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).
Comparison to complex algebraic varieties (GAGA):
Introductions include
Generalization of smooth complex analytic spaces to smooth $p$-adic analytic spaces is discussed in
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
Last revised on May 22, 2017 at 08:20:41. See the history of this page for a list of all contributions to it.