Bergman kernels and eigenvalue estimate of $\bar{\partial}$-laplacian

Abstract

Let $(X,\omega)$ be a compact K\"{a}hler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that $\Theta_{L,h}\geq -\varepsilon\omega$ for a small $\varepsilon>0$, $E$ be a holomorphic line bundle over $X$. For $k\in \mathbb{N}_+$, denote by $X_k:=(X,\omega^k)$ the K\"{a}hler manifold $X$ with new scaled metric $\omega^k=k\omega$. Estimates of the number of eigenvalues smaller than $\lambda$ of the $\debar$-Laplacian on forms on $X_k$ with values in $L^k\otimes E$ are presented for $0\leq \lambda<k$. In particular, when $\lambda=0$, we get a numeric bound for the cohomology groups.