The cone conjecture for Calabi-Yau pairs in dimension two

A central idea of minimal model theory as formulated by Mori is to study algebraic varieties using convex geometry. The cone of curves of a projective variety is defined as the convex cone spanned by the numerical equivalence classes of algebraic curves; the dual cone is the cone of nef line bundles. For Fano varieties (varieties with ample anticanonical bundle), these cones are rational polyhedral by the cone theorem [22, Theorem 3.7]. For more general varieties, these cones are not well understood: they can have infinitely many isolated extremal rays, or they can be “round”. Both phenomena occur among Calabi-Yau varieties such as K3 surfaces, which can be considered the next simplest varieties after Fano varieties. The Morrison-Kawamata cone conjecture would give a clear picture of the nef cone for Calabi-Yau varieties [26, 27, 19]. The conjecture says that the action of the automorphism group of the variety on the nef cone has a rational polyhedral fundamental domain. (The conjecture includes an analogous statement about the movable cone; see section 1 for details.) Thus, for Calabi-Yau varieties, the failure of the nef cone to be rational polyhedral is always explained by an infinite discrete group of automorphisms of the variety. It is not clear where these automorphisms should come from. Nonetheless, the conjecture has been proved for Calabi-Yau surfaces by Sterk, Looijenga, and Namikawa [39, 30, 19], the heart of the proof being the Torelli theorem for K3 surfaces of Piatetski-Shapiro and Shafarevich [3, Theorem 11.1]. Kawamata proved the cone conjecture for all 3-dimensional Calabi-Yau fiber spaces over a positive-dimensional base [19]. The conjecture is wide open for CalabiYau 3-folds, despite significant results by Oguiso and Peternell [35], Szendröi [41], Uehara [44] and Wilson [46]. The conjecture was generalized from Calabi-Yau varieties to klt Calabi-Yau pairs (X,∆) in [43]. Here ∆ is a divisor on X and “Calabi-Yau” means that KX + ∆ (rather than KX) is numerically trivial. In this paper we prove the cone conjecture for all klt Calabi-Yau pairs of dimension 2 (Theorem 4.1), using the geometry of groups acting on hyperbolic space and reduction to the case of K3 surfaces. This is enough to show that the conjecture is reasonable in the greater generality of pairs. More concretely, the theorem gives control over the nef cone and the automorphism group for a large class of rational surfaces, including the Fano surfaces as well as many others. In particular, we get a good description of when the Cox ring (or total coordinate ring) is finitely generated in this class of surfaces (Corollary 5.1). Thanks to Caucher Birkar, Igor Dolgachev, Brian Harbourne, Artie PrendergastSmith, and Chenyang Xu for their comments.