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.
Categories: math.AG