On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds

TL;DR

This paper proves a special case of the cone conjecture for certain log Calabi-Yau 3-folds with K3 fibrations, showing the finiteness of the action of their pseudoautomorphism groups on the movable cone faces.

Contribution

It confirms a specific case of the Kawamata--Morrison--Totaro cone conjecture for log Calabi-Yau 3-folds with K3 fibrations and vanishing third cohomology.

Findings

Pseudoautomorphism group acts with finitely many orbits on the cone faces.

Validates a case of the cone conjecture for a class of log Calabi-Yau 3-folds.

Constructs examples as mirrors of Fano 3-folds.

Abstract

Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f : Y \rightarrow \mathbb{P}^1$ with $-K_Y = f^*\mathcal{O}(1)$. We show that the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces of the movable cone if $H^3(Y,\mathbb{C})=0$, confirming a special case of the Kawamata--Morrison--Totaro cone conjecture. In [CCGK16], [P18], and [CP18], the authors construct log Calabi-Yau 3-folds with K3 fibrations satisfying the hypotheses of our theorem as the mirrors of Fano 3-folds.