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

Abstract

We establish a general Nakano-Griffiths inequality with boundary conditions and apply it to derive holomorphic Morse inequalities for domains satisfying analytic convexity assumptions. The proof of these inequalities relies on analyzing the spectral spaces of the Laplace operator with $\overline{\partial }$-Neumann boundary conditions. As an application, we obtain a criterion for Moishezon $1$-concave manifolds. We further apply the holomorphic Morse inequalities on Levi $q$-concave domains in conjunction with the Kohn-Rossi extension theorem to obtain the following result. Let $X$ be a compact complex manifold of dimension $n$ equipped with a holomorphic line bundle that is semi-positive everywhere and positive at least at one point, and let $D\subset X$ be a smooth $q$-concave domain ($1 \le q \le n-1$). We prove that every $\bar{\partial }_{b}$-closed $(0,\ell )$-form on $bD$ with values in a holomorphic vector bundle, admits a meromorphic extension to $D$ for all $q \le \ell \le n-1$.