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.

KEYWORDS:

• Campana-Peternell conjecture
• positive characteristic
• projective manifold with nef tangent bundle

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J35
• 14J45
• 14M17
• 14E30

Acknowledgments

The authors would like to thank an anonymous referee for his/her useful comments. In particular, he/she noted that the computational part of Proposition 3.3 follows from the same argument as [21, Lemma 5.1].

Additional information

Funding

The author is partially supported by JSPS KAKENHI Grant Number 21K03170 and the Sumitomo Foundation Grant Number 190170.