  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.
