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 $sec\leq -K<0$ and $D_{E}$ 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 to 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 close 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 $.