What is a good reference for the normalization of analytic spaces?

Question

I know how to define the normalization for a (algebraic) scheme, but how to define the normalization for a complex analytic space? My goto reference for analytic spaces is Griffiths' and Harris' Foundations of algebraic geometry, but I didn't find anythink in there.

Answer 1

For reduced spaces, you could look at Chapter 8 in Grauert and Remmert's Coherent analytic sheaves, or Remmert's Chapter 1, §14 in Several complex variables VII.

In the latter reference, the normalization is constructed as the analytic spectrum $$\operatorname{Specan} \widehat{\mathcal{O}}$$ where the sheaf of normalization $\widehat{\mathcal{O}}$ is the subalgebra of the sheaf $\mathcal{M}$ of meromorphic functions such that $\widehat{\mathcal{O}}_x$ is the integral closure of $\mathcal{O}_x$ in $\mathcal{M}_x$ for every $x \in X$. The fact that $\widehat{\mathcal{O}}$ is coherent is a theorem of Cartan and Oka; see §14.4 in Remmert's chapter for more historical background.