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

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 ($1\le q\le n-1$) on the boundary. We prove that every ${\bar{\mathrm{\partial }}}_b$-closed $(0,\mathrm{\ell })$-form on the boundary with values in a holomorphic vector bundle admits a meromorphic extension for all $q\le \mathrm{\ell }\le n-1$. 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 $\bar{\mathrm{\partial }}$-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.