u,v]^{1,0} \end{align*} for $(1,0)$ vector fields $u,v$. In holomorphic coordinates, its $(2,0)$ components are \begin{align*} T^p{}_{ij}=\Gamma^p_{ij}-\Gamma^p_{ji}, \qquad \Gamma^p_{ij}=\sum_q h^{p\bar q}\partial_i h_{j\bar q}. \end{align*} [/definition] The tensor $T^C$ measures the failure of the Hermitian metric to behave like a Kähler metric at first order. The next theorem is needed to make that statement precise: it identifies the torsion components with the coefficients of $d\omega$, so the analytic condition $d\omega=0$ becomes a connection-theoretic torsion condition. [quotetheorem:7037] [citeproof:7037] This criterion explains why Kähler geometry has stronger curvature symmetries, and its hypotheses are precise. The connection must be the Chern connection of a Hermitian metric: for an arbitrary metric connection with torsion, vanishing torsion is a statement about that chosen connection and need not be equivalent to $d\omega=0$. The associated form must come from the same Hermitian metric, since the proof compares the coefficients of $\partial\omega$ with the antisymmetric part of the Chern Christoffel symbols. A concrete failure occurs on the Hopf surface example below: the displayed Hermitian form has $d\omega\ne 0$, and therefore its Chern torsion is nonzero. Conversely, on any Kähler metric $d\omega=0$, so the Chern and Levi-Civita connections agree on $T^{1,0}X$. The criterion does not imply that a complex manifold admits a Kähler metric; it only tests whether a particular Hermitian metric is Kähler. To see what replaces Kähler curvature symmetries before imposing $d\omega=0$, we need the version of the first Bianchi identity that retains the torsion terms instead of suppressing them. [quotetheorem:7038] [citeproof:7038] When $T=0$, this reduces to the Riemannian first Bianchi identity. The tangent-bundle hypothesis is essential: the expression $R(u,v)w$ only makes sense when curvature acts on vector fields, while a connection on an unrelated vector bundle has curvature acting on sections of that bundle instead. The torsion terms are not cosmetic additions; they are exactly the obstruction to the cyclic curvature sum vanishing. For a concrete warning, take any Hermitian metric with $d\omega\ne0$, such as the Hopf surface metric below. Its Chern torsion is nonzero by the Kähler criterion, so the cyclic sum of Chern curvature terms is governed by derivatives and quadratic expressions in torsion rather than by the torsion-free identity. The theorem does not recover the extra Kähler symmetries by itself; those require the separate condition $T=0$, or equivalently $d\omega=0$ for the Chern connection. [example: Non-Kähler Hermitian Metric on a Hopf Surface] Let $r=|z|^2=|z_1|^2+|z_2|^2$ on $\mathbb C^2\setminus\{0\}$, so \begin{align*} \omega=i r^{-1}(dz_1\wedge d\bar z_1+dz_2\wedge d\bar z_2). \end{align*} For the dilation $F(z)=2z$, we have $F^*r=|2z|^2=4r$ and $F^*(dz_j\wedge d\bar z_j)=2dz_j\wedge 2d\bar z_j=4dz_j\wedge d\bar z_j$. Hence \begin{align*} F^*\omega=i(4r)^{-1}\sum_{j=1}^2 4dz_j\wedge d\bar z_j=\omega, \end{align*} so $\omega$ descends to the Hopf surface $X=(\mathbb C^2\setminus\{0\})/\langle z\mapsto 2z\rangle$. Now compute $d\omega$. Since $d(dz_j\wedge d\bar z_j)=0$, \begin{align*} d\omega=i\,d(r^{-1})\wedge \sum_{j=1}^2 dz_j\wedge d\bar z_j. \end{align*} Also \begin{align*} dr=\bar z_1dz_1+z_1d\bar z_1+\bar z_2dz_2+z_2d\bar z_2, \end{align*} and therefore \begin{align*} d(r^{-1})=-r^{-2}dr. \end{align*} At the point $p=(1,0)$, $r(p)=1$ and $dr_p=dz_1+d\bar z_1$, so \begin{align*} (d\omega)_p=-i(dz_1+d\bar z_1)\wedge(dz_1\wedge d\bar z_1+dz_2\wedge d\bar z_2). \end{align*} The first summand vanishes because it repeats $dz_1$ or $d\bar z_1$, while the second gives \begin{align*} (d\omega)_p=-i\,dz_1\wedge dz_2\wedge d\bar z_2-i\,d\bar z_1\wedge dz_2\wedge d\bar z_2. \end{align*} For example, evaluating on $\partial_{z_1},\partial_{z_2},\partial_{\bar z_2}$ gives $-i$, so $d\omega\ne0$. Thus the Chern torsion is nonzero by the Kähler criterion by Chern torsion. The Chern-Ricci form is still computable. Here \begin{align*} h_{i\bar j}=r^{-1}\delta_{ij}, \end{align*} so \begin{align*} \det(h_{i\bar j})=\det(r^{-1}I_2)=r^{-2}. \end{align*} Hence \begin{align*} \log\det(h_{i\bar j})=\log(r^{-2})=-2\log r. \end{align*} By the Chern-Ricci formula, \begin{align*} \operatorname{Ric}^C(\omega)=-i\partial\bar\partial\log\det(h_{i\bar j})=2i\partial\bar\partial\log r. \end{align*} This shows explicitly that Chern curvature and the Chern-Ricci form remain available in a non-Kähler Hermitian example, even though the vanishing-torsion curvature symmetries fail. [/example] This example is deliberately not a Kähler example: compact Hopf surfaces have odd first Betti number, so they cannot admit a Kähler metric. The Chern connection is therefore the natural Hermitian connection for local holomorphic calculations, while the Levi-Civita connection remains the torsion-free connection of the underlying Riemannian metric. [remark: Chern Versus Levi-Civita Viewpoints] The Levi-Civita connection is torsion-free and metric-compatible on the real tangent bundle, but it need not preserve $T^{1,0}X$ unless the metric is Kähler. The Chern connection preserves the holomorphic splitting and the Hermitian metric, but it generally has torsion. This chapter uses the Chern connection because curvature of holomorphic vector bundles, determinant metrics, and the Chern-Ricci form are holomorphic constructions. [/remark] The course avoids [Kähler identities](/theorems/3853) at this stage. The point is to keep separate the Hermitian facts that hold on every complex manifold with a Hermitian metric from the extra symmetries that appear only when $d\omega=0$. Chern-Weil theory explains how curvature produces cohomology classes independent of auxiliary choices. The next chapter applies that viewpoint to line bundles, divisors, and meromorphic data, where the abstract characteristic classes become concrete through zeros, poles, and canonical bundles. # 10. Chern-Weil Theory for Holomorphic Bundles Chern-Weil theory turns curvature, which depends on a chosen connection or Hermitian metric, into cohomology classes that do not depend on these choices. In the holomorphic setting the Chern connection supplies a canonical connection attached to a Hermitian metric, so the characteristic forms become concrete differential forms of type $(k,k)$. The chapter explains how invariant polynomials produce closed forms, why changing the metric changes them only by exact forms, and how this recovers the degree of a holomorphic line bundle on a compact Riemann surface. ## Invariant Polynomials and Characteristic Forms from Curvature The first problem is to extract scalar differential forms from the curvature of a vector bundle connection in a way that survives a change of frame. Curvature is endomorphism-valued, so applying an arbitrary polynomial to its matrix entries would depend on the chosen local trivialisation. The correct input is a polynomial on matrices that is invariant under conjugation. [definition: Invariant Polynomial] Let $r \in \mathbb N$. An invariant polynomial of degree $k$ on $\mathfrak{gl}(r,\mathbb C)$ is a complex polynomial function $P: \mathfrak{gl}(r,\mathbb C) \to \mathbb C$ that is homogeneous of degree $k$ and satisfies \begin{align*} P(gAg^{-1}) = P(A) \end{align*} for all $A \in \mathfrak{gl}(r,\mathbb C)$ and all $g \in GL(r,\mathbb C)$. [/definition] The invariance condition is exactly what lets the expression glue across frames. The main examples come from traces and determinants, because trace is unchanged under conjugation and because determinant packages the elementary symmetric polynomials in eigenvalues. [example: Trace Powers] For $A\in \mathfrak{gl}(r,\mathbb C)$ and $k\ge 1$, define $P_k(A)=\operatorname{tr}(A^k)$. The map $P_k$ is a polynomial in the entries of $A$, because each entry of $A^k$ is a sum of products of exactly $k$ entries of $A$, and the trace is the sum of the diagonal entries. For $\lambda\in\mathbb C$, \begin{align*} P_k(\lambda A)=\operatorname{tr}((\lambda A)^k)=\operatorname{tr}(\lambda^k A^k)=\lambda^k\operatorname{tr}(A^k)=\lambda^kP_k(A). \end{align*} Thus $P_k$ is homogeneous of degree $k$. Now let $g\in GL(r,\mathbb C)$. The factors in $(gAg^{-1})^k$ telescope: \begin{align*} (gAg^{-1})^k=(gAg^{-1})(gAg^{-1})\cdots(gAg^{-1})=gA(g^{-1}g)A\cdots A g^{-1}=gA^kg^{-1}. \end{align*} Using $\operatorname{tr}(BC)=\operatorname{tr}(CB)$ for square matrices, \begin{align*} P_k(gAg^{-1})=\operatorname{tr}(gA^kg^{-1})=\operatorname{tr}(A^kg^{-1}g)=\operatorname{tr}(A^k)=P_k(A). \end{align*} So $P_k$ is invariant under conjugation. These trace powers are the basic invariant polynomials appearing in the Chern character, where one sums the normalised forms $\operatorname{tr}(F^k)/k!$. [/example] Trace powers show that invariant polynomials give frame-independent scalar functions on endomorphisms. The next problem is to apply the same idea when the endomorphism is not a matrix of functions but a curvature matrix whose entries are differential forms, so we need a definition of the resulting global form. [definition: Characteristic Form Associated to an Invariant Polynomial] Let $E \to X$ be a complex vector bundle of rank $r$, let $\nabla$ be a connection on $E$, and let $F_\nabla \in A^2(X;\operatorname{End}(E))$ be its curvature. If $P$ is an invariant polynomial of degree $k$ on $\mathfrak{gl}(r,\mathbb C)$, the associated characteristic form is \begin{align*} P(F_\nabla) \in A^{2k}(X;\mathbb C), \end{align*} obtained by applying $P$ to the local curvature matrix and wedging form entries. [/definition] This definition is local in appearance, but invariant polynomials make it global. If the frame changes by a matrix $g$, the curvature matrix changes by conjugation, and the value of $P$ is unchanged. The remaining question is whether this global form has a cohomology class, which requires closedness. [quotetheorem:7039] [citeproof:7039] The invariance hypothesis is not decorative: without it, a local polynomial in the curvature matrix need not survive a change of frame. For instance, the entry function $A\mapsto A_{11}$ changes under conjugation, so applying it to local curvature matrices would generally give different local forms on overlapping trivialisations rather than a global differential form. The theorem also says only that the Chern-Weil form is closed; it does not yet identify the resulting cohomology class as independent of the connection. Closedness is the first half of Chern-Weil theory: it says the form defines a de Rham cohomology class. The next issue is whether this class remembers the auxiliary connection or only the underlying vector bundle. [quotetheorem:1545] [citeproof:1545] This independence is a statement in de Rham cohomology over $\mathbb C$; by itself it does not prove that the class comes from integral cohomology. The normalisations used for Chern classes, especially the factor $i/(2\pi)$, are what make the resulting classes match the usual integral Chern classes in standard situations. The theorem also depends on comparing connections on the same underlying bundle: it does not identify characteristic classes of unrelated bundles merely because some curvature forms happen to look similar in chosen frames. The theorem explains why curvature formulas can define topological invariants. In complex geometry the useful connections are usually Chern connections, because their curvature is compatible with the holomorphic type decomposition. [remark: Type of Chern-Weil Forms for Holomorphic Bundles] If $(E,h)$ is a Hermitian holomorphic vector bundle and $\nabla^h$ is its Chern connection, then $F_{\nabla^h}$ has type $(1,1)$. Therefore an invariant polynomial of degree $k$ gives a characteristic form of type $(k,k)$. This type statement is special to the Chern connection and is the reason Chern forms live naturally in even bidegrees. [/remark] ## Chern Forms and the Total Chern Class The characteristic forms used most often are obtained from the determinant expression that formally records the eigenvalues of curvature. The problem is to normalise the expression so that it represents integral Chern classes in de Rham cohomology and behaves correctly under direct sums. [definition: Chern Forms] Let $(E,h) \to X$ be a Hermitian holomorphic vector bundle of rank $r$, and