 Complex analytic space

A complex analytic space is a locally ringed space locally isomorphic to the common zero set of finitely many holomorphic functions on an open subset of ${\mathbb{C}}^n$, equipped with the quotient sheaf of holomorphic functions by a coherent ideal sheaf. This sheaf is part of the structure: the same underlying set can carry different analytic structures.

The cusp $y^2=x^3$ gives a reduced singular example. Its local holomorphic functions are represented by ambient holomorphic functions modulo the equation $y^2-x^3$. A complex manifold is the special smooth local model, with no singular equations needed after choosing coordinates.

General complex analytic spaces can also be nonreduced. The quotient $\mathbb{C}\{z\}/(z^2)$ has underlying set consisting of one point, but retains a nonzero nilpotent class of $z$. This distinguishes an analytic space from only a zero set. The notion extends complex geometry to singularities and infinitesimal structure that ordinary manifold charts cannot describe.