Semi-stable higgs bundles and flat bundles over non-K\"ahler manifolds

Research paper by Changpeng Pan, Chuanjing Zhang, Xi Zhang

Indexed on: 12 Nov '19Published on: 08 Nov '19Published in: arXiv - Mathematics - Differential Geometry

Abstract

In this paper, we study Higgs bundles and flat bundles over non-K\"ahler manifolds. Suppose $(X,\omega)$ is a compact Hermitian manifold of dimension $n$ satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$, we prove that every semi-stable Higgs bundle $(E,\bar{\partial}_{E},\theta)$ over $(X,\omega)$ with vanishing first and second Chern numbers is an extension of Higgs-Hermitian flat bundles. Moreover, suppose $(X,\omega)$ also satisfies $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for every $[\eta]\in H^{0,1}(X)$, we prove that there is an equivalence of categories between the category of poly-stable Higgs bundles with vanishing Chern numbers and the category of semi-simple flat bundles. At last, in rank $2$ case, we show that there is an one-to-one correspondence between the moduli space of semi-stable Higgs bundles with vanishing Chern numbers and the moduli space of flat bundles.