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

Abstract

Let 𝘠 be a smooth projective 3-fold admitting a K3 fibration 𝘧:𝘠→ ℙ¹ with -𝘒ᵧ = = 𝘧*𝔒(1). We show that the pseudo-automorphism group of 𝘠 acts with finitely many orbits on the codimension one faces of the movable cone if 𝘏³ (𝘠, 𝘊) = 0, confirming a special case of the Kawamata-MorrisonTotaro cone conjecture. In Coates-Corti-Galkin-Kasprzyk 2016, Przyjalkowski 2018, and CheltsovPrzyjalkowski 2018, the authors construct log Calabi-Yau 3-folds with K3 fibrations satisfying the hypotheses of our theorem as the mirrors of Farro 3-folds.