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Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$ be a holomorphic map of degree one. How to prove that for each $x\in N$ the set $f^{-1}(x)$ is simply-connected?

Added. The above statement would follow from a different one:
There is an $\varepsilon$-neighbourhood $U_x$ of $x$ such that $f^{-1}(x)$ is the deformation retract of $f^{-1}(U_x)$. This second statement seems very plausible but I don't know how to prove it.

Indeed, since $U_x$ and $f^{-1}(U_x)$ are birational, they have same fundamental group (Griffiths Harris page 474), i.e $f^{-1}(U_x)$ is simply connected.

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Dmitri, I think your second statement will hold for any proper map f between manifolds... with some argument like, on general grounds there is an open subset U of f^{-1}(x) with the inclusion a homotopy equivalence; then if U_x is an epsilon-neighborhood of x in N-f(M-U) we will have f^{-1}(x) inside U_x inside U giving the deformation retraction up to homotopy. – Dustin Clausen Dec 30 '10 at 22:07
A suggestion about the second statement : follow the gradient of $d_x^2\circ f$, with $d_x$ the distance to $x\in N$ with respect to a suitable (real analytic, hermitian, kaehler?) metric on $N$, the gradient being taken with respect to another such metric on $M$. – BS. Dec 31 '10 at 10:20
Isn't Milnor's curve selection lemma adapted to this situation ? Namely, if $(z_n)$ is a sequence of critical points in $f^{-1}(U\setminus x)$, with $f(z_n)\to x$, one may assume by properness that $z_n\to z\in f^{-1}(x)$, and by selection lemma there is an analytic curve $\gamma:[0,\delta[\to M$ of critical points with $\gamma(0)=z$ and $f(\gamma(t))\neq x$ for $t>0$. It remains to rule this out... – BS. Dec 31 '10 at 15:47
but this seems easy : $t\mapsto d_x^2(f(\gamma(t)))$ has to be constant, which is absurd. – BS. Dec 31 '10 at 15:55
To conclude the argument, it would suffice to know that $f^{-1}(x)$ is a retract (not necessarily by deformation) of some neighborhood in $M$. But this surely results from sharp results on the structure of analytic sets (triangulability - in a "relative" situation), – BS. Dec 31 '10 at 16:13
up vote 5 down vote accepted

[This is just a recap/editing of comments that seem to give an answer]

A suggestion about the second statement : follow the gradient of $\phi=d^2_x∘f$, where $d_x$ is the distance to $x\in N$ with respect to a real analytic metric on N, the gradient being taken with respect to another such metric on M.

For this to work, one has first to rule out the possibility that $\phi$ has critical points arbitrarily near $f^{-1}(x)$, but not in it (note that each $z \in f^{-1}(x)$ is a critical point of $\phi$). But the curve selection lemma (CSL) is precisely adapted to this situation. Namely, if $(z_n)$ is a sequence of critical points of $\phi$ in $f^{−1}(N∖x)$, with $f(z_n)\to x$, one may assume by properness of $f$ that $z_n\to z\in f^{−1}(x)$. Then the curve selection lemma (applied to the semi-analytic set $Z=Crit(\phi)\setminus f^{−1}(x)$, and the point $z$ in its closure) gives an analytic curve $\gamma:[0,\delta[\to M$ of critical points of $\phi$ with $\gamma(0)=z$ and $f(\gamma(t))\neq x$ for $t>0$. But this is absurd : $t\mapsto \phi(\gamma(t))$ would have to be constant.

To conclude the argument, one may resort to Lojasiewicz inequality (itself a deep consequence of CSL) $|\mathrm{grad} \phi(z)| \geq c \phi(z)^\alpha$ if $\phi(z)<\epsilon$, for some $\alpha\in[0,1[$, $c>0$, $\epsilon >0$. This implies that the gradient trajectories have uniformly bounded lengths since they also are those of $\phi^{1-\alpha}$, which has gradient norm $\geq(1-\alpha)c$ on $\phi^{-1}(]0,\epsilon[)$.

This implies that the neighborhood $\phi^{-1}([0,\epsilon[)$ deformation retracts to $f^{-1}(x)=\phi^{-1}(0)$.

It must be said, however, that this is only a small fragment of the theory of (semi-,sub-) analytic sets developed by Lojasiewicz, Hironaka, etc (with previous work by Whitney and Thom) since the 60's, and that I am not competent to retrace here the story of its development, nor to give proper attributions. In particular, I don't know to what extent it relies on Hironaka's resolution of singularities.

Some references I've found are Milnor's 1969 Singular points of complex hypersurfaces where I first learned CSL (in the semi-algebraic context), Denef and van den Dries 1988 p-adic and Real Subanalytic Sets, Bierstone and Milman's 1988 Semianalytic and subanalytic sets, Kurdyka's 1998 On gradients of functions definable in o-minimal structures. Two papers by Lojasiewicz seem also very relevant (but not easily accessible).

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Thanks a lot! I'll accept this answer, since this resolves also the original question. – Dmitri Jan 1 '11 at 15:16

This is probably not the most optimal way to do it, but this was what came to mind right away. Also, this is in some sense more general than what you ask and in some sense it is less. Finally, it is not a complete solution, but it might give you some ideas.

So, let $M$ and $N$ be complex projective manifolds.

According to Shokurov's rational connectedness conjecture the set $f^{-1}(x)$ is rationally chain connected. This was proved by Hacon and McKernan here. Rationally connected manifolds are simply connected. This is due to Campana and Kollár-Miyaoka-Mori and can be found for example in this book.

You are not there yet, but I think you should be able to prove that the graph of the components of $f^{-1}(x)$ should be a tree and that the individual components are simply connected using the above argument. Check Kollár's article in the book as well.

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Sándor thanks! This looks indeed as a possible approach. – Dmitri Dec 30 '10 at 19:40

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