On diagonal pluriharmonic metrics of $G$-Higgs bundles

Abstract

Let $(E,\Phi )\to (X,{\omega }_X)$ be a Higgs bundle over a compact Kähler manifold. We suppose that the holomorphic vector bundle $E$ decomposes into a direct sum of holomorphic line bundles. In this paper, we give the necessary and sufficient condition for the existence of a diagonal metric which is a solution to the Hermitian-Einstein equation. Our theorem can easily be generalized to $G$-Higgs bundles. We also describe the relationship between the stability condition and our condition using the torus action on the space of Higgs fields.

v4:Many changes. The statement about harmonic maps has been removed from the main theorem. 7 pages. Comments are welcome

References in corpus (2)

• G2 geometry and integrable systems
• Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space