On Kodaira type vanishing for Calabi-Yau threefolds in positive characteristic

TL;DR

This paper investigates Kodaira vanishing for Calabi-Yau threefolds in positive characteristic, providing bounds and conditions under which vanishing holds, especially for certain special varieties and those with specific Picard properties.

Contribution

It establishes new conditions and bounds for Kodaira vanishing on Calabi-Yau threefolds in positive characteristic, including cases for Schr"oer and Schoen varieties and those with Picard varieties lacking p-torsion.

Findings

Kodaira vanishing holds for Schr"oer and Schoen varieties.

Provides a lower bound for $h^1(X, L^{-1})$ when non-zero.

Modified Raynaud-Mukai construction does not yield counterexamples.

Abstract

We consider Calabi-Yau threefolds $X$ over an algebraically closed field $k$ of characteristic $p>0$ that are not liftable to characteristic $0$ or liftable ones with $p=2$. It is unknown whether Kodaira vanishing holds for these varieties. In this paper, we give a lower bound of $h^1(X, L^{-1})=\dim_k H^1(X, L^{-1})$ if $L$ is an ample divisor with $H^1(X, L^{-1})\ne{0}$. Moreover, we show that a Kodaira type vanishing holds if $X$ is a Schr\"oer variety or a Schoen variety, which extends the similar result given in my previous paper for the Hirokado variety. We show that such kind of vanishing holds for Calabi-Yau manifold whose Picard variety has no $p$-torsion. Also we show that a modified Raynaud-Mukai construction does not produce any counter-example to Kodaira vanishing.