let $F_h$ denote the curvature of its Chern connection. The Chern forms $c_k(E,h) \in A^{k,k}(X)$ are defined by \begin{align*} \det\left(I + \frac{i}{2\pi}F_h\right) &= 1 + c_1(E,h) + \cdots + c_r(E,h). \end{align*} The total Chern form is \begin{align*} c(E,h) := 1+c_1(E,h)+\cdots+c_r(E,h). \end{align*} [/definition] The factor $i/(2\pi)$ is the standard normalisation in complex geometry. With this convention, the de Rham class of $c_k(E,h)$ is expected to be the image of the integral Chern class, but this expectation first needs the determinant components to be closed forms. [quotetheorem:7040] The determinant construction is essential here: its homogeneous pieces are invariant polynomials, so the preceding theorem applies. A non-invariant expression built from selected entries of the curvature matrix would not even define a global form, and therefore could not be expected to be closed globally. Closedness also does not make $c_k(E,h)$ independent of the metric as a form; different metrics usually give different representatives, and only their cohomology classes agree. Closed Chern-Weil forms still depend on a chosen Hermitian metric as differential forms, so they are not yet invariants of the holomorphic bundle alone. The problem is to keep the part that survives a change of metric while discarding the representative-level variation. Passing to de Rham cohomology does this: exact changes disappear, and the resulting classes can be named without mentioning $h$. [definition: Chern Classes of a Holomorphic Vector Bundle] Let $E \to X$ be a holomorphic vector bundle. For any Hermitian metric $h$ on $E$, define \begin{align*} c_k(E) := [c_k(E,h)] \in H^{2k}_{\mathrm{dR}}(X;\mathbb C), \qquad c(E):=1+c_1(E)+\cdots+c_r(E). \end{align*} [/definition] This is well-defined because changing $h$ changes the Chern connection but not the de Rham class of the associated Chern-Weil form. Computations may therefore use whichever metric makes the curvature manageable. [example: The Hyperplane Bundle on Projective Space] On the affine chart $U_0=\{Z_0\ne 0\}$, write $z_j=Z_j/Z_0$ and let $e_0$ be the standard holomorphic frame of $\mathcal O(1)$. For the dual Fubini-Study metric, \begin{align*} |e_0|_{h_{\mathrm{FS}}}^2=(1+|z_1|^2+\cdots+|z_n|^2)^{-1}. \end{align*} Using the local curvature formula for a Hermitian line bundle, \begin{align*} F_{h_{\mathrm{FS}}}=-\partial\bar\partial\log |e_0|_{h_{\mathrm{FS}}}^2=-\partial\bar\partial\log\left((1+\sum_{j=1}^n |z_j|^2)^{-1}\right). \end{align*} Since $\log(a^{-1})=-\log a$ for $a>0$, \begin{align*} F_{h_{\mathrm{FS}}}=\partial\bar\partial\log\left(1+\sum_{j=1}^n |z_j|^2\right). \end{align*} With the convention \begin{align*} \omega_{\mathrm{FS}}=i\partial\bar\partial\log\left(1+\sum_{j=1}^n |z_j|^2\right), \end{align*} the first Chern form is therefore \begin{align*} c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{i}{2\pi}F_{h_{\mathrm{FS}}}=\frac{1}{2\pi}\omega_{\mathrm{FS}}. \end{align*} To see the normalisation, restrict to a projective line $\mathbb{CP}^1\subset \mathbb{CP}^n$. On its affine coordinate $z$, \begin{align*} \partial\log(1+|z|^2)=\frac{\bar z}{1+|z|^2}\,dz. \end{align*} Applying $\bar\partial$ gives \begin{align*} \partial\bar\partial\log(1+|z|^2)=-\frac{1}{(1+|z|^2)^2}\,dz\wedge d\bar z. \end{align*} Thus \begin{align*} \frac{1}{2\pi}\omega_{\mathrm{FS}}=\frac{i}{2\pi}\partial\bar\partial\log(1+|z|^2)=\frac{i}{2\pi}\frac{-dz\wedge d\bar z}{(1+|z|^2)^2}. \end{align*} In polar coordinates $z=re^{i\theta}$ one has $i\,dz\wedge d\bar z=2r\,dr\wedge d\theta$, so \begin{align*} \int_{\mathbb{CP}^1}c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{1}{\pi}\int_0^{2\pi}\int_0^\infty \frac{r}{(1+r^2)^2}\,dr\,d\theta. \end{align*} Since \begin{align*} \int_0^\infty \frac{r}{(1+r^2)^2}\,dr=\frac{1}{2}, \end{align*} the integral is $1$. Hence $c_1(\mathcal O(1))$ is the positive generator of $H^2(\mathbb{CP}^n;\mathbb Z)$ under the de Rham comparison map. For $\mathcal O(m)=\mathcal O(1)^{\otimes m}$, tensor powers add first Chern classes, so \begin{align*} c_1(\mathcal O(m))=m\,c_1(\mathcal O(1)). \end{align*} [/example] The hyperplane bundle gives a concrete generator for computations on projective space, but most bundles do not split visibly into line bundles on the original space. To compute universal identities for higher-rank bundles, the next result explains why it is enough to work after a pullback where the bundle splits. [quotetheorem:7041] The course uses this principle as a computational device rather than proving it in full; the proof belongs to the topology of flag bundles. Its role here is to justify manipulations with formal Chern roots $x_j=c_1(L_j)$ while remembering that the original bundle need not split on $X$ itself. The point is not that every bundle is secretly a sum of line bundles on the original space, but that universal Chern-class identities can be checked after a pullback where such a splitting exists and then descended back to $X$. This is why computations may be done as if the Chern roots were genuine first Chern classes, provided the final expression is symmetric in the roots. The next projective-space calculation illustrates this method: the Euler sequence reduces the total Chern class to the behaviour of line bundles, and the resulting polynomial identity is intrinsic even though the intermediate roots are formal. [example: Total Chern Class of the Tangent Bundle of Projective Space] Let $H=c_1(\mathcal O(1))\in H^2(\mathbb{CP}^n;\mathbb Z)$. The Euler sequence \begin{align*} 0\to \mathcal O \to \mathcal O(1)^{\oplus(n+1)} \to T\mathbb{CP}^n \to 0 \end{align*} is a short exact sequence of holomorphic vector bundles, so the *Whitney product formula* gives \begin{align*} c(\mathcal O(1)^{\oplus(n+1)})=c(\mathcal O)\,c(T\mathbb{CP}^n). \end{align*} The structure sheaf $\mathcal O$ has total Chern class $c(\mathcal O)=1$, hence \begin{align*} c(T\mathbb{CP}^n)=c(\mathcal O(1)^{\oplus(n+1)}). \end{align*} For a direct sum, total Chern classes multiply across the summands: \begin{align*} c(\mathcal O(1)^{\oplus(n+1)})=c(\mathcal O(1))^{n+1}. \end{align*} Since $\mathcal O(1)$ is a line bundle, its total Chern class has only the degree $0$ and degree $2$ terms: \begin{align*} c(\mathcal O(1))=1+c_1(\mathcal O(1))=1+H. \end{align*} Therefore \begin{align*} c(T\mathbb{CP}^n)=(1+H)^{n+1}. \end{align*} Expanding by the [binomial theorem](/theorems/750) gives \begin{align*} (1+H)^{n+1}=\sum_{k=0}^{n+1}\binom{n+1}{k}H^k. \end{align*} Here $H^k\in H^{2k}(\mathbb{CP}^n;\mathbb Z)$, so terms with $k>n$ lie above the top cohomological degree $2n$ and vanish. Thus, for $0\le k\le n$, the degree-$2k$ component is \begin{align*} c_k(T\mathbb{CP}^n)=\binom{n+1}{k}H^k. \end{align*} Taking $k=1$ gives \begin{align*} c_1(T\mathbb{CP}^n)=\binom{n+1}{1}H=(n+1)H. \end{align*} The Euler sequence therefore reduces the tangent bundle computation to the binomial expansion of $(1+H)^{n+1}$, truncated above degree $2n$. [/example] The Euler sequence illustrates how total Chern classes turn exact sequences into computable identities. For direct sums, another invariant is often more efficient because it converts direct sums into ordinary sums; this motivates the Chern character. [definition: Chern Character Form] Let $(E,h)$ be a Hermitian holomorphic vector bundle with Chern curvature $F_h$. The Chern character form is \begin{align*} \operatorname{ch}(E,h) := \operatorname{tr}\left(\exp\left(\frac{i}{2\pi}F_h\right)\right) = \sum_{k\ge 0}\frac{1}{k!}\operatorname{tr}\left(\left(\frac{i}{2\pi}F_h\right)^k\right). \end{align*} [/definition] Like the Chern forms, the Chern character form is closed and has a metric-independent de Rham class. Its advantage is functorial behaviour with direct sums. [example: Chern Character of a Direct Sum of Line Bundles] Choose Hermitian metrics $h_j$ on the line bundles $L_j$, and give $E=L_1\oplus\cdots\oplus L_r$ the direct sum metric. Write \begin{align*} \Omega_j=\frac{i}{2\pi}F_{h_j} \end{align*} so that $[\Omega_j]=x_j=c_1(L_j)$. In the direct sum frame, the curvature matrix of $E$ is block diagonal: \begin{align*} \frac{i}{2\pi}F_E=\operatorname{diag}(\Omega_1,\dots,\Omega_r). \end{align*} For a diagonal matrix of forms, powers remain diagonal: \begin{align*} \left(\operatorname{diag}(\Omega_1,\dots,\Omega_r)\right)^m=\operatorname{diag}(\Omega_1^m,\dots,\Omega_r^m). \end{align*} Therefore the exponential series gives \begin{align*} \exp\left(\frac{i}{2\pi}F_E\right)=\operatorname{diag}(e^{\Omega_1},\dots,e^{\Omega_r}). \end{align*} Taking the trace and then passing to cohomology, \begin{align*} \operatorname{ch}(E)=e^{x_1}+\cdots+e^{x_r}. \end{align*} Expanding each exponential, \begin{align*} e^{x_j}=1+x_j+\frac{x_j^2}{2!}+\frac{x_j^3}{3!}+\cdots. \end{align*} Summing over $j$ gives \begin{align*} \operatorname{ch}(E)=r+\sum_{j=1}^r x_j+\frac{1}{2}\sum_{j=1}^r x_j^2+\frac{1}{6}\sum_{j=1}^r x_j^3+\cdots. \end{align*} The ordinary Chern classes are the elementary symmetric polynomials in the Chern roots: \begin{align*} c_1(E)=\sum_j x_j,\qquad c_2(E)=\sum_{a<b}x_ax_b,\qquad c_3(E)=\sum_{a<b<c}x_ax_bx_c. \end{align*} For example, \begin{align*} c_1(E)^2=\left(\sum_j x_j\right)^2=\sum_j x_j^2+2\sum_{a<b}x_ax_b=\sum_j x_j^2+2c_2(E), \end{align*} so $\sum_j x_j^2=c_1(E)^2-2c_2(E)$. Thus the first terms may be rewritten as \begin{align*} \operatorname{ch}(E)=r+c_1(E)+\frac{1}{2}\left(c_1(E)^2-2c_2(E)\right)+\cdots. \end{align*} The splitting principle justifies this calculation as the universal formula for arbitrary vector bundles, because any universal identity in the Chern roots descends to an identity in the Chern classes of $E$. [/example] ## First Chern Class, Line Bundles, and Degree on Curves For line bundles the full Chern-Weil construction becomes especially concrete: the curvature is an ordinary scalar two-form. The guiding question is how this curvature integral records the topological degree of a holomorphic line bundle on a compact Riemann surface. [definition: First Chern Form of a Hermitian Line Bundle] Let $(L,h)\to X$ be a Hermitian holomorphic line bundle, and let $F_h$ be the curvature of its Chern connection. The first Chern form is \begin{align*} c_1(L,h)=\frac{i}{2\pi}F_h. \end{align*} [/definition] The definition identifies the first Chern form with scalar curvature, but it does not yet tell us how to compute that curvature from a local metric. To connect the construction with complex analysis, we need the formula in a local holomorphic frame. [quotetheorem:3840] [citeproof:3840] The holomorphic frame hypothesis matters. If the frame is changed by a nowhere-vanishing holomorphic function $g$, then $\log|gs|_h^2=\log|s|_h^2+\log|g|^2$, and $\partial\bar\partial\log|g|^2=0$ away from zeros and poles, so the curvature expression is unchanged on the overlap. For a non-holomorphic change of frame this cancellation would fail, which is why the formula belongs to holomorphic line bundles with the Chern connection rather than to arbitrary smooth frames. This formula is the bridge from complex analysis to topology. It says that a logarithmic metric weight produces a curvature representative of $c_1(L)$. [example: The Tautological Computation on Projective Space] On the affine chart $U_0=\{Z_0\ne 0\}\subset \mathbb{CP}^n$, write $z_j=Z_j/Z_0$ and set \begin{align*} \varphi(z)=1+|z_1|^2+\cdots+|z_n|^2. \end{align*} For the dual Fubini-Study metric on $\mathcal O(1)$, choose the holomorphic frame $e_0$ dual to the tautological frame over $U_0$; then \begin{align*} |e_0|_{h_{\mathrm{FS}}}^2=\varphi(z)^{-1}. \end{align*} Using the local curvature formula for a Hermitian line bundle, \begin{align*} F_{h_{\mathrm{FS}}}=-\partial\bar\partial\log |e_0|_{h_{\mathrm{FS}}}^2. \end{align*} Since $\log(\varphi^{-1})=-\log\varphi$, this becomes \begin{align*} F_{h_{\mathrm{FS}}}=-\partial\bar\partial(-\log\varphi)=\partial\bar\partial\log\varphi. \end{align*} Therefore \begin{align*} c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{i}{2\pi}F_{h_{\mathrm{FS}}}=\frac{i}{2\pi}\partial\bar\partial\log(1+|z_1|^2+\cdots+|z_n|^2). \end{align*} With the convention \begin{align*} \omega_{\mathrm{FS}}=i\partial\bar\partial\log(1+|z_1|^2+\cdots+|z_n|^2), \end{align*} we get \begin{align*} c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{1}{2\pi}\omega_{\mathrm{FS}}. \end{align*} Thus the curvature computation produces the