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

Abstract

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 $\dim H^q(X,L^k\otimes E\otimes \mathcal{I}(h_L^k))\le C{k}^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.

arXiv comment27 pages, submitted first time on November 28, 2017, second time on February 12,2018. Comments welcome

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Cited by in corpus (2)

• Cohomology dimension growth for Nakano $q$-semipositive line bundles
• An eigenvalue estimate for the $\bar{\partial }$-Laplacian associated to a nef line bundle