ag.algebraic geometry - Birational invariants and fundamental groups

The paper of Keum and Zhang

"Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds",
Journal of Pure and Applied Algebra vol. 170

provides some answers to Tom's question "What happens to the fundamental group of a singular variety when removing the singular points?"

It appears that the fundamental group of the smooth part can be quite complicate also in simple situations. For instance, one of the results in the paper is the following:

"THEOREM. Let X be a $K3$ surface with at worst Du Val singularities (then X is still simply connected), and let $X^0$ be its smooth part. The number $c$=#(Sing $X$) is bounded by $16$, and if $c=16$ then $pi_1(X^0)$ is infinite".

So, given for instance a quartic surface $X subset mathbb{P}^3$ with $16$ nodes (a Kummer surface) we have

$pi_1(X)={1}$, but $pi_1(X^0)$ is infinite!

This follows from the fact that $X^0$ has an étale $mathbb{Z}_2$-cover $Y^0 to X^0$, where $Y^0$ is an Abelian surface minus 16 points.

For smaller values of $c$, the group $pi_1(X^0)$ is finite, but not trivial in general.

In higher dimension, there is the following

"CONJECTURE. Let $V$ be a $mathbb{Q}$-Fano $n$-fold. Then the topological fundamental group $pi_1(V^0)$ of the smooth part $V^0$ of $V$ is finite".

(a normal variety $V$ with at worst log terminal singularities is $mathbb{Q}$-Fano if, by definition, the anti-canonical divisor $−K_V$ is $mathbb{Q}$-Cartier and ample).