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.