Complex Analytic Spaces

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In this section, we will introduce the so-called complex analytic spaces using ([46]) as our reference.

Definition 4.1. Let $K$ be a ring. A $K$-locally ringed space is a locally ringed space $(X,\mathscr{O}_{X})$ equipped with a $K$-algebra structure $p$ in $\mathscr{O}_{X}$. A $K$-algebra structure $p$ in $\mathscr{O}_{X}$ is a collection of ring homomorphisms $p_{U}:K\rightarrow\mathscr{O}_{X}(U)$, one for each open $U\subset X$, such that for every $U_{1}\supset U_{2}$ the following diagram commutes:

If $z\in\mathscr{O}_{X}(U)$ and $a\in K$, we usually write $az$ instead of $p_{U}(a)z$.

Let $(X',\mathscr{O}_{X'})$ be another $K$-locally ringed space. A morphism of $K$-locally ringed space or simply a $K$-morphism from $(X,\mathscr{O}_{X})$ to $(X',\mathscr{O}_{X'})$ is a morphism $(f,f^{*})$ of locally ringed spaces such that $$ f^{*}_{U'}\circ p'_{U'}=p_{f^{-1}(U')} $$ for every open set $U'\subset X'$.

Example 4.2. The complex number space $$ \mathbb{C}^{n}=\{(\xi_{1},...,\xi_{n})\mid\xi_{i}\in\mathbb{C}\textrm{ for }1\leq i\leq n\}$$ will be viewed as a $\mathbb{C}$-locally ringed space with the natural structure as follows:

The underlying topological space is $\mathbb{C}^{n}$ equipped with the standard topology. And the structure sheaf $\mathscr{O}_{\mathbb{C}^{n}}$ is the sheaf of homological functions. Namely, for each open set $U\subset\mathbb{C}^{n}$, $\mathscr{O}_{\mathbb{C}^{n}}(U)$ is the $\mathbb{C}$-algebra of all holomorphic functions on $U$, where the elements of $\mathbb{C}$ are identified with constant functions. Moreover, any open subset $V$ of $\mathbb{C}^{n}$ will be also viewed as a $\mathbb{C}$-locally ringed space with the structure sheaf $\mathscr{O}_{V}$ obtained as the restriction of $\mathscr{O}_{\mathbb{C}^{n}}$ to $V$.

Example 4.3. Let $V$ be an open subset in $\mathbb{C}^{n}$ and let $(h_{1},...,h_{m})$ be a finite system of holomorphic functions in $V$. Let $$S=\{\xi\in V\mid h_{i}(\xi)=0,1\leq i\leq m\},$$ with the induced topology from $\mathbb{C}^{n}$. The quotient sheaf of $\mathscr{O}_{V}$ by the ideal generated by $(h_{1},...,h_{m})$ is concentrated on $S$. We will denote by $\mathscr{O}_{S}$ the restriction of this sheaf to $S$. Then we obtain a $\mathbb{C}$-locally ringed space $(S,\mathscr{O}_{S})$ with the $\mathbb{C}$-algebra structure $\rho$ which is the identification of constants with constant functions.

Definition 4.4. A complex analytic space or simply a $\mathbb{C}$-space is a locally ringed space $(X,\mathscr{O}_{X})$ such that every point in $X$ admits an open neighborhood $U$ such that $(U,\mathscr{O}_{X|U})$ is $\mathbb{C}$-isomorphic to some $(S,\mathscr{O}_{S})$ described in Example 4.3.

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