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

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.