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

TL;DR

This paper solves the prescribed Hermitian-Yang-Mills tensor problem on compact Kähler manifolds, establishing existence and uniqueness results using a new comparison theorem, and deriving related Chern number inequalities.

Contribution

It introduces a novel comparison theorem for Hermitian-Yang-Mills tensors and applies it to solve the prescribed tensor problem with implications for Chern number inequalities.

Findings

Existence and uniqueness of Hermitian metrics satisfying prescribed Hermitian-Yang-Mills tensors.

Development of a new comparison theorem for Hermitian-Yang-Mills tensors.

Derivation of quantitative Chern number inequalities for bundles and manifolds.

Abstract

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact K\"ahler 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}\sqrt{-1} R^{h_0} $ is positive definite. Then for any Hermitian positive definite tensor $ P\in \Gamma\left(M,E^*\otimes \overline E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$\Lambda_{\omega_g} \sqrt{-1} R^h=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact K\"ahler manifolds.