Fubini-Study representative of the hyperplane class, with the normalisation matching $c_1(\mathcal O(1))$. [/example] The projective computation shows that integrating first Chern forms can produce integer topological data. This raises the problem of naming the integer obtained from a compact complex curve and a holomorphic line bundle. The definition below fixes that invariant by using any Chern-Weil representative of the first Chern class. [definition: Degree of a Holomorphic Line Bundle] Let $X$ be a compact connected Riemann surface and let $L\to X$ be a holomorphic line bundle. The degree of $L$ is \begin{align*} \deg L := \int_X c_1(L,h) \end{align*} for any Hermitian metric $h$ on $L$. [/definition] The curvature integral is a good intrinsic definition only if it matches the divisor-theoretic degree already attached to line bundles of the form $\mathcal O(D)$. Without that comparison, the analytic integral could be a parallel invariant unrelated to zeros and poles. The issue is therefore to prove that curvature, after the standard normalization, counts the same integer as the divisor data. [quotetheorem:7042] Compactness and the absence of boundary are essential in the metric-independence part of this argument. When the metric changes, the curvature representative changes by an exact form, and [Stokes' theorem](/theorems/1530) makes the integral of that exact form vanish only on a compact boundaryless surface. On a non-compact curve, or on a surface with boundary, boundary behaviour and decay conditions must be specified before the same integral can be treated as a topological degree. This theorem is the curvature interpretation of degree: positivity or negativity of curvature gives geometric information about the line bundle. It is also the simplest instance of the broader theme that curvature forms encode intersection-theoretic data. [example: Degree of the Hyperplane Bundle on the Projective Line] On $\mathbb{CP}^1$, use the affine coordinate $z=re^{i\theta}$ on $\mathbb C\subset\mathbb{CP}^1$. For the Fubini-Study metric on $\mathcal O(1)$, the preceding computation gives \begin{align*} c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{i}{2\pi}\frac{dz\wedge d\bar z}{(1+|z|^2)^2}. \end{align*} Since $z=re^{i\theta}$, one has $i\,dz\wedge d\bar z=2r\,dr\wedge d\theta$, so \begin{align*} c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{1}{\pi}\frac{r}{(1+r^2)^2}\,dr\wedge d\theta. \end{align*} Therefore \begin{align*} \deg\mathcal O(1)=\int_{\mathbb{CP}^1}c_1(\mathcal O(1),h_{\mathrm{FS}})=\frac{1}{\pi}\int_0^{2\pi}\int_0^\infty \frac{r}{(1+r^2)^2}\,dr\,d\theta. \end{align*} For the radial integral, set $u=1+r^2$, so $du=2r\,dr$. Then \begin{align*} \int_0^\infty \frac{r}{(1+r^2)^2}\,dr=\frac{1}{2}\int_1^\infty u^{-2}\,du=\frac{1}{2}. \end{align*} Substituting this value gives \begin{align*} \deg\mathcal O(1)=\frac{1}{\pi}\cdot 2\pi\cdot \frac{1}{2}=1. \end{align*} For $m\in\mathbb Z$, the line bundle $\mathcal O(m)$ has $c_1(\mathcal O(m))=m\,c_1(\mathcal O(1))$, so linearity of integration gives \begin{align*} \deg\mathcal O(m)=\int_{\mathbb{CP}^1}c_1(\mathcal O(m))=m\int_{\mathbb{CP}^1}c_1(\mathcal O(1))=m. \end{align*} Thus the curvature definition of degree recovers the divisor convention: a section of $\mathcal O(m)$ has divisor of total degree $m$. [/example] The chapter closes the loop between local Hermitian geometry and global topology. Curvature matrices produce closed forms through invariant polynomials; those forms have cohomology classes independent of metric; and for line bundles on curves the integral of curvature is the classical degree. Divisors and meromorphic sections give a more algebraic way to see the same line bundles and curvature classes that appeared in Chern-Weil theory. With that correspondence established, the course is ready to synthesize local complex analysis and global geometry into a unified picture. # 11. Holomorphic Line Bundles, Divisors, and Meromorphic Data After Chern-Weil theory connected curvature with cohomology classes, meromorphic data provide the parallel algebraic source of the same line bundles. Meromorphic data are the first point where holomorphic geometry behaves like algebraic geometry. A holomorphic function on a compact complex manifold is often too rigid, but a meromorphic function may have prescribed zeros and poles along hypersurfaces. This chapter packages such zero and pole data into divisors, turns divisors into holomorphic line bundles, and then relates the resulting bundles to canonical forms, hypersurfaces, and curvature currents. ## Meromorphic Functions and Divisors The guiding question is how to record the zeros and poles of a meromorphic function without choosing coordinates. In one complex variable, the order of vanishing of a holomorphic function at a point is intrinsic. On an $n$-dimensional complex manifold, the codimension-one analogue is a hypersurface, and the order is measured by the exponent of a local defining equation. [definition: Meromorphic Function] Let $X$ be a complex manifold. A meromorphic function on $X$ is a collection of quotients $f_i = g_i/h_i$ on open sets $U_i$, where $g_i,h_i \in \mathcal O_X(U_i)$ and $h_i$ is not identically zero on any connected component, such that $f_i=f_j$ as quotients on $U_i \cap U_j$. [/definition] This definition allows poles, but only in the controlled way forced by holomorphic denominators. The equality condition says that the quotient is not a coordinate artefact; it defines a global section of the sheaf of meromorphic functions. [example: Meromorphic Functions on Projective Space] Let $F$ and $G$ be homogeneous polynomials of the same degree $d$ on $\mathbb C^{n+1}$, with $G$ not identically zero. For any $\lambda \in \mathbb C^*$ and any homogeneous coordinates $[Z_0:\cdots:Z_n]$, homogeneity gives \begin{align*} F(\lambda Z_0,\dots,\lambda Z_n)=\lambda^d F(Z_0,\dots,Z_n) \end{align*} and \begin{align*} G(\lambda Z_0,\dots,\lambda Z_n)=\lambda^d G(Z_0,\dots,Z_n). \end{align*} Hence, wherever $G(Z)\ne 0$, \begin{align*} \frac{F(\lambda Z_0,\dots,\lambda Z_n)}{G(\lambda Z_0,\dots,\lambda Z_n)} = \frac{\lambda^d F(Z_0,\dots,Z_n)}{\lambda^d G(Z_0,\dots,Z_n)} = \frac{F(Z_0,\dots,Z_n)}{G(Z_0,\dots,Z_n)}. \end{align*} Thus the quotient is unchanged by rescaling homogeneous coordinates and defines a meromorphic function on $\mathbb{CP}^n$. On the affine chart $U_0=\{Z_0\ne 0\}$, write $z_j=Z_j/Z_0$. Then \begin{align*} [Z_0:\cdots:Z_n]=[1:z_1:\cdots:z_n], \end{align*} so the local representative is \begin{align*} \frac{F(1,z_1,\dots,z_n)}{G(1,z_1,\dots,z_n)}. \end{align*} Near a smooth point of an irreducible component $Y$ of $\{F=0\}$ or $\{G=0\}$, choose a local defining equation $s$ for $Y$. If \begin{align*} F=s^a u \end{align*} and \begin{align*} G=s^b v \end{align*} with $u$ and $v$ holomorphic and nonvanishing along $Y$, then locally \begin{align*} \frac{F}{G}=s^{a-b}\frac{u}{v}. \end{align*} Therefore $Y$ appears as a zero with multiplicity $a-b$ when $a>b$, as a pole with multiplicity $b-a$ when $b>a$, and does not appear in the divisor when $a=b$. The zeros and poles are therefore exactly the projective hypersurface components cut out by $F$ and $G$, counted with these local multiplicities. [/example] This example shows that zeros and poles naturally live on hypersurfaces rather than on isolated points in higher dimension. To make that bookkeeping independent of a chosen equation such as $F/G$, we need a global object that lists hypersurfaces with integer multiplicities. This motivates the following definition. [definition: Divisor] Let $X$ be a complex manifold. A divisor on $X$ is a locally finite formal integer linear combination \begin{align*} D = \sum_Y m_Y Y, \end{align*} where $Y$ ranges over irreducible analytic hypersurfaces in $X$ and $m_Y \in \mathbb Z$. [/definition] Local finiteness means that each point has a neighbourhood meeting only finitely many hypersurfaces appearing with nonzero coefficient. The positive and negative parts are written \begin{align*} D^+ &= \sum_{m_Y>0} m_Y Y, & D^- &= \sum_{m_Y<0} (-m_Y)Y, \end{align*} so that $D=D^+-D^-$. Since meromorphic functions produce zeros and poles simultaneously, we next single out the divisors that arise from one global meromorphic function. [definition: Principal Divisor] Let $X$ be a complex manifold and let $f$ be a nonzero meromorphic function on $X$. The principal divisor of $f$ is \begin{align*} (f)=\sum_Y \operatorname{ord}_Y(f)Y, \end{align*} where $\operatorname{ord}_Y(f)$ is the vanishing order of $f$ along the irreducible hypersurface $Y$. [/definition] Principal divisors are the divisors that come from a single global meromorphic function. The next structural question is whether the multiplicative law for meromorphic functions becomes an additive law for their divisors. The relevant statement is the basic calculus of orders along hypersurfaces. [quotetheorem:7043] [citeproof:7043] This theorem is the bridge between multiplicative meromorphic functions and additive divisor calculus. The nonzero hypothesis is essential: the zero function has infinite vanishing order along every hypersurface, so it does not define a divisor in this finite-integer sense. The result also explains why divisors should be compared modulo principal divisors when the underlying question is about meromorphic functions. [example: Principal Divisors on $\mathbb{CP}^1$] Let $z$ be the affine coordinate on $\mathbb C \subset \mathbb{CP}^1$, and fix $a\in \mathbb C$. Near the finite point $a$, the coordinate $u=z-a$ is a local defining coordinate, and the function is exactly \begin{align*} z-a=u. \end{align*} Thus $\operatorname{ord}_{a}(z-a)=1$. If $b\in \mathbb C$ with $b\ne a$, then $z-a=(b-a)+(z-b)$ has nonzero value $b-a$ at $b$, so $\operatorname{ord}_{b}(z-a)=0$. To compute the order at infinity, use the coordinate $w=1/z$ near $\infty$, so $w=0$ is the point $\infty$. Then \begin{align*} z-a=\frac{1}{w}-a=\frac{1-aw}{w}=w^{-1}(1-aw). \end{align*} The factor $1-aw$ is holomorphic and nonzero at $w=0$, since $1-a\cdot 0=1$. Therefore $\operatorname{ord}_{\infty}(z-a)=-1$, and hence \begin{align*} (z-a)=[a]-[\infty]. \end{align*} More generally, let $p,q\in \mathbb C[z]$ be nonzero polynomials. At a finite point $b\in \mathbb C$, write \begin{align*} p(z)=(z-b)^r u(z) \end{align*} and \begin{align*} q(z)=(z-b)^s v(z), \end{align*} where $u(b)\ne 0$ and $v(b)\ne 0$. Then \begin{align*} \frac{p(z)}{q(z)}=(z-b)^{r-s}\frac{u(z)}{v(z)}, \end{align*} so the coefficient of $[b]$ in the divisor is $r-s$. At infinity, if $\deg p=m$ and $\deg q=n$, then in the coordinate $w=1/z$, \begin{align*} \frac{p(1/w)}{q(1/w)}=w^{n-m}\frac{w^m p(1/w)}{w^n q(1/w)}. \end{align*} The functions $w^m p(1/w)$ and $w^n q(1/w)$ are holomorphic at $w=0$ and take the nonzero values equal to the leading coefficients of $p$ and $q$. Thus the coefficient of $[\infty]$ is $n-m$. So the divisor of $p/q$ is its finite zero divisor minus its finite pole divisor, with the additional term $(\deg q-\deg p)[\infty]$. [/example] ## The Line Bundle Associated to a Divisor The next problem is to convert divisor data into a holomorphic line bundle. This conversion is powerful because sections of the line bundle can then be treated with the tools of vector bundles, metrics, connections, and curvature. Suppose $D$ is a divisor. On a sufficiently small open set $U_i$, choose a meromorphic function $f_i$ whose divisor on $U_i$ is $D|_{U_i}$. On overlaps $U_i \cap U_j$, the quotient $f_i/f_j$ has divisor zero, hence is a nowhere-zero holomorphic function. These quotients are transition functions for a line bundle. [definition: Line Bundle of a Divisor] Let $X$ be a complex manifold and let $D$ be a divisor. Choose an open cover $(U_i)$ and local meromorphic functions $f_i$ with $(f_i)=D|_{U_i}$. The holomorphic line bundle $\mathcal O(D)$ is the line bundle with transition functions \begin{align*} g_{ij}=f_i/f_j \in \mathcal O_X^*(U_i \cap U_j). \end{align*} [/definition] Different choices of the local equations $f_i$ produce isomorphic line bundles, because changing $f_i$ by a nowhere-zero holomorphic factor changes the transition functions by a coboundary. The next question is whether this construction respects addition of divisors and whether principal divisors carry no new bundle twisting. This compatibility is the content of the next result. [quotetheorem:7044] The