Local Demailly-Bouche's holomorphic Morse inequalities

TL;DR

This paper develops local versions of Demailly-Bouche's holomorphic Morse inequalities for Hermitian manifolds, applicable regardless of compactness, by adapting Berman's method to the non-compact setting.

Contribution

It introduces local holomorphic Morse inequalities on Hermitian manifolds with foliated curvature kernels, extending previous global results to non-compact cases.

Findings

Local inequalities hold on any Hermitian manifold, compact or not.

The method adapts Berman's approach to non-compact manifolds.

Provides asymptotic bounds for cohomology groups in new settings.

Abstract

Let $(X,\omega)$ 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 $x\in X$ 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 $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) 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.