Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains
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| Format: | Preprint |
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2025
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| _version_ | 1866918042714243072 |
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| author | Liu, Bingxiao Marinescu, George Wang, Huan |
| author_facet | Liu, Bingxiao Marinescu, George Wang, Huan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains Liu, Bingxiao Marinescu, George Wang, Huan Complex Variables Differential Geometry 32W10 (Primary) 32W05, 32F10, 32D15, 32A22 (Secondary) 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. |
| title | Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains |
| topic | Complex Variables Differential Geometry 32W10 (Primary) 32W05, 32F10, 32D15, 32A22 (Secondary) |
| url | https://arxiv.org/abs/2506.00879 |