Complex analytic space

Summary

A complex analytic space, or complex space, is a locally ringed space that is locally modelled on an analytic set. The model is the common zero set of finitely many holomorphic functions on an open subset of $\C^n$. Its structure sheaf is the quotient of the sheaf of holomorphic functions by the ideal that these functions generate. Complex manifolds and complex analytic varieties are complex spaces. The structure sheaf may have nilpotent elements, which record multiplicities, as for the double point $z^2 = 0$. Complex spaces are the analytic counterpart of schemes and the setting of the global theory of several complex variables.