Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914773629665280 |
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| author | Chiang, Yueh-Lin |
| author_facet | Chiang, Yueh-Lin |
| contents | Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ over a complex manifold $M$ which may not be compact, we generalize the scaling method in arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and spectral kernels with respect to the space of global holomorphic sections of $L_k$ with $(0,q)$-forms. We derive the leading term of the Bergman and spectral kernels under the local convergence assumption in the sequence of Chern curvatures $c_1(L_k,h_k)$, inspired by arXiv:2012.12019. The manifold $M$ may be non-Kähler and $c_1(L_k,h_k)$ may be negative or degenerate. Moreover, we establish the $L_k$-asymptotic version of Demailly's holomorphic Morse inequalities as an application to compact complex manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles Chiang, Yueh-Lin Complex Variables Differential Geometry Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ over a complex manifold $M$ which may not be compact, we generalize the scaling method in arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and spectral kernels with respect to the space of global holomorphic sections of $L_k$ with $(0,q)$-forms. We derive the leading term of the Bergman and spectral kernels under the local convergence assumption in the sequence of Chern curvatures $c_1(L_k,h_k)$, inspired by arXiv:2012.12019. The manifold $M$ may be non-Kähler and $c_1(L_k,h_k)$ may be negative or degenerate. Moreover, we establish the $L_k$-asymptotic version of Demailly's holomorphic Morse inequalities as an application to compact complex manifolds. |
| title | Spectral Kernels and Holomorphic Morse Inequalities for Sequence of Line Bundles |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2404.18079 |