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

TL;DR

This paper explores the relationship between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic, establishing new implications and applications including a proof of Kawamata-Viehweg vanishing and results related to Fujita's conjecture.

Contribution

It demonstrates the derivation of Bogomolov's instability from the Miyaoka-Sakai theorem and introduces a partial version that still implies key vanishing theorems, with applications to del Pezzo surfaces and Fujita's conjecture.

Findings

Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem

A partial version of Miyaoka-Sakai theorem suffices for Mumford-Ramanujam vanishing

New proof of Kawamata-Viehweg vanishing on del Pezzo surfaces

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.