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

TL;DR

This paper investigates the positive characteristic analogue of the Campana-Peternell conjecture, proving that certain Fano 4-folds with nef tangent bundles and higher Picard number are homogeneous.

Contribution

It provides an affirmative result for Fano 4-folds with nef tangent bundle in positive characteristic, extending the conjecture beyond characteristic zero.

Findings

Fano 4-folds with nef tangent bundle and Picard number > 1 are homogeneous in positive characteristic

Supports the positive characteristic version of the Campana-Peternell conjecture

Advances understanding of Fano varieties in algebraic geometry

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.