Hodge theory of holomorphic vector bundle on compact K\"{a}hler hyperbolic manifold

TL;DR

This paper explores the Hodge theory of holomorphic vector bundles on compact Kähler hyperbolic manifolds with negative curvature, establishing inequalities for Chern numbers and linking eigenvalues of Laplacians to the Euler characteristic.

Contribution

It introduces new Chern number inequalities for bundles over negatively curved Kähler manifolds and connects Laplacian eigenvalues with the Euler characteristic.

Findings

Established Chern number inequalities under curvature conditions.

Linked eigenvalues of Laplace-Beltrami operator to the Euler characteristic.

Identified conditions involving line bundles affecting the Euler characteristic.

Abstract

Let $E$ be a holomorphic vector bundle over a compact K\"{a}hler manifold $(X,\omega)$ with negative sectional curvature $sec\leq -K<0$, $\Delta_{E}$ be the Chern connection on $E$. In this article we show that if $C:=|[\Lambda,i\Theta(E)]|\leq c_{n}K$, then $(X,E)$ satisfy a family of Chern number inequalities. The main idea in our proof is study the $L^{2}$ $\bar{\partial}_{\tilde{E}}$-harmonic forms on lifting bundle $\tilde{E}$ over the universal covering space $\tilde{X}$. We also observe that there is a closely relationship between the eigenvalue of the Laplace-Beltrami operator $\Delta_{\bar{\partial}_{\tilde{E}}}$ and the Euler characteristic of $X$. Precisely, if there is a line bundle $L$ on $X$ such that $\chi^{p}(X,L^{\otimes m})$ is not constant for some integers $p\in[0,n]$, then the Euler characteristic of $X$ satisfies $(-1)^{n}\chi(X)\geq (n+1)+\lfloor\frac{c_{n}K}{2nC} \rfloor$.