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

Abstract

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.

Reductive quotients of klt singularities

“…However, klt type singularities are better behaved than general rational singularities. For instance, we know that the fundamental group of a klt type singularity is finite [66,11] and it satisfies the Jordan property [12], while the fundamental group of a rational singularity can be an arbitrary Q-superperfect group [36]. In a similar vein, klt type singularities are local versions of Mori dream spaces [13], i.e., their local Cox ring is finitely generated, while this is no longer the case for general rational singularities.…”

“…Thus, the finiteness of the iteration follows from the finiteness of the chain of finite abelian covers by the groups A pnq in the bottom row of the diagram. But these covers ramify only over the singular locus, so by [66,28,11] (depending on the context) they will eventually become trivial.…”

Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

“…Although the conjecture is largely unknown in general, the profinite completion of π 1 (X reg ) is known to be finite: in fact, it is proved in [Xu14, Theorem 2] and [GKP16b, Theorem 1.13] that for X weak log Fano, the étale fundamental group of X reg is finite. Recently the author is informed that this conjecture is settled by [Bra20].…”

“…We will see in §6 that Conjecture 2 implies Conjecture 3. Recently the author is informed that Part (i) of the Conjecture 3 is settled by [Bra20]. In the sequel let us briefly explain the ideas of the proof of Theorem A and Theorem B:…”

The Jordan property for local fundamental groups

“…It is natural to ask whether it is possible to show that the local (resp., regional) fundamental group of a klt singularity carries any further structure besides its finiteness. In [Bra20, Corollary 6], it is shown that all finite groups can appear as regional fundamental groups of quotient singularities. At a first glance, it would then seem that such question is untenable.…”