A complex manifold begins with a tension between two kinds of geometry. Locally, it should look like an open subset of $\mathbb{C}^n$, so that holomorphic functions, [power series](/page/Power%20Series), and complex derivatives make sense. Globally, it may bend, wrap, and carry topology that no single open subset of $\mathbb{C}^n$ can carry. The theory asks for a way to do complex analysis on spaces that are not themselves domains in $\mathbb{C}^n$. The naive idea is to take a [smooth manifold](/page/Smooth%20Manifold) of real dimension $2n$ and declare pairs of real coordinates to be complex coordinates. This fails because smooth changes of coordinates need not preserve holomorphicity. A smooth coordinate change can mix $z$ and $\bar z$, and after that a function that was holomorphic in one chart need not be holomorphic in another. Complex manifolds are built by forbidding exactly that failure: coordinate changes are required to be holomorphic. [example: A Smooth Surface With a Bad Complex Coordinate Change] Let $X=\mathbb{C}$ as a [topological space](/page/Topological%20Space). Take $U=V=\mathbb{C}$, let $\varphi:U\to\mathbb{C}$ be $\varphi(p)=p$, and let $\psi:V\to\mathbb{C}$ be $\psi(p)=\overline{p}$. As real coordinate maps on $\mathbb{R}^2$, these are smooth because if $p=x+iy$, then $\varphi(x,y)=(x,y)$ and $\psi(x,y)=(x,-y)$. The transition map from the $\varphi$-coordinate to the $\psi$-coordinate is computed by first writing the inverse of $\varphi$ as $\varphi^{-1}(z)=z$, and then applying $\psi$: \begin{align*} (\psi\circ\varphi^{-1})(z)=\psi(z)=\overline{z}. \end{align*} This map is not complex differentiable at any point. Indeed, at $a\in\mathbb{C}$ the difference quotient for $g(z)=\overline{z}$ is \begin{align*} \frac{g(a+h)-g(a)}{h}=\frac{\overline{a+h}-\overline{a}}{h}=\frac{\overline{h}}{h}. \end{align*} For real $h=t\ne0$, this quotient is \begin{align*} \frac{\overline{t}}{t}=\frac{t}{t}=1. \end{align*} For imaginary $h=it$ with $t\ne0$, it is \begin{align*} \frac{\overline{it}}{it}=\frac{-it}{it}=-1. \end{align*} The limiting quotient depends on the direction of approach, so $\psi\circ\varphi^{-1}$ is not holomorphic. The same failure appears in the identity function on the underlying set. In the $\varphi$-coordinate it has coordinate expression \begin{align*} \varphi\circ\operatorname{id}_X\circ\varphi^{-1}(z)=z, \end{align*} which is holomorphic, while in the $\psi$-coordinate its expression relative to the usual target coordinate is \begin{align*} \operatorname{id}_{\mathbb{C}}\circ\operatorname{id}_X\circ\psi^{-1}(w)=\psi^{-1}(w)=\overline{w}, \end{align*} which is antiholomorphic and not holomorphic by the same quotient computation. Thus arbitrary smooth real coordinates do not preserve [complex differentiability](/page/Complex%20Differentiability), and holomorphic transition maps are necessary for a complex atlas. [/example] The point of the definition is not only to say what the local models are. It also says which coordinate changes are allowed, and therefore which local analytic constructions survive when transported from one chart to another. The holomorphic transition condition is the mechanism that lets local complex analysis become global complex geometry. ## Coordinate Compatibility To make complex analysis intrinsic, each point must first have a coordinate neighbourhood looking like $\mathbb{C}^n$. Without this local model there is no place to write power series, partial derivatives, or holomorphic coordinate functions. [definition: Complex Chart] Let $X$ be a topological space. A complex chart of complex dimension $n$ on $X$ is a pair $(U,\varphi)$ such that $U\subset X$ is open and $\varphi:U\to \varphi(U)\subset \mathbb{C}^n$ is a homeomorphism from $U$ onto an open subset $\varphi(U)$ of $\mathbb{C}^n$. [/definition] A chart turns points near $p\in X$ into complex coordinate vectors. The next problem is compatibility: two charts may describe the same point with different coordinate vectors, and the change of coordinates must preserve holomorphic functions rather than only continuity or smoothness. [definition: Holomorphically Compatible Charts] Let $(U,\varphi)$ and $(V,\psi)$ be complex charts of complex dimension $n$ on a topological space $X$. They are holomorphically compatible if, whenever $U\cap V\ne\varnothing$, the transition maps $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to \psi(U\cap V)$ and $\varphi\circ\psi^{-1}:\psi(U\cap V)\to \varphi(U\cap V)$ are holomorphic maps between open subsets of $\mathbb{C}^n$. [/definition] Compatibility is the local rule, but a space needs