On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

Abstract

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $\Gamma$, and obtain an asymptotic estimate for the $\Gamma$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^k\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $\Gamma$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manifolds.

Cited by in corpus (3)

• Cohomology dimension growth for Nakano $q$-semipositive line bundles
• The growth of dimension of cohomology of semipositive line bundles on Hermitian manifolds
• On Bergman kernel functions and weak Morse inequalities