theorem gives every divisor both a line bundle and a distinguished section, but that section is in general only meromorphic: along the pole part of $D$ the local equations $f_i$ have poles, so the section blows up. For the construction to produce genuine holomorphic geometry — a section that vanishes along the prescribed hypersurfaces and has no poles anywhere — we must first single out the divisors whose coefficients carry no pole data at all. The next definition isolates exactly that class, which is the input the following theorem needs to manufacture a holomorphic section. [definition: Effective Divisor] A divisor $D=\sum_Y m_Y Y$ on a complex manifold $X$ is effective if $m_Y \ge 0$ for every irreducible hypersurface $Y$. [/definition] Effectivity removes the pole obstruction, but one still has to check that the local defining equations glue to a global holomorphic section of the associated line bundle. This is not automatic from the divisor alone: the local equations change by units on overlaps, and those units must match the transition functions of $\mathcal O(D)$. The point of the result below is to identify the canonical section produced by this gluing and to determine its zero divisor. [quotetheorem:7045] [citeproof:7045] This construction is the model for many later arguments: a geometric locus is replaced by the zero set of a section of a line bundle. The effectivity hypothesis is exactly what prevents poles; for instance, on $\mathbb{CP}^1$ the divisor $-[0]$ corresponds to a meromorphic section with a simple pole at $0$, not to a holomorphic section vanishing somewhere. The bundle remembers the twisting needed to make the local defining equations glue. [example: The Bundle $\mathcal O(k)$ from Divisors on $\mathbb{CP}^1$] Let first take $D=k[\infty]$ on $\mathbb{CP}^1$, and use the two affine charts $U_0=\mathbb C_z$ and $U_\infty=\mathbb C_w$ with $w=1/z$. On $U_0$, the divisor has no support, so take the local equation \begin{align*} f_0=1. \end{align*} On $U_\infty$, the point $\infty$ is $w=0$, so a local equation for $k[\infty]$ is \begin{align*} f_\infty=w^k. \end{align*} On the overlap $U_0\cap U_\infty=\mathbb C^*$, the transition function of $\mathcal O(D)$ is \begin{align*} g_{0\infty}=\frac{f_0}{f_\infty}=\frac{1}{w^k}=z^k. \end{align*} This is the standard transition function for $\mathcal O(k)$ on $\mathbb{CP}^1$, so \begin{align*} \mathcal O(k[\infty])\cong \mathcal O(k). \end{align*} The distinguished section has local representatives $1$ on $U_0$ and $w^k$ on $U_\infty$. Since $w=0$ is the point $\infty$, the function $w^k$ vanishes to order $k$ there and has no other zeros on $U_\infty$. Thus the divisor of the distinguished section is exactly $k[\infty]$; on the affine chart $\mathbb C_z$ the same section is represented by the constant polynomial $1$, and its zero appears only after changing to the coordinate $w=1/z$ near infinity. For a general effective divisor \begin{align*} D=m_1[a_1]+\cdots+m_r[a_r]+m_\infty[\infty], \end{align*} with $a_j\in\mathbb C$ and \begin{align*} m_1+\cdots+m_r+m_\infty=k, \end{align*} take \begin{align*} P(z)=\prod_{j=1}^r (z-a_j)^{m_j}. \end{align*} At each finite point $a_j$, the polynomial $P$ has vanishing order $m_j$. If $M=m_1+\cdots+m_r$, then $P$ has degree $M$, so in the coordinate $w=1/z$ near infinity, \begin{align*} P(1/w)=w^{-M}\prod_{j=1}^r(1-a_jw)^{m_j}. \end{align*} The factor $\prod_{j=1}^r(1-a_jw)^{m_j}$ is holomorphic and nonzero at $w=0$, so $P$ has a pole of order $M$ at $\infty$. Therefore \begin{align*} (P)=m_1[a_1]+\cdots+m_r[a_r]-M[\infty]. \end{align*} Since $k=M+m_\infty$, we get \begin{align*} (P)+k[\infty]=m_1[a_1]+\cdots+m_r[a_r]+m_\infty[\infty]=D. \end{align*} The divisor-line bundle construction sends addition of divisors to tensor product and principal divisors to the identity line bundle, hence \begin{align*} \mathcal O(D)\cong \mathcal O((P))\otimes \mathcal O(k[\infty])\cong \mathcal O(k). \end{align*} Thus on $\mathbb{CP}^1$, the associated line bundle of an effective divisor depends only on its total degree. [/example] ## Canonical and Anticanonical Bundles Divisors become especially important when they interact with differential forms. The central question is how holomorphic volume forms transform under coordinate changes, and what bundle records that transformation. [definition: Canonical Bundle] Let $X$ be a complex manifold of complex dimension $n$. The canonical bundle of $X$ is \begin{align*} K_X = \Lambda^n (T^{1,0}X)^*. \end{align*} Its holomorphic sections are holomorphic $(n,0)$-forms. [/definition] The canonical bundle measures the availability of holomorphic volume forms. If it is isomorphic to $X\times \mathbb C$, the manifold has a nowhere-zero holomorphic $(n,0)$-form; if it is positive or negative in a suitable sense, it strongly constrains the geometry. Since dualising reverses that sign, we also need the opposite line bundle. This motivates the following definition. [definition: Anticanonical Bundle] Let $X$ be a complex manifold. The anticanonical bundle is the dual line bundle \begin{align*} K_X^{-1}=\operatorname{Hom}(K_X,\mathcal O_X). \end{align*} [/definition] The anticanonical bundle appears naturally when vector fields or hypersurfaces are used to produce volume forms with poles, and in projective geometry positivity of $K_X^{-1}$ is the first signal of Fano-type behaviour. But the definitions above are abstract: they say what $K_X$ and $K_X^{-1}$ are without pinning down which line bundle results in any concrete case, and without that anchor the sign conventions in $\mathcal O(\pm k)$ are impossible to use. We therefore need a worked computation in the one example where every line bundle is already labelled by an integer, projective space, so that the canonical bundle acquires a definite degree. The next theorem supplies that calibration. [quotetheorem:3879] [citeproof:3879] This computation is the prototype for extracting a line bundle from a Jacobian determinant. It does not produce a nowhere-zero holomorphic volume form on projective space; the negative degree is precisely the obstruction. In dimension one, it says $K_{\mathbb{CP}^1}\cong \mathcal O(-2)$, matching the fact that a rational differential has total divisor degree $-2$. [example: Rational Differentials on $\mathbb{CP}^1$] On the affine chart $U_0=\mathbb C_z\subset \mathbb{CP}^1$, the differential $dz$ is a holomorphic local frame for $K_{\mathbb{CP}^1}$ because $z$ is a holomorphic coordinate there. On the chart $U_\infty=\mathbb C_w$ around infinity, take $w=1/z$, so $z=w^{-1}$. Differentiating the identity $z=w^{-1}$ gives \begin{align*} dz=d(w^{-1})=-w^{-2}dw. \end{align*} Here $dw$ is a holomorphic local frame for $K_{\mathbb{CP}^1}$ near $w=0$, which is the point $\infty$. Thus, relative to the frame $dw$, the meromorphic differential $dz$ is represented by the function $-w^{-2}$. The constant factor $-1$ is holomorphic and nonzero, while $w^{-2}$ has a pole of order $2$ at $w=0$, so \begin{align*} \operatorname{ord}_{\infty}(dz)=-2. \end{align*} On $U_0$, the same differential is represented by the constant function $1$ relative to the frame $dz$, so it has no zeros or poles at any finite point. Therefore \begin{align*} \operatorname{div}(dz)=-2[\infty]. \end{align*} By the divisor-line bundle construction, a nonzero meromorphic section of a line bundle records that line bundle through its divisor, so this computation gives \begin{align*} K_{\mathbb{CP}^1}\cong \mathcal O(-2[\infty])\cong \mathcal O(-2). \end{align*} Thus the pole of $dz$ at infinity is exactly the degree $-2$ of the canonical bundle on $\mathbb{CP}^1$. [/example] ## Adjunction for Smooth Hypersurfaces The next problem is to compare canonical forms on a manifold with canonical forms on a hypersurface. A hypersurface has one missing normal direction, so its canonical bundle should be obtained by restricting the ambient canonical bundle and correcting by the normal line. [definition: Normal Bundle of a Hypersurface] Let $Y \subset X$ be a smooth hypersurface in a complex manifold. The normal bundle of $Y$ in $X$ is \begin{align*} N_{Y/X}=T^{1,0}X|_Y/T^{1,0}Y. \end{align*} [/definition] If $Y$ is cut out by a section of a line bundle, the normal bundle should be the restriction of that line bundle. For a divisor hypersurface, this line bundle is $\mathcal O(Y)|_Y$, so we need to compare the derivative of the defining section with normal directions. The definition above presents the normal bundle as an abstract quotient of tangent spaces, which gives no handle on its degree or its sections. To compute with it we need to recognise it as a line bundle we already control. The expectation that the defining section's derivative trivialises the normal direction must be turned into a precise isomorphism, and the following theorem provides it by identifying $N_{Y/X}$ with the restriction of the divisor line bundle $\mathcal O(Y)$. [quotetheorem:7046] [citeproof:7046] With the normal bundle now identified, the missing differential in the ambient volume form has an explicit home, and the smoothness assumption is what makes that identification valid: at a nodal or cuspidal hypersurface there is no honest vector-bundle quotient $T^{1,0}X|_Y/T^{1,0}Y$ over the singular point, so the identification must be replaced by a sheaf-theoretic version. But knowing the normal bundle alone does not yet tell us the canonical bundle of $Y$, which is what governs holomorphic forms on the hypersurface itself. We still need a formula that produces $K_Y$ from data on the ambient manifold, so that hypersurface invariants can be read off from $X$. The next theorem is exactly that bridge: it expresses $K_Y$ through $K_X$ and the divisor line bundle by taking determinants in the tangent exact sequence and substituting the normal bundle identification. [quotetheorem:3878] [citeproof:3878] Adjunction turns ambient computations into intrinsic computations on hypersurfaces. It does not apply unchanged to singular hypersurfaces: for example, a nodal plane curve has a dualising sheaf governed by the singularity, not simply the canonical bundle of a smooth Riemann surface obtained by restricting $K_{\mathbb{CP}^2}\otimes \mathcal O(C)$. In projective space, the smooth formula lets us compute canonical bundles of smooth hypersurfaces from their degree. [example: Canonical Bundle of a Smooth Plane Curve] Let $C \subset \mathbb{CP}^2$ be a smooth plane curve cut out by one homogeneous polynomial of degree $d$. By *Canonical Bundle of Projective Space* with $n=2$, \begin{align*} K_{\mathbb{CP}^2}\cong \mathcal O(-2-1)=\mathcal O(-3). \end{align*} The defining polynomial of $C$ is a section of $\mathcal O(d)$ whose zero divisor is $C$, so by the divisor-line bundle construction, \begin{align*} \mathcal O(C)\cong \mathcal O(d). \end{align*} Now apply *[Adjunction Formula](/theorems/3878) for a Smooth Hypersurface*: \begin{align*} K_C\cong (K_{\mathbb{CP}^2}\otimes \mathcal O(C))|_C. \end{align*} Substituting the two identifications above gives \begin{align*} K_C\cong (\mathcal O(-3)\otimes \mathcal O(d))|_C. \end{align*} The tensor product rule for the line bundles $\mathcal O(k)$ is $\mathcal O(a)\otimes\mathcal O(b)\cong\mathcal O(a+b)$, hence \begin{align*} \mathcal O(-3)\otimes\mathcal O(d)\cong\mathcal O(d-3). \end{align*} Therefore \begin{align*} K_C\cong \mathcal O(d-3)|_C. \end{align*} For a smooth cubic, $d=3$, so \begin{align*} K_C\cong \mathcal O(3-3)|_C=\mathcal O(0)|_C\cong C\times \mathbb C. \end{align*} For a smooth quartic, $d=4$, so \begin{align*} K_C\cong \mathcal O(4-3)|_C=\mathcal O(1)|_C. \end{align*} Thus the degree of the defining plane curve shifts the ambient canonical bundle $\mathcal O(-3)$ by exactly $d$. [/example] The plane-curve calculation is the two-dimensional case of a general hypersurface computation. Once the ambient canonical bundle is known, only the degree of the defining equation remains to be inserted. This motivates the next example. [example: Hypersurfaces in Projective Space] Let $Y \subset \mathbb{CP}^n$ be a smooth hypersurface cut out by one homogeneous polynomial of degree $d$. By *Canonical Bundle of Projective Space*, \begin{align*} K_{\mathbb{CP}^n}\cong \mathcal O(-n-1). \end{align*} The defining polynomial is a holomorphic section of $\mathcal O(d)$ whose zero divisor is $Y$, so by the divisor-line bundle construction, \begin{align*} \mathcal