Normalization of complex analytic spaces from a global viewpoint

TL;DR

This paper investigates the algebraic and topological properties of the global analytic functions on the normalization of reduced complex analytic spaces, extending previous results to the general case and analyzing real structures and their complexifications.

Contribution

It generalizes the characterization of the normalization's global functions from irreducible Stein spaces to all reduced complex spaces and explores the interplay between real structures, complexification, and normalization.

Findings

Normalization is determined by the ring of global functions for Stein spaces.

The paper extends previous irreducible Stein space results to the general case.

Complexification of real structures relates to normalization under coherence conditions.

Abstract

In this work we study some algebraic and topological properties of the ring ${\mathcal O}(X^\nu)$ of global analytic functions of the normalization $(X^\nu,{\mathcal O}_{X^\nu})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize ${\mathcal O}(X^\nu)$ in terms of the (topological) completion of the integral closure $\overline{{\mathcal O}(X)}^\nu$ of the ring ${\mathcal O}(X)$ of global holomorphic functions on $X$ (inside its total ring of fractions) with respect to the usual Fr\'echet topology of $\overline{{\mathcal O}(X)}^\nu$. This shows that not only the Stein space $(X,{\mathcal O}_X)$ but also its normalization is completely determined by the ring ${\mathcal O}(X)$ of global analytic functions on $X$. This result was already proved in 1988 by Hayes-Pourcin when $(X,{\mathcal O}_X)$ is an irreducible Stein space whereas in this paper we afford the general case. We also analyze the real underlying structures $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ and $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$ and its normalization $(X^\nu,{\mathcal O}_{X^\nu})$. We prove that the complexification of $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ provides the normalization of the complexification of $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ if and only if $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ 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.