A Holomorphic Splitting Theorem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912432766582784 |
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| author | Song, Miao |
| author_facet | Song, Miao |
| contents | A long-term project is to construct a complete Calabi-Yau metric on the complement of the anticanonical divisor in a compact Kähler manifold $\oM$. We focus on the case where this smooth divisor has multiplicity 2 and is itself a compact Calabi-Yau manifold. Firstly we solved the Monge-Ampère equation when the Ricci potiential is of $O(r^{-1})$ decay on the generalized $ALG$ manifolds. Then we used the solution to this Kähler Ricci flat metric to prove a holomorphic splitting theorem: If $K_{\oM}=\calo(-2D)$, where $D$ can be realized as a smooth Calabi-Yau manifold, and if $\calo_{3D}(D)$ is trivial, then this Kähler manifold $\oM$ is biholomorphic to $\bbp^1\times D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Holomorphic Splitting Theorem Song, Miao Differential Geometry A long-term project is to construct a complete Calabi-Yau metric on the complement of the anticanonical divisor in a compact Kähler manifold $\oM$. We focus on the case where this smooth divisor has multiplicity 2 and is itself a compact Calabi-Yau manifold. Firstly we solved the Monge-Ampère equation when the Ricci potiential is of $O(r^{-1})$ decay on the generalized $ALG$ manifolds. Then we used the solution to this Kähler Ricci flat metric to prove a holomorphic splitting theorem: If $K_{\oM}=\calo(-2D)$, where $D$ can be realized as a smooth Calabi-Yau manifold, and if $\calo_{3D}(D)$ is trivial, then this Kähler manifold $\oM$ is biholomorphic to $\bbp^1\times D$. |
| title | A Holomorphic Splitting Theorem |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.13517 |