$G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929325826113536 |
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| author | Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan |
| author_facet | Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan |
| contents | Let $M$ be a complex manifold with boundary $X$, which admits a holomorphic Lie group $G$-action preserving $X$. We establish a full asymptotic expansion for the $G$-invariant Bergman kernel under certain assumptions. As an application, we get $G$-invariant version of Fefferman's result about regularity of biholomorphic maps on strongly pseudoconvex domains of $\mathbb C^n$. Moreover, we show that the Guillemin-Sternberg map on a complex manifold with boundary is Fredholm by developing reduction to boundary technique, which establish ``quantization commutes with reduction" in this case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00601 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan Complex Variables Let $M$ be a complex manifold with boundary $X$, which admits a holomorphic Lie group $G$-action preserving $X$. We establish a full asymptotic expansion for the $G$-invariant Bergman kernel under certain assumptions. As an application, we get $G$-invariant version of Fefferman's result about regularity of biholomorphic maps on strongly pseudoconvex domains of $\mathbb C^n$. Moreover, we show that the Guillemin-Sternberg map on a complex manifold with boundary is Fredholm by developing reduction to boundary technique, which establish ``quantization commutes with reduction" in this case. |
| title | $G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2305.00601 |