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arXiv 2105.03364math.DG
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紧Kähler双曲流形上全纯向量丛的Hodge理论

Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold

Teng Huang

备注：
32 pages, Appeared in IMRN
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摘要：

设$E$是具有负截面曲率$sec\leq -K<0$的紧致Kähler流形$(X,ω)$上的全纯向量丛，$Δ_{E}$是$E$上的Chern联络.本文证明了如果$C:=|[Λ,iΘ(E)]|\leq c_{n}K$，则$(X,E)$满足一族陈数不等式.本文主要研究泛覆盖空间$\tilde{X}$上提升丛$\tilde{E}$上的$L^{2}$ $\bar{\partial}_{\tilde{E}}$ -调和形式。我们还观察到Laplace-Beltrami算子$Δ_{\bar{\partial}_{\tilde{E}}}$的本征值与$X$的欧拉特征线之间有密切的关系。准确地说，如果在$X$上存在一个线丛$L$，使得$χ^{p}(X,L^{\otimes m})$对于某些整数$p\in[0,n]$不是常数，则$X$的欧拉特征线满足$(-1)^{n}χ(X)\geq (n+1)+\lfloor\frac{c_{n}K}{2nC} \rfloor$。

英文摘要：

Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(X,ω)$ with negative sectional curvature $sec\leq -K<0$, $Δ_{E}$ be the Chern connection on $E$. In this article we show that if $C:=|[Λ,iΘ(E)]|\leq c_{n}K$, 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 $Δ_{\bar{\partial}_{\tilde{E}}}$ and the Euler characteristic of $X$. Precisely, if there is a line bundle $L$ on $X$ such that $χ^{p}(X,L^{\otimes m})$ is not constant for some integers $p\in[0,n]$, then the Euler characteristic of $X$ satisfies $(-1)^{n}χ(X)\geq (n+1)+\lfloor\frac{c_{n}K}{2nC} \rfloor$.

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