An eigenvalue estimate for the $\bar{\partial }$-Laplacian associated to a nef line bundle

Abstract

We study the $\bar{\partial }$-Laplacian on forms taking values in $L^k$, a high power of a nef line bundle on a compact complex manifold, and give an estimate of the number of the eigenforms whose corresponding eigenvalues smaller than or equal to $\lambda$. In particular, the $\lambda =0$ case gives an asymptotic estimate for the order of the corresponding cohomology groups. It helps to generalize the Grauert--Riemenschneider conjecture. At last, we discuss the $\lambda =0$ case on a pseudo-effective line bundle.

30 pages. We fix an error before and add more things. Comments are welcome