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In this paper, we study the dimension of cohomology of semipositive linebundles over Hermitian manifolds, and obtain an asymptotic estimate for thedimension of the space of harmonic $(0,q)$-forms with values in high tensorpowers of a semipositive line bundle when the fundamental estimate holds. Asapplications, we estimate the dimension of cohomology of semipositive linebundles on $q$-convex manifolds, pseudo-convex domains, weakly $1$-completemanifolds and complete manifolds. We also obtain the estimate of cohomology oncompact manifolds with semipositive line bundles endowed with a Hermitianmetric with analytic singularities and the related vanishing theorems.

The growth of dimension of cohomology of semipositive line bundles on Hermitian manifolds