enough charts to describe every point. This motivates the atlas: it is the global bookkeeping device that covers the space while imposing the same analytic rule on every overlap. [definition: Complex Atlas] A complex atlas of complex dimension $n$ on a topological space $X$ is a family of complex charts $\{(U_i,\varphi_i)\}_{i\in I}$ such that $X=\bigcup_{i\in I}U_i$ and every pair $(U_i,\varphi_i)$, $(U_j,\varphi_j)$ is holomorphically compatible. [/definition] Different atlases can encode the same complex geometry, so using a particular list of charts would make the structure depend on presentation rather than on the analytic coordinate system itself. The obstruction is that one can always add more compatible charts, or choose a smaller covering, without changing which coordinate expressions count as holomorphic. To make the complex structure independent of this choice of presentation, atlases must be compared by the charts they mutually allow. [definition: Equivalent Complex Atlases] Let $\mathcal{A}$ and $\mathcal{B}$ be complex atlases of complex dimension $n$ on a topological space $X$. The atlases $\mathcal{A}$ and $\mathcal{B}$ are equivalent if every chart in $\mathcal{A}$ is holomorphically compatible with every chart in $\mathcal{B}$. [/definition] ## Definition The preceding definitions separate local coordinates, compatibility, and independence from presentation. A complex manifold packages all three: it is a topological space that can be covered by complex coordinates, with exactly the holomorphic coordinate changes needed to make complex analysis intrinsic. [definition: Complex Manifold] A complex manifold of complex dimension $n$ is a Hausdorff, second-countable topological space $X$ equipped with an equivalence class of complex atlases of complex dimension $n$. [/definition] Equivalently, a complex manifold may be described as a smooth manifold $X$ of real dimension $2n$ together with a complex atlas whose transition maps are holomorphic. The equivalence is useful because many geometric constructions start from a smooth manifold, while holomorphic geometry is encoded by the special coordinate changes. The number $n$ is the complex dimension. The same space has real dimension $2n$ after forgetting the complex coordinates. This doubling is not cosmetic: it is the reason complex curves are real surfaces, complex surfaces have real dimension $4$, and analytic constraints are much stronger than smooth real constraints. [remark: Complex Dimension and Real Dimension] If $X$ is a complex manifold of complex dimension $n$, then the coordinate maps also make $X$ into a smooth manifold of real dimension $2n$. The smooth transition maps are obtained by viewing holomorphic maps between open subsets of $\mathbb{C}^n$ as smooth maps between open subsets of $\mathbb{R}^{2n}$. [/remark] The first examples are the ones where no gluing is needed. They matter because every complex manifold is required to resemble them near each point. [example: Domains in $\mathbb{C}^n$] Let $\Omega\subset\mathbb{C}^n$ be open, with the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{C}^n$. The map $\operatorname{id}_\Omega:\Omega\to\Omega$ satisfies $\operatorname{id}_\Omega(p)=p$ for every $p\in\Omega$, so it is a homeomorphism from $\Omega$ onto the open subset $\Omega\subset\mathbb{C}^n$. Thus $(\Omega,\operatorname{id}_\Omega)$ is a complex chart of complex dimension $n$. The one-chart family $\{(\Omega,\operatorname{id}_\Omega)\}$ covers $\Omega$ because its only chart domain is all of $\Omega$. There are no distinct pairs of charts to compare, and the self-transition is \begin{align*} \operatorname{id}_\Omega\circ \operatorname{id}_\Omega^{-1}=\operatorname{id}_\Omega. \end{align*} Since $\operatorname{id}_\Omega(z)=z$ is holomorphic on $\Omega$, the compatibility condition is satisfied. Also, $\Omega$ is Hausdorff and second-countable because it is an open subspace of $\mathbb{C}^n$. Therefore every domain in $\mathbb{C}^n$ is a complex manifold of complex dimension $n$, with its global topology and local analytic coordinates coming from the same ambient Euclidean space. [/example] A less flat example comes from projective geometry. It is the standard model showing that compact complex manifolds exist, even though compact open subsets of $\mathbb{C}^n$ do not. [example: The Riemann Sphere] Let $\mathbb{CP}^1$ be the set of complex lines through the origin in $\mathbb{C}^2$, written as $[z_0:z_1]$ with $(z_0,z_1)\ne(0,0)$ and $[z_0:z_1]=[\lambda z_0:\lambda z_1]$ for $\lambda\in\mathbb{C}\setminus\{0\}$. The two standard open sets \begin{align*} U_0=\{[z_0:z_1]:z_0\ne0\},\qquad U_1=\{[z_0:z_1]:z_1\ne0\} \end{align*} cover $\mathbb{CP}^1$, because at least one of $z_0,z_1$ is nonzero. Define \begin{align*} \varphi_0([z_0:z_1])=\frac{z_1}{z_0},\qquad \varphi_1([z_0:z_1])=\frac{z_0}{z_1}. \end{align*} These formulas are independent of the chosen representative: if $[z_0:z_1]=[\lambda z_0:\lambda z_1]$, then \begin{align*} \frac{\lambda z_1}{\lambda z_0}=\frac{z_1}{z_0},\qquad \frac{\lambda z_0}{\lambda z_1}=\frac{z_0}{z_1}. \end{align*} The inverse maps are explicit: \begin{align*} \varphi_0^{-1}(w)=[1:w],\qquad \varphi_1^{-1}(u)=[u:1]. \end{align*} Indeed, \begin{align*} \varphi_0([1:w])=\frac{w}{1}=w,\qquad \varphi_1([u:1])=\frac{u}{1}=u. \end{align*} On $U_0\cap U_1$, both coordinates are nonzero. If $w=\varphi_0([z_0:z_1])=z_1/z_0$, then $w\ne0$, and \begin{align*} (\varphi_1\circ\varphi_0^{-1})(w)=\varphi_1([1:w])=\frac{1}{w}. \end{align*} Similarly, for $u\ne0$, \begin{align*} (\varphi_0\circ\varphi_1^{-1})(u)=\varphi_0([u:1])=\frac{1}{u}. \end{align*} The reciprocal map is holomorphic on $\mathbb{C}\setminus\{0\}$, since for $w\ne0$ and $h$ small enough that $w+h\ne0$, \begin{align*} \frac{\frac{1}{w+h}-\frac{1}{w}}{h}=\frac{w-(w+h)}{h\,w(w+h)}=-\frac{1}{w(w+h)} \end{align*} and this tends to $-1/w^2$ as $h\to0$. Thus the two standard charts are holomorphically compatible and give $\mathbb{CP}^1$ a complex manifold structure of complex dimension $1$. With the [quotient topology](/page/Quotient%20Topology), $\mathbb{CP}^1$ is compact because it is the image of the compact unit sphere $S^3\subset\mathbb{C}^2$ under the surjective map $(z_0,z_1)\mapsto[z_0:z_1]$, so this example is a compact complex curve rather than a domain in $\mathbb{C}$. [/example] ## Holomorphic Maps ### Coordinate Tests Once the space itself has holomorphic coordinate changes, the next question is how maps between such spaces should be tested. A definition using a single coordinate chart would be meaningless unless it is independent of the chosen charts. The atlas condition was designed so that checking holomorphicity in local coordinates gives an intrinsic notion. [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 complex chart $(U,\varphi)$ around $p$, and every complex chart $(V,\psi)$ around $F(p)$ with $F(U)\subset V$, the coordinate representation $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to \psi(V)$ is holomorphic on its domain after restricting $U$ if necessary. [/definition] The definition quantifies over many charts, which is too much for ordinary use. The practical question is whether holomorphicity can be checked in one convenient coordinate system near each point; the next theorem says that the compatibility built into the atlases is exactly what makes this possible. [quotetheorem:9942] This criterion is useful because it separates two issues that are easy to confuse. One still has to choose charts whose domains contain the points under discussion, and one may have to shrink those domains so that the map lands in a single target chart. But after that local choice has been made, no hidden global condition remains: holomorphicity is exactly the ordinary holomorphicity of the coordinate expression. The theorem does not say that every local formula automatically glues to a global map; the formulas must still agree on overlaps and define a continuous map of the underlying spaces. Its role is instead to make later constructions practical, since submanifold charts, differentials, and sheaves of holomorphic functions can all be tested in whichever compatible coordinates make the computation transparent. ### Equivalences of Complex Manifolds After defining holomorphic maps, the next structural question is when two complex manifolds should count as the same complex-geometric object. A bijection of point sets is not enough; both the map and its inverse must preserve holomorphic coordinate systems. [definition: Biholomorphism] Let $X$ and $Y$ be complex manifolds. A biholomorphism from $X$ to $Y$ is a bijective holomorphic map $F:X\to Y$ whose inverse map $F^{-1}:Y\to X$ is holomorphic. [/definition] A biholomorphism is the correct notion of isomorphism in complex geometry. It preserves holomorphic functions, complex tangent spaces, and all coordinate-dependent analytic constructions. [example: A Coordinate Change on the Sphere] On $\mathbb{CP}^1$, write the two affine coordinates as $w=z_1/z_0$ on $U_0$ and $u=z_0/z_1$ on $U_1$. On the overlap $U_0\cap U_1$, both $z_0$ and $z_1$ are nonzero, so $w\ne0$ and \begin{align*} u=\frac{z_0}{z_1}=\frac{1}{z_1/z_0}=\frac{1}{w}. \end{align*} Thus