The cone conjecture for abelian varieties

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.