Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors I

Abstract

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(M,{\omega }_g)$. Suppose that there exists a smooth Hermitian metric $h_0$ on $E$ such that the Hermitian-Yang-Mills tensor ${\Lambda }_{{\omega }_g}\left(\sqrt{-1}{R}^{h_0}\right)$ is positive-definite. Then for any positive-definite Hermitian tensor $P\in \Gamma \left(M,E^{\ast }\otimes {\overline{E}}^{\ast }\right)$, there exists a unique smooth Hermitian metric $h$ on $E$ such that $${\Lambda }_{{\omega }_g}\left(\sqrt{-1}{R}^h\right)=P.$$ 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 Kähler manifolds.

The main theorem and Chern number inequalities are improved