Finiteness results for 3-folds with semiample anticanonical bundle

The purpose of this paper is to give some evidence for the Morrison–Kawamata cone conjecture for klt pairs. Roughly speaking, the cone conjecture predicts that in appropriate ‘Calabi–Yau-type’ situations, the groups of automorphisms and pseudo-automorphisms of a projective variety act with rational polyhedral fundamental domain on the nef and movable cones of the variety. (See Section 1 for the precise statement.) In this paper we prove some statements in this direction, in the case of a mildly singular 3-fold with semiample anticanonical bundle of positive Iitaka dimension. Let us say a bit about how such a 3-fold looks geometrically. The anticanonical bundle defines a contraction morphism X→S to a positive-dimensional base; by adjunction all smooth fibres are varieties whose canonical bundle is torsion. So the generic fibre is a point, an elliptic curve, or a Calabi–Yau surface, according as the Iitaka dimension is 3, 2, or 1. (Here a Calabi–Yau surface means an abelian, K3, Enriques or hyperelliptic surface.) If the contraction morphism is equidimensional, classification results due to Kodaira and Miranda (for fibre dimension 1) and Kulikov and Crauder–Morrison (for fibre dimension 2) give information about the singular fibres. (See [5] for details of these classification results.) We will see that all 3-folds of this kind fall inside the scope of the Morrison–Kawamata cone conjecture. Our main result is the following finiteness theorem, which can be regarded as a weak form of the conjecture for these varieties. (All varieties are assumed projective, over an algebraically closed field of characteristic zero.)