the transition map from the $w$-coordinate to the $u$-coordinate is $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}\setminus\{0\}$ with $f(w)=1/w$. The same formula is its own inverse, because for every $w\ne0$, \begin{align*} (f\circ f)(w)=f\left(\frac{1}{w}\right)=\frac{1}{1/w}=w. \end{align*} It remains to see explicitly that $f$ is holomorphic on $\mathbb{C}\setminus\{0\}$. For $w\ne0$ and $h$ small enough that $w+h\ne0$, the difference quotient is \begin{align*} \frac{f(w+h)-f(w)}{h}=\frac{\frac{1}{w+h}-\frac{1}{w}}{h}. \end{align*} Combining the two fractions gives \begin{align*} \frac{\frac{1}{w+h}-\frac{1}{w}}{h}=\frac{\frac{w-(w+h)}{w(w+h)}}{h}. \end{align*} Since $w-(w+h)=-h$, this becomes \begin{align*} \frac{\frac{w-(w+h)}{w(w+h)}}{h}=\frac{-h}{h\,w(w+h)}=-\frac{1}{w(w+h)}. \end{align*} As $h\to0$, the denominator $w(w+h)$ tends to $w^2$, so the quotient tends to $-1/w^2$. Hence $f$ is holomorphic, and since $f^{-1}=f$, its inverse is holomorphic as well. Therefore the transition map is a biholomorphism between the two overlap domains, which is exactly the compatibility needed for the two affine charts to glue as complex charts. [/example] ### Smoothness Versus Holomorphicity Complex manifolds also have underlying smooth manifolds, so every holomorphic map should be compatible with ordinary differential geometry. The useful comparison theorem is that holomorphicity automatically implies smoothness, while the converse fails because smooth maps can mix $z$ and $\bar z$. [quotetheorem:9943] This theorem explains why complex manifolds automatically belong to differential geometry. Every holomorphic problem has an underlying smooth problem, but the reverse direction is usually false. [example: Smooth but Not Holomorphic] Consider $F:\mathbb{C}\to\mathbb{C}$ given by $F(z)=\bar z$. Writing $z=x+iy$, we have \begin{align*} F(x+iy)=x-iy. \end{align*} As a real map $\mathbb{R}^2\to\mathbb{R}^2$, this is \begin{align*} F(x,y)=(x,-y). \end{align*} Its coordinate functions are $u(x,y)=x$ and $v(x,y)=-y$, both polynomial functions in $x,y$, so $F$ is smooth as a real map. Now test complex differentiability at an arbitrary point $a\in\mathbb{C}$. For $h\ne0$, the difference quotient is \begin{align*} \frac{F(a+h)-F(a)}{h}=\frac{\overline{a+h}-\bar a}{h}. \end{align*} Since $\overline{a+h}=\bar a+\bar h$, this becomes \begin{align*} \frac{\overline{a+h}-\bar a}{h}=\frac{\bar h}{h}. \end{align*} If $h=t$ with $t\in\mathbb{R}$ and $t\ne0$, then \begin{align*} \frac{\bar h}{h}=\frac{\bar t}{t}=\frac{t}{t}=1. \end{align*} If $h=it$ with $t\in\mathbb{R}$ and $t\ne0$, then $\overline{it}=-it$, so \begin{align*} \frac{\bar h}{h}=\frac{\overline{it}}{it}=\frac{-it}{it}=-1. \end{align*} The limiting quotient would have to be independent of the direction in which $h\to0$, but the real direction gives $1$ and the imaginary direction gives $-1$. Therefore $F$ is not complex differentiable at any point, hence is not holomorphic. This shows that smooth real maps are too flexible for complex geometry: they can reverse the complex coordinate by replacing $z$ with $\bar z$. [/example] ## Complex Submanifolds and Local Models ### Straightening Subsets Many spaces in complex geometry arise as zero sets of holomorphic equations. The subtle point is that not every zero set is a manifold; singular points can appear. The local model for a genuine complex submanifold is a coordinate system in which the subset is cut out by setting some complex coordinates equal to zero. [definition: Complex Submanifold] Let $X$ be a complex manifold of complex dimension $n$. A subset $Y\subset X$ is a complex submanifold of complex dimension $k$ if for every $p\in Y$ there is a complex 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] This definition deliberately uses holomorphic coordinates rather than arbitrary equations. It says that near each point of $Y$, the ambient complex manifold can be straightened so that $Y$ looks like a linear coordinate plane. [example: A Smooth Hypersurface in $\mathbb{C}^2$] Let $Y=\{(z,w)\in\mathbb{C}^2:w=z^2\}$. Define $\Phi:\mathbb{C}^2\to\mathbb{C}^2$ by \begin{align*} \Phi(z,w)=(z,w-z^2). \end{align*} Both component functions $z$ and $w-z^2$ are polynomials in the complex variables $z,w$, so $\Phi$ is holomorphic. Define $\Psi:\mathbb{C}^2\to\mathbb{C}^2$ by \begin{align*} \Psi(z,u)=(z,u+z^2). \end{align*} Its component functions are also polynomials, so $\Psi$ is holomorphic. The two maps are inverse to each other because, for every $(z,w)\in\mathbb{C}^2$, \begin{align*} (\Psi\circ\Phi)(z,w)=\Psi(z,w-z^2)=(z,(w-z^2)+z^2)=(z,w), \end{align*} and