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 ${ℂ}^n\to ℂ$. This is naturally a locally ringed space over the complex numbers $ℂ$. A complex analytic space is a locally ringed space over $ℂ$ 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).

References

• 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)