Nef cones of fiber products and an application to the Cone Conjecture

Abstract

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron--Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher dimensional analogues of the Calabi--Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi--Yau pairs, and in each dimension at least three, there exist Schoen varieties with non-polyhedral nef cone. We prove the Kawamata--Morrison--Totaro Cone Conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.

v2. Accepted by Forum Math. Sigma. Besides minor changes, the assumption of Theorem 1.4 and the assumption in Subsection 4.2 were weakened, and Example 3.6 and Lemma 4.6 were newly added