O(Y)\cong \mathcal O(d). \end{align*} Applying *Adjunction Formula for a Smooth Hypersurface* gives \begin{align*} K_Y\cong (K_{\mathbb{CP}^n}\otimes \mathcal O(Y))|_Y. \end{align*} Substituting the two identifications above gives \begin{align*} K_Y\cong (\mathcal O(-n-1)\otimes \mathcal O(d))|_Y. \end{align*} On projective space, the tensor product rule is $\mathcal O(a)\otimes \mathcal O(b)\cong \mathcal O(a+b)$, hence \begin{align*} \mathcal O(-n-1)\otimes \mathcal O(d)\cong \mathcal O((-n-1)+d). \end{align*} Since $(-n-1)+d=d-n-1$, this gives \begin{align*} K_Y \cong \mathcal O(d-n-1)|_Y. \end{align*} Thus the integer $d-n-1$ measures the canonical bundle against the ambient hyperplane class. It is negative when $d<n+1$, zero when $d=n+1$, and positive when $d>n+1$; in the middle case, \begin{align*} K_Y\cong \mathcal O(0)|_Y\cong Y\times \mathbb C. \end{align*} [/example] ## The Poincare-Lelong Formula Divisors record zeros algebraically, while curvature records line bundles differentially. The question now is how these two descriptions see the same data. The answer is the [Poincare-Lelong formula](/theorems/3850): the curvature of a singular metric detects the current of integration over the zero divisor. [definition: Current of Integration over a Divisor] Let $X$ be a complex manifold and let $D=\sum_Y m_Y Y$ be a divisor. The current of integration over $D$ is the complex-valued linear functional \begin{align*} [D]=\sum_Y m_Y [Y], \end{align*} where each $[Y]$ is the linear functional \begin{align*} [Y]: \mathcal D^{n-1,n-1}(X) \to \mathbb C, \qquad [Y](\alpha) = \int_{Y_{\mathrm{reg}}} \alpha, \end{align*} on compactly supported smooth $(n-1,n-1)$-forms $\alpha$, integrating over the smooth locus $Y_{\mathrm{reg}}$ of $Y$. [/definition] This current is the analytic replacement for the formal divisor. It allows divisors to enter equations involving differential forms, where equalities are interpreted after testing against smooth compactly supported forms. To express such an equation as curvature, we also need curvature for metrics whose weights have logarithmic singularities. This motivates the following definition. [definition: Curvature Current of a Singular Hermitian Metric] Let $L \to X$ be a holomorphic line bundle and let $h$ be a singular Hermitian metric locally written as $|e|_h^2=e^{-\varphi}$ in a holomorphic frame $e$, where $\varphi$ is locally integrable. The curvature current of $(L,h)$ is \begin{align*} \Theta_h(L)=\partial\bar{\partial}\varphi. \end{align*} [/definition] The constants used with $\partial\bar{\partial}$ vary across the literature. In these notes, the statement below records the geometric content: the logarithmic singularity of a holomorphic section contributes exactly the divisor current, plus the smooth curvature term coming from the background metric. We now have both ingredients in place: the divisor current $[D]$ as the analytic avatar of a zero locus, and a notion of curvature that tolerates the logarithmic singularities a section's norm produces. What is still missing is the equation that ties them together — a single identity showing that the $\partial\bar\partial$ of $\log|s|_h$ splits into a singular piece supported on the zeros and a smooth piece coming from the metric. Without such a statement, divisors and curvature remain two unrelated descriptions of a line bundle. The following theorem provides the bridge. [quotetheorem:3850] [citeproof:3850] The formula says that the zero set of a holomorphic section is not separate from curvature; it is the singular part of curvature for the metric obtained by dividing by the section. Both hypotheses are doing precise work. If $s$ vanished identically there would be no divisor $D=\operatorname{div}(s)$ to integrate against and the left-hand side would be identically $-\infty$, so the requirement that $s$ be a nonzero section is what makes $[D]$ a well-defined current. The smoothness of $h$ is what isolates the singular part as exactly the divisor: $-c_1(L,h)$ is then a smooth form, so the entire singular support of $\frac{i}{\pi}\partial\bar\partial\log|s|_h$ comes from $\log|f|$ and equals $[D]$; were $h$ itself singular, its own logarithmic poles would contaminate the right-hand side and the clean splitting into "divisor plus smooth curvature" would fail. This is the analytic bridge used in later positivity, Chern class, and compactness arguments. [example: Poincare-Lelong on $\mathbb C$] For the holomorphic function $f(z)=z^m$ on $\mathbb C$, the coordinate $z$ is a local defining equation for the point $0$, and \begin{align*} f(z)=z^m\cdot 1. \end{align*} The factor $1$ is holomorphic and nonzero at $0$, so $\operatorname{ord}_0(f)=m$. At every point $a\ne 0$, the value $f(a)=a^m$ is nonzero, so the vanishing order is $0$. Hence \begin{align*} \operatorname{div}(f)=m[0]. \end{align*} Use the constant global frame of the product line bundle over $\mathbb C$ and the standard flat metric, so the metric curvature term is $0$. By *Poincare-Lelong Formula*, \begin{align*} \frac{i}{\pi}\partial\bar{\partial}\log |f|=\operatorname{div}(f). \end{align*} Substituting $f=z^m$ and $\operatorname{div}(f)=m[0]$ gives \begin{align*} \frac{i}{\pi}\partial\bar{\partial}\log |z^m|=m[0]. \end{align*} On $\mathbb C^*$, the function has no zero, and \begin{align*} |z^m|=|z|^m. \end{align*} Therefore \begin{align*} \log |z^m|=m\log |z|. \end{align*} On any simply connected open subset of $\mathbb C^*$, choose a holomorphic branch of $\log z$. Then \begin{align*} \log |z|=\frac{1}{2}(\log z+\log \bar z). \end{align*} Here $\log z$ is holomorphic and $\log\bar z$ is antiholomorphic, so \begin{align*} \partial\bar{\partial}\log |z|=0. \end{align*} Thus the current has no smooth contribution away from $0$; all of its mass is concentrated at the zero, with coefficient equal to the vanishing multiplicity $m$. [/example] The local computation on $\mathbb C$ isolates the singular contribution of a zero. On projective space, the same identity also compares the divisor current to a smooth curvature form representing the same cohomology class. This motivates the next example. [example: Hyperplane Sections] Let $H \subset \mathbb{CP}^n$ be a hyperplane, and let $s_H$ be the section of $\mathcal O(1)$ whose zero set is $H$. Since $s_H$ vanishes to order $1$ along $H$ and is nonzero away from $H$, its divisor is \begin{align*} \operatorname{div}(s_H)=H. \end{align*} Equip $\mathcal O(1)$ with the Fubini-Study metric $h_{FS}$. Applying *Poincare-Lelong Formula* to $L=\mathcal O(1)$, $h=h_{FS}$, and $s=s_H$ gives \begin{align*} \frac{i}{\pi}\partial\bar{\partial}\log |s_H|_{h_{FS}} = [\operatorname{div}(s_H)]-c_1(\mathcal O(1),h_{FS}). \end{align*} Substituting $\operatorname{div}(s_H)=H$ yields \begin{align*} \frac{i}{\pi}\partial\bar{\partial}\log |s_H|_{h_{FS}} = [H]-c_1(\mathcal O(1),h_{FS}). \end{align*} Rearranging this identity gives \begin{align*} [H]-c_1(\mathcal O(1),h_{FS})=\frac{i}{\pi}\partial\bar{\partial}\log |s_H|_{h_{FS}}. \end{align*} The right-hand side is $\partial\bar{\partial}$-exact as a current, so it represents the zero cohomology class. Therefore \begin{align*} [H]=c_1(\mathcal O(1),h_{FS}) \end{align*} in cohomology. Thus the current of integration over a hyperplane and the Fubini-Study curvature form are two representatives of the same hyperplane class. [/example] ## Summary of the Chapter Divisors encode codimension-one zero and pole data, and principal divisors are the divisors of meromorphic functions. The construction $D \mapsto \mathcal O(D)$ turns this data into holomorphic line bundles, with effective divisors giving distinguished holomorphic sections. The canonical bundle records holomorphic top forms, while adjunction computes the canonical bundle of a smooth hypersurface by correcting the ambient canonical bundle with the divisor line bundle. The Poincare-Lelong formula completes the circle by identifying divisor currents with curvature terms, linking meromorphic data, line bundles, and Hermitian geometry. By now the course has assembled the analytic, bundle-theoretic, and curvature-based tools needed for the global theory. The final chapter uses them together to show how local holomorphic structure controls global geometric and topological features of complex manifolds. # 12. Synthesis: From Local Complex Analysis to Global Geometry These notes have built the basic language of complex manifolds, holomorphic vector bundles, Hermitian metrics, and curvature. This final chapter gathers those threads into the main goal of the course: to explain how local holomorphic analysis produces global geometric and topological information. The prerequisites are the Dolbeault decomposition from Chapters 2 and 3, holomorphic bundle transition functions from Chapter 4, Chern connections from Chapter 8, and de Rham cohomology. Projective space shows how a canonical metric produces positive curvature and computable Chern classes, while compact complex tori show the opposite extreme: compact complex manifolds can have flat tangent bundle and vanishing Chern classes. ## Comparing the Three Complexes The first problem is to keep track of which differential is being used. A smooth manifold has the de Rham complex, while a complex manifold has a type decomposition that separates holomorphic and antiholomorphic directions. We need the de Rham complex first because every later Dolbeault construction is a refinement of this underlying smooth complex. [definition: De Rham Complex] Let $X$ be a smooth manifold. The de Rham complex of $X$ is the cochain complex \begin{align*} 0 \longrightarrow A^0(X;\mathbb C) \xrightarrow{d} A^1(X;\mathbb C) \xrightarrow{d} A^2(X;\mathbb C) \xrightarrow{d} \cdots, \end{align*} where $A^k(X;\mathbb C)$ denotes smooth complex-valued $k$-forms and $d^2=0$. [/definition] The de Rham complex ignores the distinction between $dz_i$ and $d\bar z_i$, so it cannot by itself detect holomorphicity. On a complex manifold, the obstruction is that ordinary differential forms mix holomorphic and antiholomorphic directions, while the equation for a holomorphic function is specifically a $\bar\partial$-equation. Separating forms by type produces the complex that measures this antiholomorphic failure. [definition: Dolbeault Complex] Let $X$ be a complex manifold. For each $p \ge 0$, the Dolbeault complex in holomorphic degree $p$ is \begin{align*} 0 \longrightarrow A^{p,0}(X) \xrightarrow{\bar{\partial}} A^{p,1}(X) \xrightarrow{\bar{\partial}} A^{p,2}(X) \xrightarrow{\bar{\partial}} \cdots, \end{align*} where $A^{p,q}(X)$ denotes smooth forms of type $(p,q)$ and $\bar{\partial}^2=0$. [/definition] Because $\bar\partial$ was introduced through local coordinates, there is a genuine compatibility question on overlaps. If a change of coordinates mixed $dz$ and $d\bar z$ terms, the decomposition of $d$ into type components would not be globally meaningful. The result below is the integrability statement that rules out this problem on complex manifolds and gives the algebraic identities needed for a cochain complex. [quotetheorem:7004] [citeproof:7004] The hypotheses matter here in two ways. The complex manifold structure is what makes coordinate changes holomorphic, so the subspaces spanned by the $dz_i$ and by the $d\bar z_i$ are preserved on overlaps; on a merely smooth real manifold there is no invariant splitting into these two types. The identity $d^2=0$ is also doing real work: it is what forces $\partial^2=0$, $\bar{\partial}^2=0$, and the anticommutation relation rather than just giving two unrelated first-order operators. The theorem does not yet compute any cohomology, but it gives the differential algebra on which Dolbeault cohomology and holomorphic bundle theory are built. The theorem explains how complex coordinates refine de Rham theory, but examples are needed before moving to bundles because the formulas can otherwise look only formal. The next example grounds the operators in the affine model that underlies every chart. [example: The Operators on Complex Affine Space] On $\mathbb C^n$ write $z_j=x_j+iy_j$ and $\bar z_j=x_j-iy_j$. Then \begin{align*} dz_j=dx_j+i\,dy_j,\qquad d\bar z_j=dx_j-i\,dy_j. \end{align*} Adding and subtracting these identities gives \begin{align*} dx_j=\frac{1}{2}(dz_j+d\bar z_j),\qquad dy_j=\frac{1}{2i}(dz_j-d\bar z_j). \end{align*} For a smooth function $f:\mathbb C^n\to\mathbb C$, the