Hello,


I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex spaces such that $f$ is 1-1 over $U\subset Y$, where $U$ is open and dense, then $f$ is an isomorphism.


Could somebody help me?


Thanks in advance,
Serge



Zariski's main theorem in the complex analytic category
One reference is Proposition 14.7 in Remmert's paper *Local Theory of complex analytic spaces*, Several complex variable VII, Encyclopaedia of Math. Sci. vol **74**. For the reader's convenience I will restate the result here.


Recall that a finite, surjective, holomorphic map $\eta \colon X \to Y$ between complex spaces is called a *one sheeted (analytic) covering* if there exists a (critical) thin set $A \subset Y$ such that $\eta^{-1} A$ is thin in $X$ and $\eta \colon X \setminus \eta^{-1}(A) \to Y \setminus A$ is biholomorphic. Then Remmert's statement essentially says that normalizations "dominate" all one-sheeted coverings:



> 
> **Proposition.** Let $\eta \colon X \to Y$, $\xi \colon Z \to Y$ be one-sheeted coverings. If $X$ is normal, there exists a unique holomorphic map $g \colon X \to Z$ such that $\eta= \xi \circ g$. If moreover $Z$ is normal, the map $g$ is biholomorphic.
> 
> 
> 


In your setting, take $\eta=f$ and $g=1\_Y$. Then if both $X$ and $Y$ are normal Remmert's Proposition tells us that $f$ is a biholomorphism. 


