📝 SummarXiv

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.