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.