JournalsjemsVol. 14, No. 3pp. 605–657

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

  • Pham Huu Tiep

    University of Arizona, Tucson, USA
  • Robert M. Guralnick

    University of Southern California, Los Angeles, United States
A problem of Kollár and Larsen on finite linear groups and crepant resolutions cover
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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. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age ≤ 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 GUd(C)/GGU_{d}(\mathbb C)/G having shortest closed geodesics of bounded length, and of quotients Cd/G\mathbb C^{d}/G having a crepant resolution.

Cite this article

Pham Huu Tiep, Robert M. Guralnick, A problem of Kollár and Larsen on finite linear groups and crepant resolutions. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 605–657

DOI 10.4171/JEMS/313