Let $X^k$ be a complete intersection in $\mathbb CP^n$ of dimension $k>1$. Is it true that a smooth variety obtained by resolutions of singularities of $X$ is simply-connected?
Note that in the case $X^k$ is smooth itself, $\pi_1(X^k)=0$.
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No. For example, take $X$ to be the projective cone in $\mathbb P^3$ of a smooth plane curve $C$ of degree at least $3$. Then blowing up the vertex gives a resolution that is a $\mathbb P^1$-bundle over $C$, and the fundamental group is that of $C$.
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$\begingroup$ thanks a lot! Do you think there can be any positive statement in this direction? $\endgroup$ Commented Nov 1, 2012 at 0:47