Fano $4$-folds with nef tangent bundle in positive characteristic

Abstract

In characteristic 0, the Campana-Peternell conjecture claims that the only smooth Fano variety with nef tangent bundle should be homogeneous. In this paper, we study the positive characteristic version of the Campana-Peternell conjecture. In particular, we give an affirmative answer for Fano 4-folds with nef tangent bundle and Picard number greater than one.