We study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal... Besides, estimates for the modified Dirac operator on Nakano $q$-positive line bundle on almost complex manifolds are given. (read more)