Stable Higgs Bundles on ruled surfaces

TL;DR

This paper establishes a correspondence between (semi)stable Higgs bundles on a curve and their pullbacks to a ruled surface, showing an isomorphism of moduli spaces and existence of non-trivial stable Higgs bundles.

Contribution

It proves that pullback preserves (semi)stability for Higgs bundles on ruled surfaces and characterizes the inverse, leading to an isomorphism of moduli spaces.

Findings

Pullback of (semi)stable Higgs bundles remains (semi)stable.

Existence of non-trivial stable Higgs bundles on ruled surfaces over curves of genus ≥ 2.

An isomorphism between moduli spaces of Higgs bundles on the curve and the ruled surface.

Abstract

Let $\pi : X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $\pi$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,\theta)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= \pi^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,\psi)$ of degree $d$ on $C$ whose pullback under $\pi$ is isomorphic to $(V,\theta)$. As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on $X$ whenever $g(C)\geq 2$ and the base field is $\mathbb{C}$.