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Definition 2.1.1 (holomorphic atlas)

[definition] A holomorphic atlas on a differentiable manifold is an atlas $\{(U_i, \phi_i)\}$ of the form $\phi_i:U_i \simeq \phi_i(U_i) \subseteq \CC^n$, such that the transition functions

are holomorphic. The pair $(U_i, \phi_i)$ is called a holomorphic chart. Two holomorphic atlases $\{(U_i, \phi_i)\}$ and $\{(U_j’, \phi_j’)\}$ are called equivalent if all maps $\phi_i \circ \phi_j’^{-1}$ are holomorphic. [/definition]

Definition 2.1.2 (complex manifold)

[definition] A complex manifold $X$ of dimension $n$ is a real differentiable manifold of dimension $2n$ endowed with an equivalence class of holomorphic atlases. [/definition]

Topological convention

In addition to the above, we say that $X$ is connected, compact, simply-connected, etc. if the underlying topological manifold has said property.

Dimension naming convention

We also impose the convention that complex manifolds of dimension one (two, three, …) is called a complex curve (a complex surface, a complex threefold, …, respectively).

Definition 2.1.3 (holomorphic functions $X \to \CC$)

[definition] A holomorphic function on a complex manifold $X$ is a function $f: X \to \CC$ such that $f\circ \phi_{i}^{-1} : \phi_i(U_i) \to \CC$ is holomorphic for any chart $(U_i, \phi_i)$ of a holomorphic atlas in the equivalence class defining $X$. [/definition]

Definition 2.1.4 (sheaf of holomorphic functions)

[definition] Let $X$ be a complex manifold. By $\mathcal{O}_X$ we denote the sheaf of holomorphic functions on $X$, i.e., for any open subsets $U \subseteq X$ one has

[/definition]

Stalk isomorphism $\mathcal{O}_{X,x} \cong \mathcal{O}_{\CC^n, 0}$

There is an obvious isomorphism (as $\CC$-algebras) of stalks $\mathcal{O}_{X,x} \cong \mathcal{O}_{\CC^n, 0}$ by taking a holomorphic chart $(U, \phi)$ with $x \in U$ and $\phi(x) = 0$ by pulling back via $\phi$.

Quotient field of $\mathcal{O}_{X, x}$

Since $\mathcal{O}_{X, x} \cong \mathcal{O}_{\CC^n, 0}$, and the latter is an integral domain (in fact, its a local UFD by Proposition 1.1.15), we can take its quotient field which we denote by $Q(\mathcal{O}_{X, x})$. Thus, if $U \subseteq X$ is an open connected subset, then holomorphic functions $g, h : U\to \CC$, with $h \not\equiv 0$, define elements $g, h \in \mathcal{O}_{X, x}$ and $g/h \in Q(\mathcal{O}_{X, x})$ for all $x \in U$.

Proposition 2.1.5

[proposition] Let $X$ be a compact connected complex manifold. Then $\Gamma(X, \mathcal{O}_X) = \CC$, i.e. any global holomorphic function on $X$ is constant. [/proposition]

[proof] Let $f \in \mathcal{O}_X$ and $x \in X$ be a point where $f$ attains its maximum. There is a chart $(U, \phi)$ around $x$ in a holomorphic atlas in the equivalence class defining $X$. Then

is holomorphic. Taking $U_i$ to be sufficiently small, we can assume that $U_i$ is connected and so $f\circ \phi^{-1}$ is locally constant by the maximum principle on $\phi(U_i)$. But since $X$ is connected, it follows that $f$ is constant. [/proof]

Contrast to real differentiable manifolds

This result is obviously false compared to real differentiable manifolds. An counterexample is to consider $S^1$ as a real differentiable manifold. Then the map $f:S^1 \to \RR$ defined by $f(e^{i\theta}) = \sin\theta$ is smooth.

Indeed, we can cover $S^1$ by the sets

where $\phi_i(e^{i\theta}) = \theta$ are the chart maps. Now since

and

are smooth, $f$ is smooth and so the claim follows.