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.

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

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 ${\displaystyle {{𝒪}}_X}$ be the restriction to ${\displaystyle X}$ of ${\displaystyle {{𝒪}}_U/(f_1,\dots ,f_k)}$, where ${\displaystyle {{𝒪}}_U}$ is the sheaf of holomorphic functions on ${\displaystyle U}$. Then the locally ringed ${\displaystyle \mathbb{ℂ}}$-space ${\displaystyle (X,{{𝒪}}_X)}$ is a local model space.

A complex analytic space is a locally ringed ${\displaystyle \mathbb{ℂ}}$-space ${\displaystyle (X,{{𝒪}}_X)}$ 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.

• Analytic space

• Grauert and Remmert, Complex Analytic Spaces
• Grauert, Peternell, and Remmert, Encyclopaedia of Mathematical Sciences 74: Several Complex Variables VII

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