Frobenius type positivity of Hodge bundles and applications

Abstract

We define two new Frobenius type positivity notions, $F^t$-ampleness and $F^tGG$-ampleness, for coherent sheaves, show that they are stronger than ordinary ampleness and establish some of their basic properties. The analogous Frobenius type semipositivity notions have also been established. We prove the $F^1GG$-semipositivity of Hodge bundles for semistable morphisms of varieties. This strengthens and generalizes various previous semipositivity results of Fujita, Kawamata, Kollár, Viehweg, Fujino and others. As a consequence we obtain the positivity of top Chern characters of Hodge bundles and Kawamata-Viehweg-Kollár type vanishing theorems for some semistable morphisms. Our results are also applied to study the images of Fano varieties under semistable morphisms and the slope inequality.