Fundamental Groups and Coregularity: Klt Pairs with Trivial and Nef Log-Anticanonical Class

In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we show that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ and arbitrary dimension the fundamental group ${\pi }_1^{\mathrm{r}\mathrm{e}\mathrm{g}}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We reduce this statement to klt Calabi-Yau pairs with standard coefficients of dimension $c$. In the cases $0\le c\le 2$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. Combining the reduction with the recent regular-locus theorem of Fu-Guo-Song-Wang, we prove the coregularity bound in every dimension. In particular, for a projective klt pair $(X,D)$ with $-(K_X+D)$ nef, the orbifold fundamental group of the smooth locus is virtually abelian of rank at most $2\mathrm{d}\mathrm{i}\mathrm{m}X$.