Curve semistable Higgs bundles and smooth projective varieties whose canonical bundle is ample

TL;DR

This paper proves the stability and semistability of Higgs bundles on smooth projective varieties with ample canonical bundle, establishes a new proof of the Guggenheimer-Yau inequality, and demonstrates algebraic hyperbolicity with a novel lower bound.

Contribution

It provides new proofs of stability, semistability, and the Guggenheimer-Yau inequality for Higgs bundles on varieties with ample canonical bundle, and introduces a unique algebraic hyperbolicity result.

Findings

Higgs bundle stability on varieties with ample canonical bundle

Equality case implies vanishing discriminant class and curve semistability

New lower bound on a constant related to algebraic hyperbolicity

Abstract

Considering the so-called Simpson system on smooth projective varieties, defined over an algebraically closed field of characteristic 0, whose canonical bundle is ample, I give another proof the stability of this Higgs bundle, from which follows another proof of the Guggenheimer-Yau inequality. Where the equality holds, I prove that the discriminant class of the Simpson system vanishes and this Higgs bundle is curve semistable. This result follows from the study of the relations between ampleness and numerically nefness for Higgs bundles which "feel" the Higgs field and (semi)stability. Moreover, I obtain another proof of algebraic hyperbolicity of these varieties which furnishes a lower bound on a real positive constant related to this property; to best of my knowledge, this is the first and unique result of this type.