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

TL;DR

This paper investigates the asymptotic growth of the von Neumann dimension of harmonic spaces associated with semipositive line bundles over covering manifolds, providing local estimates and asymptotic formulas.

Contribution

It introduces new asymptotic estimates for the von Neumann dimension of harmonic spaces of line bundle valued forms on covering manifolds with semipositive line bundles.

Findings

Asymptotic estimate for the von Neumann dimension of harmonic spaces.

Local pointwise norm estimates for harmonic forms.

Estimation of the reduced $L^2$-Dolbeault cohomology dimension.

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.