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. 



Topology of the preimage of a point for degree one holomorphic maps
[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*](http://books.google.com/books?id=mBwmipQl0ZoC&lpg=PA112&dq=analytic%2520curve%2520selection%2520lemma&hl=fr&pg=PA112#v=onepage&q&f=false) (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](http://books.google.com/books?id=cuAAFbBYlSsC&lpg=PA25&dq=curve%2520selection%2520lemma&hl=fr&pg=PA25#v=onepage&q=curve%2520selection%2520lemma&f=false)* where I first learned
CSL (in the semi-algebraic context), Denef and van den Dries 1988 *[p-adic and Real Subanalytic Sets](http://p-adic%20and%20Real%20Subanalytic%20Sets/)*,
Bierstone and Milman's 1988 *[Semianalytic and subanalytic sets](http://archive.numdam.org/article/PMIHES_1988__67__5_0.djvu)*, Kurdyka's 1998 *[On gradients of functions definable in o-minimal structures](http://archive.numdam.org/article/AIF_1998__48_3_769_0.djvu)*. [Two](http://www.ams.org/mathscinet-getitem?mr=160856) [papers](http://www.ams.org/mathscinet-getitem?mr=771152) by Lojasiewicz seem also very relevant (but not easily accessible).


