Stable Higgs bundles over positive principal elliptic fibrations

TL;DR

This paper classifies stable Higgs bundles on positive principal elliptic fibrations over compact Kähler orbifolds, showing they are pullbacks of stable Higgs bundles on the base, twisted by line bundles.

Contribution

It extends previous results on stable vector bundles to stable Higgs bundles, providing a complete description in the setting of positive principal elliptic fibrations.

Findings

Stable Higgs bundles on M are pullbacks of stable Higgs bundles on X.

Every stable Higgs bundle on M has the form (L⊗Π*B₀, Π*Φₓ).

The classification parallels the vector bundle case, incorporating Higgs fields.

Abstract

Let $M$ be a compact complex manifold of dimension at least three and $\Pi : M\rightarrow X$ a positive principal elliptic fibration, where $X$ is a compact K\"ahler orbifold. Fix a preferred Hermitian metric on $M$. In \cite{V}, the third author proved that every stable vector bundle on $M$ is of the form $L\otimes \Pi^*B_0$, where $B_0$ is a stable vector bundle on $X$, and $L$ is a holomorphic line bundle on $M$. Here we prove that every stable Higgs bundle on $M$ is of the form $(L\otimes \Pi^*B_0,\Pi^*\Phi_X)$, where $(B_0, \Phi_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $M$.