Is restriction of projection a covering map? Let $X$ be a compact complex manifold of dimension $k$ and $\Gamma \subset X \times X$ be an irreducible analytic subset of dimension $k$. Assume that projection on the second coordinate $\Pi_2|_{\Gamma}$ is surjective. Does there exist (analytic) zariski open subset $\Omega$ of $X$ such that $(\Gamma \cap \Pi_2^{-1}(\Omega), \Omega, \Pi_2)$ is a covering map?
I guess answer is YES but I don’t know how to prove it. Any help or reference is welcome!
 A: I will use some standard facts about analytic subsets of complex manifolds which could be found for instance in: 
Chirka, E. M., Complex analytic sets. Translated from the Russian by R. A. M. Hoksbergen, Mathematics and Its Applications: Soviet Series, 46. Dordrecht etc.: Kluwer Academic Publishers. xix, 372 p. Dfl. 195.00;  (1989). ZBL0683.32002.
The proof of the claim breaks in several steps: 


*

*$\Gamma$ contains a nowhere dense analytic subset $S$ such that $Y=\Gamma -S$ is a smooth complex manifold (of dimension $k$). 

*The restriction of $\Pi_2$ to $Y$ is locally biholomorphic away from a nowhere dense analytical subset $Z$. The union $S\cup Z$ is an analytical subset in $X\times X$. 

*The image $\Pi_2(Z\cup S)$ is a proper analytical subset $W\subset X$. 

*The preimage $\Pi_2^{-1}(W)$ is an analytical subset in $X\times X$. 

*The restriction of $\Pi_2$ to $M=Y - \Pi_2^{-1}(W)$ is a proper map (in view of compactness of $X$) which is locally biholomorphic on $M$. 

*The intersection $V=\Pi_2^{-1}(W)\cap \Gamma$ is a proper analytical subset. 

*The restriction of $\Pi_2$ to $M$ is a covering map to its image $\Omega$ (Ehresmann's "Stack of Records" theorem).  The image $\Omega= X- \Pi_2(W)$ is Zariski open and dense (in your terminology) in $X$. 
