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The cone conjecture for abelian varieties

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journal contribution
posted on 13.07.2015, 12:41 by Artie PrendergastArtie Prendergast
The purpose of this paper is to write down a complete proof of the Morrison-Kawamata cone conjecture for abelian varieties. The conjecture predicts, roughly speaking, that for a large class of varieties (including all smooth varieties with numerically trivial canonical bundle) the automorphism group acts on the nef cone with rational polyhedral fundamental domain. (See Section 1 for a precise statement.) The conjecture has been proved in dimension 2 by Sterk-Looijenga, Namikawa, Kawamata, and Totaro [Ste85, Nam85, Kaw97, Tot 10], but in higher dimensions little is known in general. Abelian varieties provide one setting in which the conjecture is tractable, because in this case the nef cone and the automorphism group can both be viewed as living inside a larger object, namely the real endomorphism algebra. In this paper we combine this fact with known results for arithmetic group actions on convex cones to produce a proof of the conjecture for abelian varieties.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Mathematical Sciences (Japan)

Volume

19

Issue

2

Pages

243 - 261

Citation

PRENDERGAST-SMITH, A., 2012. The cone conjecture for abelian varieties. Journal of Mathematical Sciences (University of Tokyo), 19 (2), pp. 243 - 261

Publisher

University of Tokyo

Version

SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2012

Notes

This article was published in the Journal of Mathematical Sciences [University of Tokyo] and is available here with the kind permission of the publisher.

ISSN

1340-5705

Language

en

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