Kodaira type vanishing theorem for the Hirokado variety

TL;DR

This paper proves a specific Kodaira vanishing result for the Hirokado variety, a non-liftable Calabi-Yau threefold in characteristic 3, demonstrating a form of vanishing despite the failure of classical methods.

Contribution

It establishes a Kodaira type vanishing theorem for the Hirokado variety, a case where traditional approaches do not apply due to non-liftability.

Findings

Proves $H^1(X, L^{-1})=0$ for certain ample line bundles on the Hirokado variety

Shows vanishing holds despite the variety's non-liftability to characteristic 0 or $W_2$

Provides insight into Kodaira vanishing in positive characteristic for non-liftable varieties

Abstract

The Hirokado variety is a Calabi-Yau threefold in characteristic 3 that is not liftable either to characteristic~0 or the ring $W_2$ of the second Witt vectors. Although Deligne-Illusie-Raynaud type Kodaira vanishing cannot be applied, we show that $H^1(X, L^{-1})=0$, for an ample line bundle such that $L^3$ has a non-trivial global section, holds for this variety.