Curve semistable Higgs bundles and smooth projective varieties whose canonical bundle is ample
Armando Capasso
TL;DR
The paper develops a positivity framework for Higgs bundles on smooth projective varieties with ample canonical bundle, centered on the Simpson system $\mathfrak{S}=(S,\varphi)$ with $S=\Omega^1_X\oplus\mathcal{O}_X$. It provides a new proof of the stability of $\mathfrak{S}$ under $K_X$-polarization and derives the Guggenheimer–Yau inequality, with equality implying discriminant vanishing and curve semistability. Through the notions of $H$-ample/$H$-nef Higgs bundles and the Higgs-Grassmann machinery, it deduces ampleness of $\Omega^1_X$ in the extremal case and analyzes twisted systems $\mathfrak{S}_{\beta}$ to obtain nef/ample results for $\Omega^1_X(-\beta K_X)$ and concrete genus bounds for curves on $X$. The work further establishes algebraic hyperbolicity with an explicit lower bound and, in the complex setting, uniformization by the complex ball, tying positivity of Higgs bundles to strong geometric constraints and hyperbolicity. It also clarifies the relationship between cotangent ampleness and the positivity of the Simpson system, highlighting cases where $GY(X)=0$ cannot occur.
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.