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 $H_2(Y\setminus D)$ 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.

26 pages, 12 figures. Comments welcome!

References in corpus (2)

• Homological mirror symmetry for log Calabi-Yau surfaces
• Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves