ABSTRACT. We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action , and obtain an asymptotic estimate for thedimension of the harmonic space with respect to the tensor times in the holomorphic line bundle Ä ª and the type´Ò Õµ of the differential form, when Ä is semipositive. In particular, we estimate the -dimension of the corresponding reduced Ä ¾ -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.

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In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic´¼ Õµ-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we estimate the dimension of cohomology of semipositive line bundles on Õ-convex manifolds, pseudo-convex domains, weakly ½-complete manifolds and complete manifolds. We also obtain the estimate of cohomology on compact manifolds with semipositive line bundles endowed with a Hermitian metric with analytic singularities and the related vanishing theorems.

We prove a Bochner–Kodaira–Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR $$\mathbb {R}$$ R -action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.