Semistable Higgs bundles on elliptic surfaces

TL;DR

This paper studies Higgs bundles on elliptic surfaces, proving conditions under which the Higgs field is vertical or scalar, and characterizes slope-semistable Higgs bundles with vanishing discriminant via pull-backs from curves.

Contribution

It establishes new criteria for the structure of Higgs fields on elliptic surfaces based on spectral curve properties and stability conditions.

Findings

If the spectral curve is reduced, then the Higgs field is vertical.

Under certain stability and spectral curve conditions, the Higgs field is scalar.

Provides a characterization of slope-semistable Higgs bundles with vanishing discriminant.

Abstract

We analyze Higgs bundles $(V,\phi)$ on a class of elliptic surfaces $\pi:X\to B$, whose underlying vector bundle $V$ has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of $V$ is reduced, then $\phi$ is vertical, while if $V$ is fiberwise regular with reduced (resp. integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of $B$ and the degree of the fundamental line bundle of $\pi$ (resp., if the fundamental line bundle is sufficiently ample), then $\phi$ is scalar. We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to $X$.