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

TL;DR

This paper provides an eigenvalue estimate for the $ar{ullpartial}$-Laplacian on high tensor powers of nef line bundles, offering asymptotic bounds that generalize the Grauert--Riemenschneider conjecture and extend to pseudo-effective line bundles.

Contribution

It introduces a new eigenvalue estimate for the $ar{ullpartial}$-Laplacian on nef line bundles, advancing understanding of cohomology asymptotics and conjecture generalizations.

Findings

Eigenvalue estimates for high tensor powers of nef line bundles

Asymptotic bounds for cohomology group dimensions

Extension of results to pseudo-effective line bundles

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.