On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic

Abstract

In this paper, we study the equivalence between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic. We show that Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem. Conversely, it implies a partial version of the Miyaoka-Sakai theorem that lacks the vanishing conclusion. This partial version is still sufficient to deduce the Mumford-Ramanujam vanishing theorem. Additionally, we identify a class of surfaces in positive characteristic for which the Miyaoka-Sakai theorem (or a weaker variant), or the Kawamata-Viehweg vanishing theorem holds. In particular, we present a new proof of the Kawamata-Viehweg vanishing theorem on smooth del Pezzo surfaces. As an application of the Miyaoka-Sakai theorem, we obtain Reider-type results concerning Fujita's conjecture.