nLab complex analytic space

Idea

The notion of complex analytic space is the notion of analytic space in complex geometry; the generalization of the notion of complex manifold to spaces with singularities.

Definition

A complex analytic test space is a common vanishing locus of a set of holomorphic functions $\mathbb{C}^n \to \mathbb{C}$. This is naturally a locally ringed space over the complex numbers $\mathbb{C}$. A complex analytic space is a locally ringed space over $\mathbb{C}$ that is locally isomorphic to such a complex analytic test space.

Properties

Local contractibility

A smooth complex analytic space is locally isomorphic to a polydisc and hence locally contractible. See also (Berkovich, p.2).

GAGA theorems

Comparison to complex algebraic varieties (GAGA):

• comparison theorem (étale cohomology)

• Chow's theorem

Related concepts

• complex analytic topology

• comparison theorem (etale cohomology)

• real analytic space

References

Introductions include

• Brian Osserman, Complex varieties and the analytic topology (pdf)

Generalization of smooth complex analytic spaces to smooth $p$-adic analytic spaces is discussed in

• Vladimir Berkovich, Smooth $p$-adic analytic spaces (Berkovich spaces) are locally contractible (pdf)

Discussion in higher geometry/higher algebra (derived complex analytic spaces) is in

• Jacob Lurie, section 4.4. of Structured Spaces, 2008

• Jacob Lurie, sections 11 and 12 of Closed Immersions