ordinary real differential is \begin{align*} df=\sum_{j=1}^n \frac{\partial f}{\partial x_j}\,dx_j+\sum_{j=1}^n \frac{\partial f}{\partial y_j}\,dy_j. \end{align*} Substituting the formulas for $dx_j$ and $dy_j$ into this expression gives \begin{align*} df=\sum_{j=1}^n \left(\frac{1}{2}\frac{\partial f}{\partial x_j}+\frac{1}{2i}\frac{\partial f}{\partial y_j}\right)dz_j+\sum_{j=1}^n \left(\frac{1}{2}\frac{\partial f}{\partial x_j}-\frac{1}{2i}\frac{\partial f}{\partial y_j}\right)d\bar z_j. \end{align*} Since $1/i=-i$, the coefficients are \begin{align*} \frac{\partial f}{\partial z_j}=\frac{1}{2}\left(\frac{\partial f}{\partial x_j}-i\frac{\partial f}{\partial y_j}\right),\qquad \frac{\partial f}{\partial \bar z_j}=\frac{1}{2}\left(\frac{\partial f}{\partial x_j}+i\frac{\partial f}{\partial y_j}\right). \end{align*} Therefore \begin{align*} df=\sum_{j=1}^n \frac{\partial f}{\partial z_j}\,dz_j+\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\,d\bar z_j. \end{align*} The $(0,1)$ part is \begin{align*} \bar{\partial}f=\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}\,d\bar z_j, \end{align*} so $\bar{\partial}f=0$ exactly means $\partial f/\partial\bar z_j=0$ for every $j$. In one variable, if $f=u+iv$, this equation is \begin{align*} 0=\frac{\partial f}{\partial \bar z}=\frac{1}{2}\left(u_x+iv_x+i u_y-v_y\right)=\frac{1}{2}\left((u_x-v_y)+i(v_x+u_y)\right), \end{align*} which is equivalent to $u_x=v_y$ and $u_y=-v_x$, the Cauchy-Riemann equations. Thus $\bar{\partial}$ extracts precisely the antiholomorphic part of the differential, and holomorphicity is detected locally by the vanishing of those coefficients. [/example] Scalar forms are not enough for geometry because holomorphic vector bundles carry sections whose coefficients change by transition matrices. This creates the next problem: we need an antiholomorphic differential that differentiates coefficients but also respects holomorphic changes of frame. [definition: Bundle-Valued Dolbeault Complex] Let $E \to X$ be a holomorphic vector bundle over a complex manifold. The bundle-valued Dolbeault complex is \begin{align*} 0 \longrightarrow A^{p,0}(X,E) \xrightarrow{\bar{\partial}_E} A^{p,1}(X,E) \xrightarrow{\bar{\partial}_E} A^{p,2}(X,E) \xrightarrow{\bar{\partial}_E} \cdots, \end{align*} where $A^{p,q}(X,E)$ denotes smooth $E$-valued forms of type $(p,q)$ and $\bar{\partial}_E^2=0$. [/definition] For the bundle-valued complex to be meaningful, its degree-zero kernel must recover the holomorphic sections one started with. The possible obstruction is that $\bar\partial_E s=0$ is written after choosing holomorphic frames, while holomorphicity of a section is a coordinate-free gluing condition. The result below identifies these two descriptions and shows that no extra flatness condition has been introduced. [quotetheorem:7047] [citeproof:7047] The holomorphic-frame hypothesis is essential: in a non-holomorphic smooth frame, the equation for the coefficient functions would acquire extra transition terms and would no longer be the scalar equation $\bar{\partial}s_i=0$. The theorem also clarifies what degree zero remembers: the kernel of $\bar{\partial}_E$ is not an arbitrary space of flat sections, but precisely the original sheaf of holomorphic sections. It does not say that higher-degree $\bar{\partial}_E$-closed forms are globally exact; the failure of exactness in positive degree is exactly the cohomological information used later. Once holomorphic sections are characterized by a $\bar\partial_E$ equation, standard bundle operations become a source of possible inconsistency. Tensor products, duals, exterior powers, and pullbacks all change local frames and coefficient functions, so the differential equation must transform with the expected Leibniz signs and type restrictions. The following compatibility result is what allows Dolbeault complexes to be used functorially in these constructions. [quotetheorem:7048] [citeproof:7048] This compatibility explains why Dolbeault complexes behave functorially, but the hypotheses are substantial. If $f:Y\to X$ is only smooth, then $f^*$ need not preserve type: the pullback of a $(0,1)$-form on $X$ can acquire a $(1,0)$ component on $Y$, so the displayed identity with $\bar{\partial}$ has no reason to hold. Likewise, the sign in the tensor formula is the ordinary graded sign coming from the total form degree; omitting it would make the formula false already for scalar-valued forms of odd total degree. The theorem also has a precise limitation. It says that the differential operators commute with standard holomorphic bundle operations; it does not say that the induced maps are isomorphisms on cohomology, nor that a smooth splitting of a holomorphic construction becomes holomorphic. Those stronger conclusions require additional exactness or vanishing input. Before using the compatibility in exact sequences, we need to name the cohomology measured by these complexes. [definition: Dolbeault Cohomology with Coefficients] Let $E\to X$ be a holomorphic vector bundle over a complex manifold. The Dolbeault cohomology of $X$ with coefficients in $E$ is \begin{align*} H^q(X,E)=\frac{\ker\left(\bar{\partial}_E:A^{0,q}(X,E)\to A^{0,q+1}(X,E)\right)} {\operatorname{im}\left(\bar{\partial}_E:A^{0,q-1}(X,E)\to A^{0,q}(X,E)\right)} \end{align*} for $q\ge 0$, with the denominator taken to be $0$ when $q=0$. [/definition] This definition records the failure of the bundle-valued Dolbeault complex to be exact in degree $q$. The next example is needed to show how a short exact sequence of bundles produces a cohomological connecting map rather than merely a sequence of vector spaces. [example: Short Exact Sequence and Dolbeault Cohomology] Let \begin{align*} 0\longrightarrow E'\xrightarrow{i}E\xrightarrow{p}E''\longrightarrow 0 \end{align*} be a short exact sequence of holomorphic vector bundles on $X$. For each $q$, applying smooth $(0,q)$-forms gives \begin{align*} 0\longrightarrow A^{0,q}(X,E')\xrightarrow{i}A^{0,q}(X,E)\xrightarrow{p}A^{0,q}(X,E'')\longrightarrow 0. \end{align*} This sequence is exact because a smooth local splitting of $p$ lifts every $E''$-valued form to an $E$-valued form, while $\ker p=\operatorname{im} i$ fibrewise. Since $i$ and $p$ are holomorphic bundle maps, the Dolbeault operators commute with them: \begin{align*} \bar{\partial}_E(i\eta)=i(\bar{\partial}_{E'}\eta),\qquad p(\bar{\partial}_E\beta)=\bar{\partial}_{E''}(p\beta). \end{align*} Now let $[\alpha'']\in H^q(X,E'')$, represented by a $\bar{\partial}_{E''}$-closed form $\alpha''\in A^{0,q}(X,E'')$. Choose a smooth lift $\beta\in A^{0,q}(X,E)$ with $p\beta=\alpha''$. Then \begin{align*} p(\bar{\partial}_E\beta)=\bar{\partial}_{E''}(p\beta)=\bar{\partial}_{E''}\alpha''=0. \end{align*} Thus $\bar{\partial}_E\beta$ lies in $\ker p=\operatorname{im} i$, so there is a unique $\gamma\in A^{0,q+1}(X,E')$ such that \begin{align*} i\gamma=\bar{\partial}_E\beta. \end{align*} This $\gamma$ is closed, because \begin{align*} i(\bar{\partial}_{E'}\gamma)=\bar{\partial}_E(i\gamma)=\bar{\partial}_E^2\beta=0, \end{align*} and $i$ is injective. Therefore $\gamma$ defines a class $[\gamma]\in H^{q+1}(X,E')$. If another lift is chosen, it has the form $\beta+i\eta$ for some $\eta\in A^{0,q}(X,E')$. Then \begin{align*} \bar{\partial}_E(\beta+i\eta)=\bar{\partial}_E\beta+i(\bar{\partial}_{E'}\eta)=i(\gamma+\bar{\partial}_{E'}\eta), \end{align*} so the resulting class in $H^{q+1}(X,E')$ is still $[\gamma]$. Hence the rule \begin{align*} \delta([\alpha''])=[\gamma] \end{align*} is a well-defined connecting homomorphism \begin{align*} \delta:H^q(X,E'')\longrightarrow H^{q+1}(X,E'). \end{align*} Splicing these connecting maps with the maps induced by $i$ and $p$ gives the [long exact cohomology sequence](/theorems/3471), so bundle-valued Dolbeault cohomology records the same extension behavior expected from sheaf cohomology of holomorphic sections. [/example] ## Hermitian Metrics and Curvature The second problem is that a holomorphic structure gives only the $(0,1)$ part of a connection. Curvature requires a full connection, but a random completion would not reflect the holomorphic geometry. A Hermitian metric supplies the missing condition by demanding compatibility with fibrewise inner products. [definition: Hermitian Metric on a Holomorphic Vector Bundle] Let $E\to X$ be a holomorphic vector bundle. A Hermitian metric on $E$ is a smooth choice of Hermitian inner product $h_x$ on each fibre $E_x$, depending smoothly on $x\in X$. [/definition] A metric alone is still not a curvature object, and the holomorphic structure alone specifies only the antiholomorphic part $\bar\partial_E$. The obstruction is non-uniqueness: many connections could extend $\bar\partial_E$, and most would not interact correctly with the Hermitian inner products. Imposing metric compatibility singles out the canonical extension whose curvature belongs to the holomorphic Hermitian geometry. [definition: Chern Connection] Let $(E,h)\to X$ be a Hermitian holomorphic vector bundle. The Chern connection is the unique $\mathbb C$-linear operator \begin{align*} \nabla^E:C^\infty(X,E)\longrightarrow A^1(X,E) \end{align*} satisfying the Leibniz rule \begin{align*} \nabla^E(fs)=df\otimes s+f\nabla^E s \end{align*} for all $f\in C^\infty(X;\mathbb C)$ and $s\in C^\infty(X,E)$, such that its type components \begin{align*} (\nabla^E)^{1,0}:C^\infty(X,E)\to A^{1,0}(X,E),\qquad (\nabla^E)^{0,1}:C^\infty(X,E)\to A^{0,1}(X,E) \end{align*} satisfy \begin{align*} (\nabla^E)^{0,1}=\bar{\partial}_E \end{align*} and \begin{align*} d\,h(s,t)=h(\nabla^E s,t)+h(s,\nabla^E t) \end{align*} for all smooth sections $s,t$ of $E$. [/definition] The intrinsic definition does not yet tell us how to compute the connection from a local metric matrix. In a holomorphic frame the $(0,1)$ part is already fixed, but the $(1,0)$ part must be recovered from metric compatibility without violating the connection transformation law. The local formula below solves this reconstruction problem and gives the corresponding curvature matrix. [quotetheorem:7049] [citeproof:7049] The holomorphic-frame assumption is what makes the formula so simple: all $(0,1)$ connection terms disappear because the holomorphic structure has already been absorbed into $\bar{\partial}_E$. The metric matrix $H$ is not itself invariant under change of frame, but the transformation law of $H^{-1}\partial H$ is exactly the connection transformation law, so the locally computed operator is global. The theorem is computational rather than topological: it gives curvature for a chosen metric, while Chern-Weil theory below explains which part of that curvature survives metric changes. For a line bundle, the matrix formula should collapse to a scalar expression, but there is still a normalization issue: a local frame can be rescaled by a nowhere-zero holomorphic function, changing the metric weight. A useful curvature formula must be insensitive to that rescaling after applying $\partial\bar\partial$. The result below gives this frame-compatible scalar formula, which is the one used in positivity and projective-space calculations. [quotetheorem:7050] [citeproof:7050] For a line bundle, the scalar curvature formula is easy to compute locally, but it still depends on the chosen Hermitian metric and on the convention used to turn curvature into a de Rham representative. Without a fixed normalization, the same geometric object would acquire incompatible factors of $2\pi$ or $i$ in later comparisons with integral Chern classes and projective-space generators. The obstruction is therefore not computation but interpretation: curvature is a differential form, while the invariant we want is a normalized cohomology class attached to the line bundle. The following definition fixes the standard normalization so that subsequent metric-change and positivity statements all refer to the same representative convention. [definition: First Chern Form of a Hermitian Line Bundle] Let $(L,h)\to X$ be a Hermitian holomorphic line bundle with Chern curvature $\Theta_L$. The first Chern form of $(L,h)$ is \begin{align*} c_1(L,h)=\frac{i}{2\pi}\Theta_L. \end{align*} [/definition] The first Chern form is defined from a metric, so it is not yet clear that it belongs to the holomorphic line bundle rather than to the auxiliary choice of Hermitian norm. If two metrics produce unrelated cohomology classes, curvature