Local Demailly-Bouche's holomorphic Morse inequalities

الأصل · EN

Let (X,ω) be a Hermitian manifold and let (E,hᵉ), (F,hᶠ) 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 Hq(X,Eᵏ⊗ Fˡ) 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.