Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains

TL;DR

This paper develops new holomorphic Morse inequalities for Levi q-concave domains, providing extension theorems and vanishing results for certain forms on complex manifolds with boundary.

Contribution

It introduces a Nakano-Griffiths inequality with boundary conditions and applies spectral analysis of the Laplace operator to unify Morse inequalities and vanishing theorems.

Findings

Proved meromorphic extension of boundary forms under Levi q-concavity.

Established a boundary Nakano-Griffiths inequality.

Provided a geometric proof of vanishing theorems for q-concave and q-convex domains.

Abstract

We consider a compact $n$-dimensional complex manifold endowed with a holomorphic line bundle that is semi-positive everywhere and positive at least at one point. Additionally, we have a smooth domain of this manifold whose Levi form has at least $n-q$ negative eigenvalues ($1\leq q\leq n-1$) on the boundary. We prove that every $\overline{\partial}_b$-closed $(0,\ell)$-form on the boundary with values in a holomorphic vector bundle admits a meromorphic extension for all $q\leq \ell\leq n-1$. This result is an application of holomorphic Morse inequalities on Levi $q$-concave domains and the Kohn-Rossi extension theorem. We propose a proof of the Morse inequalities by utilizing the spectral spaces of the Laplace operator with $\overline{\partial}$-Neumann boundary conditions. To accomplish this objective, we establish a general Nakano-Griffiths inequality with boundary conditions. This leads to a unified approach to holomorphic Morse inequalities and a geometric proof of vanishing theorems for $q$-concave and $q$-convex manifolds or domains.