Stable Higgs bundles over positive principal elliptic fibrations

Abstract

Let $M$ be a compact complex manifold of dimension at least three and $\Pi :M\to X$ a positive principal elliptic fibration, where $X$ is a compact Kähler 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 $L\otimes {\Pi }^{\ast }{B}_0$, 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 $(L\otimes {\Pi }^{\ast }{B}_0,{\Pi }^{\ast }{\Phi }_X)$, where $(B_0,{\Phi }_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $M$.