A cone conjecture for log Calabi-Yau surfaces

TL;DR

This paper proves a version of the Morrison cone conjecture for log Calabi-Yau surfaces, showing that certain automorphism and monodromy group actions on nef cones have rational polyhedral fundamental domains, and explicitly describes nef cones when D has up to 6 components.

Contribution

It establishes the Morrison cone conjecture for log Calabi-Yau surfaces and provides explicit descriptions of nef cones for cases with up to six boundary components.

Findings

Automorphism group action admits a rational polyhedral fundamental domain.

Monodromy group action admits a rational polyhedral fundamental domain.

Nef cone of certain surfaces is rational polyhedral and explicitly described.

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 $H_2(Y \setminus D)$ 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 $\le 6$, 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.