(Co)-Higgs bundles on Non-K\"ahler Elliptic Surfaces

TL;DR

This paper investigates Higgs and co-Higgs bundles on non-Kähler elliptic surfaces, establishing existence conditions related to the base's genus and describing the structure of non-trivial bundles, including their relation to Poisson structures.

Contribution

It provides new existence criteria for stable Higgs and co-Higgs bundles on non-Kähler elliptic surfaces and characterizes non-trivial rank-2 co-Higgs bundles explicitly.

Findings

Stable Higgs bundles exist only when the base genus ≥ 2.

Stable co-Higgs bundles exist only on Hopf surfaces (genus 0 base).

Complete description of non-trivial rank-2 co-Higgs bundles as Poisson structures.

Abstract

In this paper, we study Higgs and co-Higgs bundles on non-K\"ahler elliptic surfaces. We show, in particular, that non-trivial stable Higgs bundles only exist when the base of the elliptic fibration has genus at least two and use this existence result to give explicit topological conditions ensuring the smoothness of moduli spaces of stable rank-2 sheaves on such surfaces. We also show that non-trivial stable co-Higgs bundles only exist when the base of the elliptic fibration has genus 0, in which case the non-K\"ahler elliptic surface is a Hopf surface. We then given a complete description of non-trivial co-Higgs bundles in the rank 2 case; these non-trivial rank-2 co-Higgs bundles are examples of non-trivial holomorphic Poisson structures on $\mathbb{P}^1$-bundles over Hopf surfaces.