= 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.
Back to article page
