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

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 $$\varphi \colon \pi_1(Y - B) \longrightarrow S_n$$
with transitive image, up to conjugacy in $S_n$.

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

Can someone provide a precise reference for the theorem above?

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  • $\begingroup$ So the statement is certainly true for topological covers of $Y-B$. Is the issue that you're not sure if every topological cover of a complex manifold is uniquely equipped with a complex structure compatible with the covering map? $\endgroup$
    – Will Chen
    Oct 28, 2016 at 15:24
  • $\begingroup$ The hard problem is the extension of the pull-back complex structure from the cover of $Y-B$ to a compact manifold $X$. $\endgroup$
    – Francesco Polizzi
    Oct 28, 2016 at 18:42
  • $\begingroup$ Have you looked the encyclopedia in math series vol 74 title Several Complex Variables vol 7 Chapter 4 by Dethloff and Grauert .You can get the precise reference from there. $\endgroup$
    – Mohan Ramachandran
    Oct 28, 2016 at 18:49
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    $\begingroup$ You can also look at SGA 1 page 255 in the arxiv version where a proof of Grauert Remmert theorem is given . $\endgroup$
    – Mohan Ramachandran
    Oct 28, 2016 at 18:59
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    $\begingroup$ @Francesco Polizzi Thank you .I have posted my remarks as an answer. $\endgroup$
    – Mohan Ramachandran
    Oct 28, 2016 at 21:15

1 Answer 1

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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 .

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