Local Demailly-Bouche's holomorphic Morse inequalities

2017/12/06 by Zhiwei Wang, Wang, Zhiwei
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1712.02080

openalex publication_date 2017/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X,ω) be a Hermitian manifold and let (E,h^E), (F,h^F) be two Hermitian holomorphic line bundle over X. Suppose that the maximal rank of the Chern curvature c(E) of E is r, and the kernel of c(E) is foliated, i.e. there is a foliation Y of X, of complex codimension r, such that the tangent space of the leaf at each point x∈ X is contained in the kernel of c(E). In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups H^q(X,E^k⊗ F^l) as k,l,k/l→ ∞) are presented. The local version holds on any Hermitian manifold regardless of compactness and completeness. The proof is a variation of Berman's method to derive holomorphic Morse inequalities on compact complex manifolds with boundary.

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