  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.
