Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold

Abstract

Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(X,\omega )$ with negative sectional curvature $sec\le -K<0$, ${\Delta }_E$ be the Chern connection on $E$. In this article we show that if $C:=|[\Lambda ,i\Theta (E)]|\le c_{n}K$, 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 $(-1{)}^n\chi (X)\ge (n+1)+\lfloor \frac{c_{n}K}{2nC}\rfloor$.

32 pages, Appeared in IMRN