for every $(z,u)\in\mathbb{C}^2$, \begin{align*} (\Phi\circ\Psi)(z,u)=\Phi(z,u+z^2)=(z,(u+z^2)-z^2)=(z,u). \end{align*} Thus $\Phi$ is a biholomorphism with inverse $\Psi$. Now compute the image of $Y$ under $\Phi$. If $(z,w)\in Y$, then $w=z^2$, so \begin{align*} \Phi(z,w)=\Phi(z,z^2)=(z,z^2-z^2)=(z,0). \end{align*} Therefore $\Phi(Y)\subset \mathbb{C}\times\{0\}$. Conversely, for any $(z,0)\in\mathbb{C}\times\{0\}$, \begin{align*} \Psi(z,0)=(z,0+z^2)=(z,z^2)\in Y, \end{align*} so applying $\Phi$ gives $(z,0)\in\Phi(Y)$. Hence \begin{align*} \Phi(Y)=\mathbb{C}\times\{0\}. \end{align*} The biholomorphic coordinate change $\Phi$ straightens $Y$ to the coordinate plane $\mathbb{C}\times\{0\}$, so $Y$ is a complex submanifold of $\mathbb{C}^2$ of complex dimension $1$. [/example] The failure mode is as important as the successful example. A holomorphic equation may define a set that has no manifold structure at a point because the local model changes dimension or develops a crossing. [example: A Singular Zero Set] Let $Y=\{(z,w)\in\mathbb{C}^2:zw=0\}$. If $(z_0,0)\in Y$ with $z_0\ne0$, choose the neighbourhood $|z-z_0|<|z_0|/2$. Then \begin{align*} |z|\ge |z_0|-|z-z_0|>|z_0|/2>0. \end{align*} On this neighbourhood the equation $zw=0$ therefore forces $w=0$, so $Y$ is locally the $z$-axis. Similarly, if $(0,w_0)\in Y$ with $w_0\ne0$, then near that point the inequality $|w|>0$ forces $z=0$, so $Y$ is locally the $w$-axis. At $(0,0)$ the two axes meet. We show that no neighbourhood of $(0,0)$ in $Y$ can be homeomorphic to an open subset of $\mathbb{C}$. Let $W\subset Y$ be any open neighbourhood of $(0,0)$. Since $W$ is open in the subspace topology, there is $\varepsilon>0$ such that \begin{align*} Y\cap \{(z,w): |z|^2+|w|^2<\varepsilon^2\}\subset W. \end{align*} Thus $W\setminus\{(0,0)\}$ contains points on both punctured axes. Moreover, \begin{align*} W\setminus\{(0,0)\}=\bigl(W\cap\{(z,0):z\ne0\}\bigr)\cup\bigl(W\cap\{(0,w):w\ne0\}\bigr). \end{align*} These two sets are disjoint, nonempty, and open in $W\setminus\{(0,0)\}$, because the two coordinate axes in $Y$ meet only at $(0,0)$. Hence $W\setminus\{(0,0)\}$ is disconnected. If $W$ were homeomorphic to an [open set](/page/Open%20Set) $\Omega\subset\mathbb{C}$, with $(0,0)$ corresponding to $a\in\Omega$, then some disk $D(a,r)$ would lie in $\Omega$. Its preimage would be an open neighbourhood $W'\subset W$ of $(0,0)$ in $Y$, and $W'\setminus\{(0,0)\}$ would be homeomorphic to the punctured disk $D(a,r)\setminus\{a\}$, which is connected. This contradicts the disconnection proved above. Therefore $Y$ is not locally modeled on $\mathbb{C}$ at the origin, so $Y$ is not a complex submanifold of $\mathbb{C}^2$. [/example] ### Regular Level Sets The examples suggest a usable criterion: holomorphic equations should define a submanifold when their first-order parts impose independent complex conditions. The obstruction is that the common zero set of holomorphic equations can acquire singular crossing or branching where the derivatives fail to control the local shape. A full-rank derivative condition rules out that pathology by making some variables solvable holomorphically in terms of the others. [quotetheorem:9944] This theorem is the bridge from analysis to geometry. It allows holomorphic equations with full-rank derivative to define manifolds and isolates singularities as precisely the points where the rank condition breaks. ## Tangent Spaces and Differentials ### First-Order Holomorphic Geometry A complex manifold has a smooth tangent bundle of real rank $2n$, but holomorphic coordinates give each tangent space multiplication by $i$. To differentiate holomorphic functions intrinsically, the tangent space should consist of first-order operations that see holomorphic germs rather than arbitrary smooth germs. Before defining tangent vectors by germs, fix the notation. For a complex manifold $X$, let $\mathcal{O}_X$ denote the sheaf of holomorphic functions on $X$: to each open set $U\subset X$ it assigns the ring $\mathcal{O}_X(U)$ of holomorphic functions $U\to\mathbb{C}$. For a point $p\in X$, the stalk $\mathcal{O}_{X,p}$ is the ring of germs at $p$ of such holomorphic functions, meaning functions defined on possibly smaller neighbourhoods of $p$ and identified when they agree near $p$. [definition: Holomorphic Tangent Space] Let $X$ be a complex manifold of complex dimension $n$, and let $p\in X$. The holomorphic tangent space $T_p^{1,0}X$ is the complex [vector space](/page/Vector%20Space) of all $\mathbb{C}$-linear maps $D:\mathcal{O}_{X,p}\to\mathbb{C}$ satisfying the Leibniz rule \begin{align*} D(fg)=f(p)D(g)+g(p)D(f) \end{align*} for all germs $f,g\in\mathcal{O}_{X,p}$. [/definition] In a chart $(U,\varphi)$ with coordinates $(z_1,\ldots,z_n)$, the holomorphic tangent space has basis $\partial_{z_1}|_p,\ldots,\partial_{z_n}|_p$. Holomorphic maps should transport holomorphic first-order directions to holomorphic first-order directions. This motivates a differential that is complex-linear and independent of which coordinate representation is used. [definition: Differential of a Holomorphic Map] Let $F:X\to Y$ be a holomorphic map between complex manifolds, and let $p\in X$. The holomorphic differential of $F$ at $p$ is the complex-[linear map](/page/Linear%20Map) $dF_p:T_p^{1,0}X\to T_{F(p)}^{1,0}Y$ obtained by differentiating a coordinate representation of $F$ in complex charts. [/definition] The notation $dF_p$ emphasizes the manifold-level differential. In local coordinates it is represented by the complex [Jacobian matrix](/page/Jacobian%20Matrix) of the holomorphic coordinate expression. [example: Differential of a Projective Chart Map] For the transition map $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}\setminus\{0\}$ defined by $f(w)=1/w$, compute its complex derivative at a point $w\ne0$. For $h\ne0$ small enough that $w+h\ne0$, the difference quotient is \begin{align*} \frac{f(w+h)-f(w)}{h}=\frac{\frac{1}{w+h}-\frac{1}{w}}{h}. \end{align*} Putting the two fractions over the common denominator $w(w+h)$ gives \begin{align*} \frac{\frac{1}{w+h}-\frac{1}{w}}{h}=\frac{\frac{w-(w+h)}{w(w+h)}}{h}. \end{align*} Since $w-(w+h)=-h$, this becomes \begin{align*} \frac{\frac{w-(w+h)}{w(w+h)}}{h}=\frac{-h}{h\,w(w+h)}. \end{align*} Because $h\ne0$, we cancel the factor $h$: \begin{align*} \frac{-h}{h\,w(w+h)}=-\frac{1}{w(w+h)}. \end{align*} Taking $h\to0$ gives \begin{align*} f'(w)=\lim_{h\to0}\left(-\frac{1}{w(w+h)}\right)=-\frac{1}{w^2}. \end{align*} In the one-dimensional holomorphic tangent coordinate, the differential therefore sends a tangent vector $\xi\in T_w^{1,0}(\mathbb{C}\setminus\{0\})\cong\mathbb{C}$ to \begin{align*} df_w(\xi)=f'(w)\xi=-\frac{\xi}{w^2}. \end{align*} Thus the change of tangent coordinate between the two affine charts on $\mathbb{CP}^1$ is multiplication by the complex number $-1/w^2$ at every point of the overlap, so it is complex-linear there. [/example] ### Functoriality and Local Inversion The [inverse function theorem](/theorems/51) appearing here is the one-variable holomorphic inverse theorem for open subsets of $\mathbb{C}$. It should be read as the basic local model for why a nonzero complex derivative gives a holomorphic coordinate change. Higher-dimensional chart constructions and complex submanifold straightening use analogous several-variable forms, so the one-variable theorem supplies the guiding mechanism rather than the full general tool. A differential would be of limited use if it did not interact correctly with composition. The problem is that a tangent vector can be pushed through one holomorphic map and then another, and this two-stage process must agree with differentiating the composite map directly. Without that compatibility, tangent spaces would depend on the chosen factorization of a map rather than on the map itself. [quotetheorem:3907] The chain rule makes invertibility of the differential the natural first-order test for whether a map is locally a change of complex coordinates. The corresponding local theorem is one of the main tools for building charts and submanifolds. [quotetheorem:4950] This theorem is used constantly but should be read with the right scope. It is local. It does not say that a globally injective-looking formula has no global identifications, nor does it classify the topology of the manifold. ## Holomorphic Functions and Sheaves Functions are the probes of a complex manifold. A [topological manifold](/page/Topological%20Manifold) can be studied by continuous functions, a smooth manifold by smooth functions, and a complex manifold by holomorphic functions. Since holomorphicity is local, the structure should remember holomorphic functions on every open set and the way they restrict. [definition: Sheaf of Holomorphic Functions] Let $X$ be a complex manifold. The sheaf of holomorphic functions on $X$, denoted $\mathcal{O}_X$, assigns to each open set $U\subset X$ the set $\mathcal{O}_X(U)$ of holomorphic maps $f:U\to\mathbb{C}$. For open sets $V\subset U\subset X$, the restriction map is $\rho^U_V:\mathcal{O}_X(U)\to\mathcal{O}_X(V)$, $\rho^U_V(f)=f|_V$. [/definition] The sheaf viewpoint records not