On the relative Morrison-Kawamata cone conjecture (II)

Abstract

Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of weak log canonical models, (2) the existence of a weak rational polyhedral fundamental domain under the action of birational automorphism groups, and (3) the finiteness of varieties as targets of contractions. As an application, the finiteness of weak log canonical models and the weak Morrison-Kawamata cone conjecture in relative dimensions ≤ 2 \leq 2 are established.