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

Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f:Y\to {\mathbb{P}}^1$ with $-K_Y=f^{\ast }\mathcal{O}(1)$. 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.