Liftability and vanishing theorems for Fano threefolds in positive characteristic II

TL;DR

This paper proves that smooth Fano threefolds in positive characteristic generally lift to Witt vectors and satisfy key vanishing theorems, with specific exceptions, advancing understanding of their geometric properties.

Contribution

It extends previous results by establishing liftability and vanishing theorems for a broad class of Fano threefolds, except in certain cases related to the ampleness of |-K_X| and Picard group structure.

Findings

Fano threefolds lift to Witt vectors in most cases

They satisfy Akizuki-Nakano vanishing and Hodge-de Rham degeneration

Arbitrary smooth Fano threefolds are quasi-F-split under certain conditions

Abstract

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results except when $|-K_X|$ is very ample and the Picard group is generated by $\omega_X$. To this end, we show that an arbitrary smooth Fano threefold is quasi-$F$-split when the Picard number or the Fano index is larger than one.