Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917042215452672 |
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| author | Ebenfelt, Peter Xiao, Ming Xu, Hang |
| author_facet | Ebenfelt, Peter Xiao, Ming Xu, Hang |
| contents | Let $M$ be a complete Kähler manifold, and let $(L, h) \to M$ be a positive line bundle inducing a Kähler metric $g$ on $M$. We study two Bergman kernels in this setting: the Bergman kernel of the disk bundle of the dual line bundle $(L^*, h^*)$, and the Bergman kernel of the line bundle $(L^k, h^k)$, $k\geq 1$, twisted by the canonical line bundle of $(M, g)$. We first prove a localization result for the former Bergman kernel. Then we establish a necessary and sufficient condition for this Bergman kernel to have no logarithmic singularity, expressed in terms of the Tian-Yau-Zelditch-Catlin type expansion of the latter Bergman kernel. This result, in particular, answers a question posed by Lu and Tian. As an application, we show that if $(M, g)$ is compact and locally homogeneous, then the circle bundle of $(L^*, h^*)$ is necessarily Bergman logarithmically flat. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian Ebenfelt, Peter Xiao, Ming Xu, Hang Complex Variables Differential Geometry Let $M$ be a complete Kähler manifold, and let $(L, h) \to M$ be a positive line bundle inducing a Kähler metric $g$ on $M$. We study two Bergman kernels in this setting: the Bergman kernel of the disk bundle of the dual line bundle $(L^*, h^*)$, and the Bergman kernel of the line bundle $(L^k, h^k)$, $k\geq 1$, twisted by the canonical line bundle of $(M, g)$. We first prove a localization result for the former Bergman kernel. Then we establish a necessary and sufficient condition for this Bergman kernel to have no logarithmic singularity, expressed in terms of the Tian-Yau-Zelditch-Catlin type expansion of the latter Bergman kernel. This result, in particular, answers a question posed by Lu and Tian. As an application, we show that if $(M, g)$ is compact and locally homogeneous, then the circle bundle of $(L^*, h^*)$ is necessarily Bergman logarithmically flat. |
| title | Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2510.22169 |