would be a local analytic measurement but not a stable characteristic class. The point to check is that changing the metric changes the normalized curvature only by an exact form, leaving a well-defined de Rham class. [quotetheorem:3839] [citeproof:3839] The comparison of two metrics uses the fact that their ratio is a globally defined positive function on the base, which is special to line bundles and keeps the argument elementary. The exactness conclusion says that the differential form representative changes, but its de Rham class does not; this is the first instance of the Chern-Weil principle in the chapter. It does not say that the curvature itself is metric-independent, and in positivity questions the choice of metric remains decisive. This theorem explains why local curvature computations can produce global classes, but the frame-change mechanism is worth seeing explicitly. The next example is needed because the same calculation reappears in the construction of the Fubini-Study form. [example: Curvature Under a Change of Frame] Let $L\to X$ be a holomorphic line bundle, and on an overlap let two holomorphic frames satisfy $e'=ge$, where $g$ is nowhere-vanishing and holomorphic. If $h(e,e)=e^{-\varphi}$, then by Hermitian sesquilinearity, \begin{align*} h(e',e')=h(ge,ge)=g\bar g\,h(e,e)=|g|^2e^{-\varphi}. \end{align*} Writing this again as $h(e',e')=e^{-\varphi'}$ gives \begin{align*} e^{-\varphi'}=|g|^2e^{-\varphi}=e^{\log |g|^2-\varphi}=e^{-(\varphi-\log |g|^2)}. \end{align*} Hence \begin{align*} \varphi'=\varphi-\log |g|^2. \end{align*} The local curvature expression in the new frame is therefore \begin{align*} \bar{\partial}\partial\varphi'=\bar{\partial}\partial\varphi-\bar{\partial}\partial\log |g|^2. \end{align*} It remains to compute the second term. Since $g$ is holomorphic, $\bar{\partial}g=0$, and since $g$ is nowhere-vanishing, \begin{align*} \partial\log |g|^2=\partial\log(g\bar g)=\frac{\partial g}{g}+\frac{\partial\bar g}{\bar g}=\frac{\partial g}{g}. \end{align*} Here $\partial\bar g=0$ because $\bar g$ depends antiholomorphically. Now \begin{align*} \bar{\partial}\left(\frac{\partial g}{g}\right)=\bar{\partial}(g^{-1})\wedge \partial g+g^{-1}\bar{\partial}\partial g. \end{align*} The first term vanishes because \begin{align*} \bar{\partial}(g^{-1})=-g^{-2}\bar{\partial}g=0. \end{align*} The second term vanishes because $\bar{\partial}\partial g=-\partial\bar{\partial}g=0$. Thus \begin{align*} \bar{\partial}\partial\log |g|^2=0, \end{align*} and consequently \begin{align*} \bar{\partial}\partial\varphi'=\bar{\partial}\partial\varphi. \end{align*} So changing a holomorphic frame changes the local metric weight by the pluriharmonic function $\log |g|^2$, whose mixed derivative vanishes; this is why the curvature forms computed in different holomorphic frames agree on overlaps. [/example] ## Chern-Weil Classes and Global Invariants The third problem is to pass from a single line bundle to arbitrary rank. Curvature is now matrix-valued, so taking its entries separately would depend on the frame. We need invariant polynomials in the curvature matrix, and the determinant is the polynomial that packages all Chern classes at once. [definition: Total Chern Form] Let $(E,h)\to X$ be a Hermitian holomorphic vector bundle of rank $r$, and let $\Theta_E$ be the curvature matrix of its Chern connection in a local holomorphic frame. The total Chern form of $(E,h)$ is the inhomogeneous differential form \begin{align*} c(E,h)=\det\left(I+\frac{i}{2\pi}\Theta_E\right)=1+c_1(E,h)+\cdots+c_r(E,h). \end{align*} [/definition] The determinant expression is designed to survive a change of holomorphic frame, but frame invariance alone is not enough for a characteristic class. The coefficients must also be closed forms, and their cohomology classes must remain unchanged when the Hermitian metric, hence the Chern connection and curvature matrix, is varied. The theorem below supplies exactly these two missing facts: closedness and metric independence for the determinant coefficients. [quotetheorem:7051] [citeproof:7051] The theorem depends on two pieces of structure: a Chern connection, which produces curvature satisfying the Bianchi identity, and an invariant polynomial, which removes dependence on the chosen holomorphic frame. If the determinant coefficients are replaced by a non-invariant polynomial in the matrix entries, the resulting expression changes under a holomorphic frame change by conjugation and therefore does not define a global form on the bundle. If the connection structure is discarded and an arbitrary matrix-valued $2$-form is substituted for curvature, the Bianchi identity is absent; for example, on a coordinate ball a matrix-valued $2$-form $A$ with $\operatorname{tr}(A)=x_1\,dx_2\wedge dx_3$ satisfies $d\operatorname{tr}(A)=dx_1\wedge dx_2\wedge dx_3$. Thus both actual hypotheses of the Chern-Weil construction are doing work: invariant polynomials give global frame-independent expressions, and curvature of a connection gives the closedness mechanism. There is also a limitation in the other direction. The theorem identifies de Rham classes represented by curvature polynomials, but it does not classify holomorphic bundles and it does not make the curvature form itself canonical. Different Hermitian metrics generally produce different representatives of the same class, and bundles with the same Chern classes can still be non-isomorphic. Metric independence is therefore a statement about the cohomology class, not about the full differential-geometric data carried by the connection. Chern-Weil theory turns curvature into classes, but it does not by itself explain how those classes behave when a bundle is built from smaller pieces. In examples such as projective space, tangent bundles and tautological bundles are related by short exact sequences rather than by isolated formulas. To compute with such sequences, one needs a rule saying how the total Chern class of the middle bundle decomposes into the classes of the subbundle and quotient. [quotetheorem:7052] [citeproof:7052] The hypotheses in the Whitney formula are restrictive in exactly the way computations require. Exactness says that $E''$ is the quotient bundle $E/E'$ and not merely another bundle of complementary rank; without this relation, ranks alone give no multiplicative relation among Chern classes. The holomorphic vector-bundle condition is also part of the input to the Chern connection and the Dolbeault-compatible curvature forms used here. A concrete failure mode is to replace a short exact sequence by an arbitrary smooth splitting $E\cong E'\oplus F$ that does not respect a chosen holomorphic quotient: the smooth direct-sum decomposition does not identify the holomorphic bundle $F$ with $E''$, so the Chern-class computation no longer describes the original holomorphic extension. The Whitney formula explains how to manipulate Chern classes algebraically, but it does not yet say what geometric information these classes carry. The most exposed test is the top Chern class of a tangent bundle: it has the correct degree to integrate over a compact complex manifold, so it should produce a number. The result below identifies that number with the Euler characteristic, showing that the top Chern class records a classical topological invariant rather than just a formal coefficient. [quotetheorem:7053] This is the Chern-Gauss-Bonnet theorem in the complex setting. It explains why the top Chern class is more than a formal coefficient: for tangent bundles, it recovers a classical topological invariant. ## The Computation Package for Projective Space The fourth problem is to test the abstract machinery on a compact example where every object is explicit. Projective space supplies compatible charts, canonical line bundles, a natural Hermitian metric, and a tangent-bundle exact sequence. We begin with the space itself because all later bundles are defined from its homogeneous coordinates. [definition: Complex Projective Space] Complex projective space $\mathbb{CP}^n$ is the set of complex lines through the origin in $\mathbb C^{n+1}$: \begin{align*} \mathbb{CP}^n=\left(\mathbb C^{n+1}\setminus\{0\}\right)/\mathbb C^*, \end{align*} where $\lambda\in\mathbb C^*$ acts by scalar multiplication. Homogeneous coordinates are written $[Z_0:\cdots:Z_n]$. [/definition] The standard affine charts $U_j=\{Z_j\ne 0\}$ make the quotient into a complex manifold, but vector bundles over it require compatible fibre data. The next definition is needed because the tautological bundle records the line represented by a projective point and generates the basic projective line bundles. [definition: Tautological Line Bundle] The tautological line bundle $\mathcal O(-1)\to \mathbb{CP}^n$ is the holomorphic line bundle whose fibre over $[Z]\in\mathbb{CP}^n$ is the line $\mathbb C Z\subset\mathbb C^{n+1}$. [/definition] Its dual is the hyperplane bundle $\mathcal O(1)$, and the transition functions are the first place where homogeneous coordinates become bundle data. The next example is needed because it verifies holomorphicity and fixes the convention for $\mathcal O(1)$ used in the Chern class computation. [example: Transition Functions of Projective Line Bundles] On $U_j=\{Z_j\ne 0\}$, define a holomorphic frame $e_j$ of $\mathcal O(-1)$ by \begin{align*} e_j([Z])=\frac{1}{Z_j}(Z_0,\dots,Z_n)\in \mathbb C Z. \end{align*} On $U_j\cap U_k$, the corresponding $k$-frame is \begin{align*} e_k([Z])=\frac{1}{Z_k}(Z_0,\dots,Z_n). \end{align*} Since the two vectors are scalar multiples of the same nonzero vector $Z=(Z_0,\dots,Z_n)$, we compare them by factoring $1/Z_k$ through $1/Z_j$: \begin{align*} e_k([Z])=\frac{1}{Z_k}Z=\frac{Z_j}{Z_k}\frac{1}{Z_j}Z=\frac{Z_j}{Z_k}e_j([Z]). \end{align*} Thus, with the convention $e_k=g_{kj}e_j$, the transition function for $\mathcal O(-1)$ from the $j$-frame to the $k$-frame is \begin{align*} g_{kj}=\frac{Z_j}{Z_k}. \end{align*} This function is holomorphic and nowhere zero on $U_j\cap U_k$ because both $Z_j$ and $Z_k$ are nonzero there. Let $e_j^*$ be the dual frame of $\mathcal O(1)$, so $e_j^*(e_j)=1$. If $e_k=g_{kj}e_j$, then \begin{align*} e_k^*=\frac{1}{g_{kj}}e_j^* \end{align*} because \begin{align*} \left(\frac{1}{g_{kj}}e_j^*\right)(e_k)=\left(\frac{1}{g_{kj}}e_j^*\right)(g_{kj}e_j)=e_j^*(e_j)=1. \end{align*} Therefore the transition function for $\mathcal O(1)$ is \begin{align*} g_{kj}^{-1}=\frac{Z_k}{Z_j}. \end{align*} The ratios $Z_j/Z_k$ and $Z_k/Z_j$ are holomorphic on the overlap, so both line bundles are holomorphic; passing to the dual replaces each transition function by its inverse. [/example] Transition functions describe the holomorphic bundle, but they do not provide a curvature form until a metric has been chosen. On projective space the metric must be written in affine coordinates while still respecting the homogeneous scaling that defines points of $\mathbb{CP}^n$. The local function below is the standard logarithmic weight that packages this scaling-invariant metric data on the chart $U_0$. [definition: Fubini-Study Potential] On the affine chart $U_0\cong\mathbb C^n$ with coordinates $z=(z_1,\dots,z_n)$, the Fubini-Study potential is the smooth function $\rho:U_0\to\mathbb R$ defined by \begin{align*} \rho(z)=\log(1+|z_1|^2+\cdots+|z_n|^2). \end{align*} [/definition] The potential itself is chart-dependent, so it cannot be the global object on $\mathbb{CP}^n$. The obstruction is that a projective point has many homogeneous representatives, and changing affine charts changes the logarithmic weight by terms coming from holomorphic transition functions. The invariant object should therefore be a differential form that ignores these harmless changes. Applying $\partial\bar{\partial}$ is precisely the operation that removes them and produces the canonical projective curvature form. [definition: Fubini-Study Form] The Fubini-Study form on $\mathbb{CP}^n$ is the global differential form $\omega_{FS}\in A^{1,1}(\mathbb{CP}^n)$ whose representative on the affine chart $U_0\cong\mathbb C^n$ is \begin{align*} \omega_{FS}=\frac{i}{2\pi}\partial\bar{\partial}\log(1+|z_1|^2+\cdots+|z_n|^2). \end{align*} [/definition] The form has been constructed analytically from a local potential, but its role in the Chern-Weil package is not settled until we know which line-bundle class it represents. The possible ambiguity is a sign and