M (bbixob) wrote in geometry_qa,

naive defitinion of normalisation of complex manifolds

I need a reference to these simple statements about normalisation of complex spaces;
are they correct at all ? I am willing to assume everything lies a very nice ambient space, say
holomorphically convex manifold.

\def\hatZ{{\hat Z}}
Let $\nnn:\hatZ \ra Z$ be the normalisation of a variety $Z$, and
let $\Zz k(\C) = \{z\in Z(\C): \#\nnn\inv(z)\geq k\}$ be a closed
subvariety of points which have $k$ preimages or more.

\begin{fact}\label{fact:zknrm} $\Zz k$ is a closed subvariety of $Z$, for all $k$;
$\Zz {\deg \nnn + 1}$ is empty, $\Zz 1 = Z$. \end{fact} \bp
Reference!!!! \ep

\begin{fact}\label{fact:214} Around each point $z\in Z(\C)$ there exists a
sufficiently small neighbourhood $z\in V$ open in complex topology
such that
$$V\cap \Zz k= \bigcup\limits_{i_1<..<i_k} V_{i_1}\cap ...\cap V_{i_k},$$ where $$Z\cap V=V_1\cup ...\cup V_n$$ is the irreducible decomposition of $Z\cap V$.\end{fact} \bp not [Grauert, ???] [Abhyankar, Local Analytic Geometry, p.402, Cl 7] \ep
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