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

2019/05/09 by Zhiwei Wang, Xiangyu Zhou, Wang, Zhiwei +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1905.03473

openalex publication_date 2019/05/09 · openalex created_date 2019/05/16 · openalex updated_date 2026/07/28

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 ∂∂-formula and Berndtsson's eigenvalue estimate of ∂-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,ω), if h_L is a singular metric with algebraic singularities, then dim H^q(X,L^k⊗ E⊗ I(h_L^k))≤ 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.

Citations

• The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles
• An extension theorem of Ohsawa-Takegoshi type for sections of a vector bundle
• An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities
• Cohomology ofq-convex spaces in top degrees

Related

• An eigenvalue estimate for the ∂-Laplacian associated to a nef line bundle
• On Junyan Cao's vanishing theorem for pseudoeffective line bundles
• The ∂-cohomology groups, holomorphic Morse inequalities, and finite type conditions
• Pseudo-effective line bundles on compact Kähler manifolds
• Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles