Asymptotic estimate of cohomology groups valued in pseudo-effective line bundles

In this paper, we study questions of Demailly and Matsumura on the asymptotic behavior of dimensions of cohomology groups for high tensor powers of (nef) pseudo-effective line bundles over non-necessarily projective algebraic manifolds. By generalizing Siu's $\partial\overline{\partial}$-formula and Berndtsson's eigenvalue estimate of $\overline{\partial}$-Laplacian and combining Bonavero's technique, we obtain the following result: given a holomorphic pseudo-effective line bundle $(L, h_L)$ on a compact Hermitian manifold $(X,\omega)$, if $h_L$ is a singular metric with algebraic singularities, then $\dim H^{q}(X,L^k\otimes E\otimes \mathcal{I}(h_L^{k}))\leq Ck^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.