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

@article{Wu2020AnEE, title={An eigenvalue estimate for the \$\bar\{\partial\}\$-Laplacian associated to a nef line bundle}, author={Jingcao Wu}, journal={arXiv: Complex Variables}, year={2020} }

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…

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It is a sequel to (Wu in arXiv:2003.05187). In that paper, we introduce a notion called modified ideal sheaf in order to make an asymptotic estimate for the order of the cohomology group. Here we…

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