Normalization of Complex Analytic Spaces from a Global Viewpoint

Abstract

In this work, we study some algebraic and topological properties of the ring of global analytic functions on the normalization of a reduced complex analytic space . If is a Stein space, we characterize in terms of the (topological) completion of the integral closure of the ring of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Fréchet topology of . This shows that not only the Stein space but also its normalization is completely determined by the ring of global analytic functions on X. This result was already proved in 1988 by Hayes–Pourcin when is an irreducible Stein space, whereas in this paper we afford the general case. We also analyze the real underlying structures and of a reduced complex analytic space and its normalization . We prove that the complexification of provides the normalization of the complexification of if and only if is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization.