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

Abstract

Let $(X,\omega )$ be a compact Kähler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that ${\Theta }_{L,h}\ge -\epsilon \omega$ for a small $\epsilon >0$, $E$ be a holomorphic line bundle over $X$. For $k\in {\mathbb{N}}_{+}$, denote by $X_k:=(X,{\omega }^k)$ the Kähler 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 $0\le \lambda <k$. In particular, when $\lambda =0$, we get a numeric bound for the cohomology groups.

arXiv commentThis paper has been withdrawn by the author due to a crucial error in the estimate of s(r)