I need to use the following result (that I'm pretty sure is true):



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> **Theorem.** Let $Y$ be a compact complex manifold and $B \subset Y$ be a connected submanifold of codimension one. Then isomorphism classes of connected analytic covers of degree $n$ $$f \colon X \longrightarrow Y,$$ branched at most over $B$, correspond to group homomorphisms $$\varphi \colon \pi\_1(Y - B) \longrightarrow S\_n$$  
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>  with transitive image, up to conjugacy in $S\_n$.
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When $Y$ is a compact Riemann surface this is very classical and a proof can be found for instance in R. Miranda's book *Algebraic curves and Riemann surfaces*, Chapter III.


In the general case, I was told that this must be contained somewhere in Grauert and Remmert's work. I tried to find it in their celebrated paper *Komplexe Räume* (Math. Ann. **136**, 1958), but it is written in German and it is very long and technical, so it is difficult (at least for me) to extract from there the statement that I want. 


So my question is



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> Can someone provide a precise reference for the theorem above? 
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> 
> 



Monodromy representations and branched covers
For a modern treatment of the Grauert Remmert argument see Chapter 4 in the book Several Complex Variables vol 7 by Dethloff and Grauert . For an alternate proof using resolution of singularities see SGA 1 page 255 in the arxiv version . The idea of all these proofs is to restrict to the local case . In the case you are interested in, the local problem is filling a finite covering of a product of a polydisc with a punctured disc .