only global functions but all local holomorphic functions and how they glue. This raises a rigidity question: if two holomorphic functions agree on a genuine open piece of a connected complex manifold, how far does that agreement propagate? [quotetheorem:9945] The identity theorem shows how rigid holomorphic functions are. Agreement on any nonempty open region propagates across a connected complex manifold, so local analytic data can determine global data. [example: Holomorphic Functions on $\mathbb{CP}^1$] Let $f:\mathbb{CP}^1\to\mathbb{C}$ be holomorphic. In the affine chart $U_0=\{[z_0:z_1]:z_0\ne0\}$ with coordinate $w=z_1/z_0$, define \begin{align*} F(w)=f([1:w]). \end{align*} Because $f$ is holomorphic on $\mathbb{CP}^1$ and $w\mapsto[1:w]$ is the inverse of the chart map on $U_0$, the function $F:\mathbb{C}\to\mathbb{C}$ is entire. Now use the second chart $U_1=\{[z_0:z_1]:z_1\ne0\}$ with coordinate $u=z_0/z_1$, whose inverse is $u\mapsto[u:1]$. Set \begin{align*} G(u)=f([u:1]). \end{align*} The function $G$ is holomorphic near $u=0$, hence continuous at $0$. Therefore there is $\delta>0$ such that $|u|<\delta$ implies \begin{align*} |G(u)-G(0)|<1. \end{align*} For such $u$, the triangle inequality gives \begin{align*} |G(u)|\le |G(0)|+1. \end{align*} If $|w|>1/\delta$, then $|1/w|<\delta$, and the projective equality \begin{align*} [1:w]=[1/w:1] \end{align*} gives \begin{align*} F(w)=f([1:w])=f([1/w:1])=G(1/w). \end{align*} Hence for $|w|>1/\delta$, \begin{align*} |F(w)|\le |G(0)|+1. \end{align*} On the closed disk $|w|\le 1/\delta$, the [continuous function](/page/Continuous%20Function) $|F|$ attains a maximum, so $F$ is bounded there as well. Combining the bounded disk with the bounded exterior region shows that $F$ is a bounded entire function. By *[Liouville's theorem](/theorems/38)*, $F$ is constant; say $F(w)=c$ for all $w\in\mathbb{C}$. Thus $f([z_0:z_1])=c$ whenever $z_0\ne0$. For the remaining point $[0:1]$, the formula above gives $G(u)=F(1/u)=c$ for all $0<|u|<\delta$, and continuity of $G$ at $0$ gives \begin{align*} f([0:1])=G(0)=c. \end{align*} Therefore every [holomorphic function](/page/Holomorphic%20Function) on $\mathbb{CP}^1$ is constant, reflecting the rigidity imposed by compactness in complex geometry. [/example] The example suggests the general compact case: compactness forces the modulus of a holomorphic function to attain a maximum, and a complex manifold has enough local charts to apply the [maximum modulus principle](/theorems/491) near that point. This motivates the global maximum principle for compact connected complex manifolds. [quotetheorem:9946] This theorem is often the first surprise in the subject. Compactness, which is useful in smooth geometry for extracting extrema, becomes a severe restriction in holomorphic geometry. ## Almost Complex Structures and Integrability For a smooth manifold with an almost complex structure $J$, the obstruction to coming from holomorphic coordinates is measured by the Nijenhuis tensor. This tensor is written $N_J$; it is a bilinear operation on vector fields built from $J$ and their Lie brackets. The condition $N_J=0$ means that this obstruction vanishes everywhere, which is the integrability condition tested in the Newlander-Nirenberg theorem. There is another way to ask for complex coordinates. Instead of beginning with charts, start with a smooth real manifold of dimension $2n$ and put multiplication by $i$ on each tangent space. The hard question is whether those pointwise complex vector spaces come from actual holomorphic coordinate charts. [definition: Almost Complex Structure] Let $M$ be a smooth manifold of real dimension $2n$. An almost complex structure on $M$ is a smooth bundle map $J:TM\to TM$ such that $J_p^2=-\operatorname{id}_{T_pM}$ for every $p\in M$. [/definition] Every complex manifold has such a structure, since in a holomorphic coordinate chart $J$ acts as multiplication by $i$ on each complex coordinate direction. The converse is a separate problem: tangent-space complex multiplication may fail to come from holomorphic coordinates, so the missing condition deserves its own name. [definition: Integrable Almost Complex Structure] Let $M$ be a smooth manifold with an almost complex structure $J$. The almost complex structure $J$ is integrable if there exists a complex manifold structure on $M$ whose induced tangent-space multiplication by $i$ is $J$. [/definition] Integrability is the boundary between complex linear algebra on tangent spaces and genuine complex analysis on the manifold. The central question is how to detect this boundary