Counterexamples to Kodaira’s vanishing and Yau’s inequality in positive characteristics

Abstract

We generalize Tango’s theorem on the Frobenius map of the first cohomology groups to higher-dimensional algebraic varieties in characteristic $p>0$. As an application we construct counterexamples to Kodaira vanishing in higher dimension and prove the Ramanujam-type vanishing on surfaces which are not of general type when $p\ge 5$.