Positivity for Higgs vector bundles: criteria and applications

TL;DR

This paper reviews positivity notions for Higgs bundles on smooth projective varieties, establishes criteria for ampleness and nefness, and applies these to show the cotangent bundle of certain minimal surfaces is ample.

Contribution

It introduces criteria for positivity of Higgs bundles and applies them to prove ampleness of cotangent bundles for specific minimal surfaces of general type.

Findings

Criteria of Barton-Kleiman type for Higgs bundle positivity

Cotangent bundle of certain minimal surfaces is ample

Stability of Simpson systems for varieties over algebraically closed fields

Abstract

Working in the category of smooth projective varieties over an algebraically closed field of characteristic 0, we review notions of ampleness and numerical nefness for Higgs bundles which "feel" the Higgs field and formulate criteria of the Barton-Kleiman type for these notions. We give an application to minimal surfaces of general type that saturate the Miyaoka-Yau inequality, showing that their cotangent bundle is ample. This will use results by Langer that imply that also for varieties over algebraically closed field of characteristic zero the so-called Simpson system is stable.