without already knowing the desired complex coordinates; the Newlander-Nirenberg theorem gives the intrinsic answer. [quotetheorem:7003] This theorem explains why complex manifolds are not merely smooth manifolds with a choice of $J$. The obstruction is differential, not pointwise. In higher dimensions, a tangent-level complex structure may fail to come from holomorphic coordinates. [example: Complex Structures on Real Surfaces] Let $M$ be an oriented smooth surface and let $J:TM\to TM$ be an almost complex structure, so $J_p^2=-\operatorname{id}_{T_pM}$ for every $p\in M$. If $v\in T_pM$ is nonzero, then $v$ and $J_pv$ are linearly independent: otherwise $J_pv=\lambda v$ for some real $\lambda$, and applying $J_p$ again gives \begin{align*} -v=J_p^2v=J_p(\lambda v)=\lambda J_pv=\lambda^2v. \end{align*} Since $v\ne0$, this would imply $\lambda^2=-1$, impossible over $\mathbb{R}$. Thus $(v,J_pv)$ is a basis of $T_pM$; saying that $J$ is compatible with the orientation means that this ordered basis is positively oriented for every nonzero $v$. Starting from any Riemannian metric $h$ on $M$, define \begin{align*} g(u,v)=\frac{1}{2}\bigl(h(u,v)+h(Ju,Jv)\bigr). \end{align*} Then $g$ is symmetric and positive definite because both $h(u,u)$ and $h(Ju,Ju)$ are positive when $u\ne0$. Also, \begin{align*} g(Ju,Jv)=\frac{1}{2}\bigl(h(Ju,Jv)+h(J^2u,J^2v)\bigr). \end{align*} Since $J^2=-\operatorname{id}$, we have $J^2u=-u$ and $J^2v=-v$, so \begin{align*} g(Ju,Jv)=\frac{1}{2}\bigl(h(Ju,Jv)+h(-u,-v)\bigr)=\frac{1}{2}\bigl(h(Ju,Jv)+h(u,v)\bigr)=g(u,v). \end{align*} Thus $J$ preserves $g$-lengths. Taking $v=u$ in the identity with $u$ replaced by $Ju$ gives \begin{align*} g(Ju,u)=g(J(Ju),Ju)=g(-u,Ju)=-g(u,Ju). \end{align*} Since $g$ is symmetric, $g(u,Ju)=g(Ju,u)$, hence \begin{align*} g(Ju,u)=-g(Ju,u), \end{align*} so $g(Ju,u)=0$. Therefore $J$ is rotation by $90$ degrees with respect to the conformal class of $g$ and the chosen orientation. Conversely, an oriented conformal class determines $J$ by declaring $Ju$ to be the positively oriented vector perpendicular to $u$ with the same length; replacing $g$ by $e^{2\phi}g$ preserves perpendicularity, equal lengths, and orientation, so the same $J$ is obtained. In real dimension $2$, every almost complex structure is integrable by the two-dimensional case of the *Newlander-Nirenberg theorem*. Thus Riemann surfaces are precisely the complex one-dimensional version of oriented conformal surface geometry: the analytic coordinate charts are forced by the same data that records oriented angle and scale-free length. [/example] ## Beyond and Connected Topics Complex manifolds are the entry point to several larger theories. In several complex variables, domains in $\mathbb{C}^n$ provide the local analytic model, but the global theory studies how holomorphic functions, convexity, and extension phenomena behave under changes of complex coordinates. The natural continuation is [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy). Sheaf theory turns the local nature of holomorphic functions into an algebraic tool. The sheaf $\mathcal{O}_X$ is the first example, and coherent sheaves, Stein manifolds, and Cartan theorems build from it. A course-level continuation is [Several Complex Variables II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory). Complex geometry adds vector bundles, metrics, curvature, and positivity. On a complex manifold, holomorphic line bundles and Hermitian metrics connect analysis to cohomology and projective embeddings. These themes continue in [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). When a complex manifold has a real hypersurface boundary, holomorphic geometry induces CR geometry on the boundary. This is the setting for extension theorems, pseudoconvexity, and [boundary regularity](/theorems/99). The relevant continuation is [Several Complex Variables V: CR Geometry and Boundary Behavior](/page/Several%20Complex%20Variables%20V%3A%20CR%20Geometry%20and%20Boundary%20Behavior). ## References Androma, [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy). Androma, [Several Complex Variables II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory). Androma, [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). Androma, [Several Complex Variables V: CR Geometry and Boundary Behavior](/page/Several%20Complex%20Variables%20V%3A%20CR%20Geometry%20and%20Boundary%20Behavior). Gunning and Rossi, *Analytic Functions of Several Complex Variables* (1965). Huybrechts, *Complex Geometry: An Introduction* (2005). Wells, *Differential Analysis on Complex Manifolds* (1980).