Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911144456749056 |
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| author | Collins, Tristan C. Guenancia, Henri |
| author_facet | Collins, Tristan C. Guenancia, Henri |
| contents | We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs $(X,D)$ such that $K_X+D=0$. More specifically, we focus on the two cases where $X$ is a Fano manifold and $D$ is either smooth or has two proportional components. Despite the existence of a complete Ricci flat Kähler metric on $X\setminus D$ in both cases, we will show that the geometric properties of the pair $(X,D)$ are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of $T_X(-\log D)$ and compactifiability of the universal cover of $X\setminus D$. When $D$ has two components we show that the universal cover of $X\setminus D$ is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover Collins, Tristan C. Guenancia, Henri Algebraic Geometry Complex Variables Differential Geometry We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs $(X,D)$ such that $K_X+D=0$. More specifically, we focus on the two cases where $X$ is a Fano manifold and $D$ is either smooth or has two proportional components. Despite the existence of a complete Ricci flat Kähler metric on $X\setminus D$ in both cases, we will show that the geometric properties of the pair $(X,D)$ are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of $T_X(-\log D)$ and compactifiability of the universal cover of $X\setminus D$. When $D$ has two components we show that the universal cover of $X\setminus D$ is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view. |
| title | Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover |
| topic | Algebraic Geometry Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2509.07508 |