Lefschetz principle-type theorems for curve semistable Higgs sheaves and applications to elliptic surfaces

TL;DR

This paper establishes Lefschetz principle-type theorems for Higgs sheaves, reducing a conjecture about their Chern classes to the complex case and proving it for elliptic surfaces.

Contribution

It introduces Lefschetz-type theorems for semistable Higgs sheaves and applies them to verify a conjecture on Chern classes over elliptic surfaces.

Findings

Reduced the conjecture to the complex case for Higgs bundles.

Proved the conjecture for elliptic surfaces.

Connected vanishing Chern classes to nilpotent, H-nflat Higgs bundles.

Abstract

I prove "Lefschetz principle"-type theorems for semistable and curve semistable Higgs sheaves on smooth projective varieties defined over an algebraically closed field of characteristic $0$. These theorems are applied to reduce a conjecture, about curve semistable Higgs bundles, from the previous general setting to the complex case. Since this conjecture is equivalent to vanishing of Chern classes of H-nflat Higgs bundles, I consider these last ones over elliptic surfaces. I reduce one more time the conjecture to nilpotent, H-nflat Higgs bundles, and I prove it on elliptic surfaces.