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, $\pi_1(X^k)=0$.
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 $C$.