Normalization of Complex Spaces

Abstract

The main goal of this chapter is to show that every reduced complex space X has, up to a canonical isomorphism, a normalization $\hat X$. This space $\hat X$will turn out to be a one-sheeted analytic covering of X. For this reason we first develop a general theory of such coverings and prove a Local Existence Theorem. This theorem easily implies OKA’s theorem that the normalization sheaf ${{\hat{\mathcal{O}}}_{X}}$ of the structure sheaf ${{\mathcal{O}}_{X}}$ is ${{\mathcal{O}}_{X}}$-coherent.