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 $(X^{\nu\,{\mathbb R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ 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.