Moduli of rank 2 Higgs sheaves on elliptic surfaces

TL;DR

This paper investigates the structure and moduli spaces of rank 2 Higgs sheaves on elliptic surfaces, establishing stability relations, decompositions, and properties of Higgs fields under various geometric conditions.

Contribution

It proves stability implications for Higgs sheaves, describes moduli space decompositions in specific cases, and analyzes Higgs field behavior related to spectral curves on elliptic surfaces.

Findings

Slope-semistability implies semistability on the generic fiber.

Moduli spaces split as products involving sheaves and forms.

Higgs fields relate to spectral curve properties.

Abstract

We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic $\neq 2$, we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular $1$-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface.