A cone conjecture for log Calabi-Yau surfaces

Abstract

We consider log Calabi-Yau surfaces (Y, D) with singular boundary. In each deformation type, there is a distinguished surface (Y e , D e ) such that the mixed Hodge structure on H 2 (Y \ D) is split. We prove that (1) the action of the automorphism group of (Y e , D e ) on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi-Yau surfaces. In addition, if the number of components of D is ≤ 6, we show that the nef cone of Y e is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.