Kawamata log terminal singularities of full rank

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.

arXiv comment35 pages

References in corpus (7)

• Finiteness of algebraic fundamental groups
• Blow-ups of canonical singularities
• ACC for minimal log discrepancies of exceptional singularities
• Terminal valuations and the Nash problem
• The local fundamental group of a Kawamata log terminal singularity is finite
• The Jordan property for local fundamental groups
• Strong $(\delta ,n)$-complements for semi-stable morphisms

Cited by in corpus (2)

• Iteration of Cox rings of klt singularities
• A geometric characterization of toric singularities