Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves

TL;DR

This paper investigates the Picard groups of spectral varieties and their implications for the geometry and non-emptiness of Hitchin fibers in moduli spaces of Higgs sheaves on surfaces.

Contribution

It establishes a Noether–Lefschetz type theorem for spectral varieties and derives criteria for the non-emptiness of Hitchin fibers on surfaces.

Findings

Picard groups of spectral varieties are computed for generic cases.

A criterion for non-empty Hitchin fibers on surfaces is provided.

The geometry of Higgs sheaf moduli spaces is analyzed as the second Chern class varies.

Abstract

We study moduli spaces of Higgs sheaves valued in line bundles and the associated Hitchin maps on surfaces. We first work out Picard groups of generic (very general) spectral varieties which holds for dimension of at least 2, i.e., a Noether--Lefschetz type theorem for spectral varieties. We then apply this to obtain a necessary and sufficient condition for the non-emptyness of generic Hitchin fibers for surfaces cases. Then we move on to detect the geometry of the moduli spaces of Higgs sheaves as the second Chern class varies.