Complex analytic space

In mathematics, a complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic spaces are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

Definition

Denote the constant sheaf on a topological space with value ${\displaystyle \mathbb {C} }$ by ${\displaystyle {\underline {\mathbb {C} }}}$. A ${\displaystyle \mathbb {C} }$-space is a locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ whose structure sheaf is an algebra over ${\displaystyle {\underline {\mathbb {C} }}}$.

Choose an open subset ${\displaystyle U}$ of some complex affine space ${\displaystyle \mathbb {C} ^{n}}$, and fix finitely many holomorphic functions ${\displaystyle f_{1},\dots ,f_{k}}$ in ${\displaystyle U}$. Let ${\displaystyle X=V(f_{1},\dots ,f_{k})}$ be the common vanishing locus of these holomorphic functions, that is, ${\displaystyle X=\{x\mid f_{1}(x)=\cdots =f_{k}(x)=0\}}$. Define a sheaf of rings on ${\displaystyle X}$ by letting ${\displaystyle {\mathcal {O}}_{X}}$ be the restriction to ${\displaystyle X}$ of ${\displaystyle {\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})}$, where ${\displaystyle {\mathcal {O}}_{U}}$ is the sheaf of holomorphic functions on ${\displaystyle U}$. Then the locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is a local model space.

A complex analytic space is a locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (X,{\mathcal {O}}_{X})}$ which is locally isomorphic to a local model space.

Morphisms of complex analytic spaces are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.