normalization issue: $\mathcal O(1)$ and $\mathcal O(-1)$ have opposite generators, and curvature conventions determine the factor multiplying $\partial\bar\partial\rho$. The theorem below fixes that identification by matching the Fubini-Study form with the hyperplane bundle's first Chern class. [quotetheorem:3848] [citeproof:3848] The standard metric and sign convention are part of the statement, not decoration. Replacing the metric changes the curvature form by an exact term, so the cohomology class remains $c_1(\mathcal O(1))$ but the displayed local representative need not be the Fubini-Study form. Reversing the convention between $\mathcal O(1)$ and $\mathcal O(-1)$ also reverses the sign of the generator. The result therefore identifies a particular geometric representative of the hyperplane class, rather than saying every curvature representative has the same positivity properties. The hyperplane class controls line bundles, but it does not by itself describe tangent vectors to projective space. The obstruction is the artificial radial direction in homogeneous coordinates: a vector in $\mathbb C^{n+1}$ can change the representative of a point without moving the projective point. To compute the tangent bundle, we need a global exact sequence that separates this radial direction from the genuine projective directions and expresses the answer using $\mathcal O(1)$. [quotetheorem:7018] [citeproof:7018] The exactness is doing more than counting ranks. The left-hand copy of $\mathcal O$ records the radial direction coming from scaling homogeneous coordinates, while the quotient records directions that genuinely move the projective point; the twist by $\mathcal O(1)$ is what turns the homogeneous coordinate derivatives into globally compatible bundle maps. This distinction prevents the sequence from being confused with the untwisted quotient description on $\mathbb C^{n+1}\setminus\{0\}$. Once the sequence is global and holomorphic, Whitney multiplicativity can convert it directly into Chern class identities. Once the Euler sequence is available, the tangent-bundle computation is no longer a question of choosing coordinates on each chart; it becomes an application of Whitney multiplicativity in the [cohomology ring](/theorems/2271) of projective space. The only remaining issue is to translate the exact sequence into powers of the hyperplane class $H$ and then discard terms above the top degree. The following result records the resulting total Chern class and the first Chern class used later in positivity and curvature-sign arguments. [quotetheorem:7054] This computation shows how exact sequences turn local projective data into global characteristic classes, but the formal expression has to be read inside $H^*(\mathbb{CP}^n;\mathbb C)$. The equality does not mean that powers above $H^n$ survive; those terms vanish because there is no cohomology above real degree $2n$. The Euler sequence is also essential: using only the rank $n$ of $T\mathbb{CP}^n$ would not determine its Chern classes, and replacing the sequence by a non-exact rank count would lose the radial direction that has been quotiented out. The next example is needed because $\mathbb{CP}^1$ lets us compare the Chern-Weil calculation with the familiar Euler characteristic of the sphere. [example: Projective Line] For $\mathbb{CP}^1$, the *Total Chern Class of Projective Space* formula gives \begin{align*} c(T\mathbb{CP}^1)=(1+H)^2. \end{align*} Expanding the square gives \begin{align*} (1+H)^2=1+2H+H^2. \end{align*} Since $\mathbb{CP}^1$ has complex dimension $1$, its real cohomology vanishes in degrees greater than $2$, so $H^2\in H^4(\mathbb{CP}^1;\mathbb C)$ is zero. Hence \begin{align*} c(T\mathbb{CP}^1)=1+2H. \end{align*} The degree-two part is therefore \begin{align*} c_1(T\mathbb{CP}^1)=2H. \end{align*} Using the normalization $\int_{\mathbb{CP}^1}H=1$, we get \begin{align*} \int_{\mathbb{CP}^1}c_1(T\mathbb{CP}^1)=\int_{\mathbb{CP}^1}2H=2\int_{\mathbb{CP}^1}H=2. \end{align*} This agrees with the *[Chern-Gauss-Bonnet Interpretation of the Top Chern Class](/theorems/7053)*, since $\mathbb{CP}^1$ is the Riemann sphere and has Euler characteristic $2$. In the affine coordinate $z$ on $U_0\subset\mathbb{CP}^1$, the Fubini-Study potential is \begin{align*} \rho(z)=\log(1+|z|^2). \end{align*} The Fubini-Study form is \begin{align*} \omega_{FS}=\frac{i}{2\pi}\partial\bar{\partial}\rho. \end{align*} By the *First Chern Class of the Hyperplane Bundle* computation, $[\omega_{FS}]=H$. Therefore \begin{align*} 2[\omega_{FS}]=2H=c_1(T\mathbb{CP}^1). \end{align*} Thus the Chern-Ricci representative of the tangent-bundle first Chern class lies in the same cohomology class as $2\omega_{FS}$, so on $\mathbb{CP}^1$ the curvature computation detects the nonzero topology measured by $\chi(\mathbb{CP}^1)=2$. [/example] ## Flat Complex Tori and the Opposite Extreme Projective space has positive curvature built into its hyperplane bundle, but compactness alone does not force positive curvature. The final example asks what happens when a compact complex manifold is built by quotienting affine space by translations. Because translations have identity derivative, the tangent bundle should retain a global constant frame. [definition: Compact Complex Torus] A compact complex torus of complex dimension $n$ is a quotient \begin{align*} X=\mathbb C^n/\Lambda, \end{align*} where $\Lambda\subset\mathbb C^n$ is a lattice of rank $2n$ over $\mathbb Z$. [/definition] The quotient inherits local holomorphic coordinates from $\mathbb C^n$, but local coordinates alone do not decide whether the tangent bundle is globally trivial. The possible obstruction is monodromy around lattice loops: after descending to the quotient, a coordinate vector field could fail to return as the same tangent vector if the deck transformations had nontrivial derivatives. For a complex torus the deck transformations are translations, so the point to check is whether their identity derivatives let the constant frame on $\mathbb C^n$ descend globally. [quotetheorem:7055] [citeproof:7055] The lattice hypothesis is essential because the quotient maps are translations, whose derivatives are the identity. For a general quotient by holomorphic automorphisms, the derivative action could twist tangent vectors and produce a nontrivial tangent bundle even if the universal cover is simple. The theorem therefore isolates the precise reason complex tori are flat: the affine coordinate vector fields are globally well-defined after quotienting. This makes tori the natural counterpoint to projective space, where homogeneous scaling creates nontrivial tangent Chern classes. The theorem gives the higher-dimensional picture, and the one-dimensional case connects it to the classical language of elliptic curves. The next example is needed to contrast the torus with $\mathbb{CP}^1$ using the same invariant $c_1(TX)$. [example: Elliptic Curve as a Flat Complex Torus] Let $E=\mathbb C/\Lambda$ and let $\pi:\mathbb C\to E$ be the quotient map. For each $\lambda\in\Lambda$, write $\tau_\lambda(z)=z+\lambda$. Since $\pi\circ\tau_\lambda=\pi$, differentiating gives \begin{align*} d\pi_{z+\lambda}\circ d(\tau_\lambda)_z=d\pi_z. \end{align*} But $d(\tau_\lambda)_z(\partial_z)=\partial_z$, so \begin{align*} d\pi_{z+\lambda}(\partial_z)=d\pi_z(\partial_z). \end{align*} Therefore the formula \begin{align*} v_{[z]}=d\pi_z(\partial_z) \end{align*} is independent of the chosen representative $z$ of $[z]\in E$ and defines a global holomorphic section $v$ of $T^{1,0}E$. The section $v$ is nowhere zero because $\pi$ is locally a biholomorphism, so each map $d\pi_z:T_z\mathbb C\to T_{[z]}E$ is an isomorphism. Hence \begin{align*} E\times\mathbb C\longrightarrow T^{1,0}E,\qquad ([z],a)\longmapsto a\,v_{[z]} \end{align*} is a holomorphic vector-bundle isomorphism. In this global frame, take the constant Hermitian metric with matrix $H=1$. Then \begin{align*} \theta=H^{-1}\partial H=1^{-1}\partial(1)=0, \end{align*} and hence \begin{align*} \Theta=\bar{\partial}\theta=\bar{\partial}(0)=0. \end{align*} Thus \begin{align*} c_1(T^{1,0}E)=\left[\frac{i}{2\pi}\Theta\right]=0. \end{align*} Topologically, a real two-dimensional torus has a cell decomposition with one $0$-cell, two $1$-cells, and one $2$-cell, so $\chi(E)=1-2+1=0$. This matches the vanishing of $c_1(T^{1,0}E)$ and contrasts with $\mathbb{CP}^1$, where the tangent bundle has first Chern class $2H$. [/example] ## What the Synthesis Gives for Later Geometry The chapter's compatibilities are the infrastructure for the next subjects. Dolbeault complexes let holomorphic data be studied through differential forms. Hermitian metrics convert holomorphic bundles into curvature objects. Chern-Weil theory converts curvature into cohomology classes that survive changes of metric and frame. The projective and torus examples also foreshadow the guiding dichotomy of complex geometry. On $\mathbb{CP}^n$, $\mathcal O(1)$ has positive curvature and the tangent bundle has positive first Chern class. On a complex torus, the tangent bundle is flat and its Chern classes vanish. Later Kähler geometry, Hodge theory, and positivity theory refine this contrast into a systematic study of how curvature, topology, and holomorphic functions constrain one another. ## Beyond and Connected Topics The local analytic material in these notes continues naturally into [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy), where domains, holomorphic functions, and several-variable phenomena are treated before the full bundle language is needed. The sheaf-theoretic side connects to [Several Complex Variables II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory), especially through Stein vanishing, coherent sheaves, and the interpretation of Dolbeault groups as gluing obstructions rather than purely local differential equations. The differential-form backbone of the course is part of the broader de Rham story developed in [Differential Forms and de Rham Cohomology](/page/Differential%20Forms%20and%20de%20Rham%20Cohomology), [Differential Forms I: Exterior Calculus](/page/Differential%20Forms%20I%3A%20Exterior%20Calculus), and [Differential Forms II: Manifolds and Cohomology](/page/Differential%20Forms%20II%3A%20Manifolds%20and%20Cohomology). Those pages provide the smooth calculus that is split here into $(p,q)$-types and then reassembled through Dolbeault cohomology. The Hermitian and curvature chapters point toward [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). In that direction, Chern connections, Chern-Weil representatives, positivity of line bundles, Kähler metrics, and Hodge theory become tools for comparing analytic, metric, and topological information on the same complex manifold. ## References - [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy), for the local holomorphic analysis and domain examples used at the start of the course. - [Several Complex Variables II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory), for sheaf cohomology, Stein vanishing, and the global context behind Dolbeault theory. - [Differential Forms and de Rham Cohomology](/page/Differential%20Forms%20and%20de%20Rham%20Cohomology), [Differential Forms I: Exterior Calculus](/page/Differential%20Forms%20I%3A%20Exterior%20Calculus), and [Differential Forms II: Manifolds and Cohomology](/page/Differential%20Forms%20II%3A%20Manifolds%20and%20Cohomology), for the smooth-form and cohomological background used throughout. - [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry), for vector bundles, connections, curvature, and characteristic-class background. - [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature), for the later Kähler, positivity, and curvature themes connected to these notes. - [Dolbeault's Theorem](/theorems/3389), [$\bar\partial$-Poincaré Lemma](/theorems/3410), [Vanishing of Dolbeault Cohomology on Stein Manifolds](/theorems/3388), and [Dolbeault Isomorphism for Vector Bundles](/theorems/3494), for the main cohomological comparison results. - [Curvature Weight Formula for Hermitian Holomorphic Line Bundles](/theorems/3840), [Well-Definedness and Metric Independence of the First Chern Form](/theorems/3839), and [Chern-Weil Representative of the First Chern Class of a Hermitian Holomorphic Line Bundle](/theorems/3848), for the line-bundle curvature and Chern-Weil computations used in the final chapters.