A problem of Kollar and Larsen on finite linear groups and crepant resolutions

TL;DR

This paper addresses a problem in algebraic geometry related to the structure of finite linear groups generated by elements with bounded age, impacting the understanding of crepant resolutions and quotient spaces.

Contribution

It solves a problem posed by Kollar and Larsen by characterizing finite irreducible linear groups generated by elements of age at most 1 and bounds their dimension.

Findings

Bound the dimension of finite irreducible linear groups generated by elements of bounded deviation.

Derive properties of symmetric spaces with short closed geodesics.

Analyze quotients of affine space with crepant resolutions.

Abstract

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi-Yau varieties. In this paper, we solve a problem raised by J. Kollar and M. Larsen on the structure of finite irreducible linear groups generated by elements of age at most 1. More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence of our main results, we derive some properties of symmetric spaces for the unitayr group having shortest closed geodesics of bounded length, and of quotients of affine space by a finite group having a crepant resolution.