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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-formula and Berndtsson’s eigenvalue estimate of $\bar{\mathrm{\partial }}$-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.