An Eigenvalue Estimate for the $\bar \partial$-Laplacian

Abstract

We study the $\bar\partial$-Laplacian on forms taking values in L^k, a high power of a semipositive line bundle over a compact complex manifold, and give an estimate of the number of eigenvalues smaller than λ. Examples show that the estimate gives the right order of magnitude in terms of the two spectral parameters k and λ.