Normalization of complex analytic spaces from a global viewpoint

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 $(X^{\nu,{\mathbb R}},{\mathcal O}{X^\nu}^{\mathbb R})$ofareducedcomplexanalyticspace$(X,{\mathcal O}X)$anditsnormalization$(X^\nu,{\mathcal O}{X^\nu})$. We prove that the complexification of $(X^{\nu,{\mathbb R}},{\mathcal O}{X^\nu}^{\mathbb R})$providesthenormalizationofthecomplexificationof$(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ifandonlyif$(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ 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.