RC-positivity, comparison theorems and prescribed Hermitian-Yang-Mills tensors II

TL;DR

This paper solves the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles on compact complex manifolds, establishing existence, uniqueness, and inequalities related to the problem.

Contribution

It provides a solution to the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles, including existence, uniqueness, and Chern number inequalities.

Findings

Existence of a unique Hermitian metric satisfying the prescribed tensor condition.

Positivity condition on the initial Hermitian-Yang-Mills tensor.

Quantitative Chern number inequalities for Higgs bundles.

Abstract

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds. Let $ (E,\theta) $ be a Higgs bundle over a compact Hermitian manifold $(M,\omega_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such that the Hermitian-Yang-Mills tensor $ \Lambda_{\omega_g}\left(\sqrt{-1} R^{D^{h_0}}\right) $ of the Higgs connection is positive definite. Then for any Hermitian positive definite tensor $ P\in \Gamma\left(M,E^*\otimes \bar E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$\Lambda_{\omega_g} \left(\sqrt{-1} R^{D^h}\right)=P.$$ We also establish quantitative Chern number inequalities for Higgs bundles.