Complex analytic space

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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[edit]

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

Choose an open subset U of some complex affine space \mathbb{C}^n, and fix finitely many holomorphic functions f_1,\dots,f_k in U. Let X=V(f_1,\dots,f_k) be the common vanishing locus of these holomorphic functions, that is, X=\{x\mid f_1(x)=\cdots=f_k(x)=0\}. Define a sheaf of rings on X by letting \mathcal{O}_X be the restriction to X of \mathcal{O}_U/(f_1, \ldots, f_k), where \mathcal{O}_U is the sheaf of holomorphic functions on U. Then the locally ringed \mathbb{C}-space (X, \mathcal{O}_X) is a local model space.

A complex analytic space is a locally ringed \mathbb{C}-space (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, it is also called holomorphic maps.

See also[edit]

References[edit]

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