The uniqueness of Poincaré type constant scalar curvature Kähler metric
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866918034530107392 |
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| author | Xu, Yulun Zheng, Kai |
| author_facet | Xu, Yulun Zheng, Kai |
| contents | Let $D$ be a smooth divisor on a closed Kähler manifold $X$. First, we prove that Poincaré type constant scalar curvature Kähler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincaré type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincaré type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The uniqueness of Poincaré type constant scalar curvature Kähler metric Xu, Yulun Zheng, Kai Differential Geometry Analysis of PDEs 32Q15, 53C55 Let $D$ be a smooth divisor on a closed Kähler manifold $X$. First, we prove that Poincaré type constant scalar curvature Kähler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincaré type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincaré type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem. |
| title | The uniqueness of Poincaré type constant scalar curvature Kähler metric |
| topic | Differential Geometry Analysis of PDEs 32Q15, 53C55 |
| url | https://arxiv.org/abs/2410.14130 |