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

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 ) = 0. Moreover, we show that a Kodaira type vanishing holds if X is a Schröer variety [21] or a Schoen variety [20], which extends the similar result given in [25] for the Hirokado variety [9]. 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 [24] does not produce any counter-example to Kodaira vanishing.

Calabi-Yau threefolds in positive characteristic

“…Moreover, we have b 3 (X) = 0 so that non-liftable to characteristic 0, X is unirational and hence simply connected and weak H 1 -Kodaira vanishing holds [44], however, according to [10], it is not liftable to W 2 .…”

“…Then it was hoped that non-liftable Calabi-Yau 3-folds on which Kodaira vanishing does not hold could be produced with this method. However, it turned out that it is impossible [42,44].…”