Kawamata log terminal singularities of full rank

TL;DR

This paper investigates high-rank Kawamata log terminal singularities, showing they degenerate to toric quotients and revealing their structure through automorphisms and affine subsets, with applications to complements and dual complexes.

Contribution

It proves that full-rank klt singularities degenerate to toric quotients and characterizes Fano varieties with large automorphisms as toric, advancing understanding of their structure.

Findings

Full-rank klt singularities degenerate to cones over toric quotients.

Fano type varieties with large automorphisms are toric.

Such Fano varieties contain affine tori as open subsets.

Abstract

We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an open affine subset isomorphic to $\mathbb{G}_m^n$. As a first application, we study complements on klt singularities of full rank. As a second application, we study dual complexes of log Calabi-Yau structures on Fano type varieties with large fundamental group of their smooth locus.