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

TL;DR

This paper establishes a decomposition theorem for nef cones of smooth fiber products over curves, describes the nef cone of Schoen varieties, and proves the Cone Conjecture for these varieties, extending previous results.

Contribution

It introduces a decomposition theorem for nef cones of fiber products and verifies the Cone Conjecture for Schoen varieties, a class of higher-dimensional Calabi--Yau analogues.

Findings

Decomposition theorem for nef cones of fiber products.

Description of nef cones of Schoen varieties.

Proof of the Cone Conjecture for Schoen varieties.

Abstract

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their N\'eron--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.