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

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 $λ$. In particular, the $λ=0$ 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 $λ=0$ case on a pseudo-effective line bundle.