The local fundamental group of a Kawamata log terminal singularity is finite

We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.

Existence of Ball Quotients Covering Line Arrangements

This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up X of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and b-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.

Finiteness of algebraic fundamental groups

We show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.