complex analytic space

*Could not include synthetic complex geometry - contents*

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.

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.

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

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, sections 11 and 12 of
*Closed Immersions*

Revised on November 26, 2013 23:40:49
by Urs Schreiber
(77.251.114.72)