Complex structures of the Gibbons-Hawking ansatz with infinite topological type
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917136517038080 |
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| author | He, Wenxin Xu, Bin |
| author_facet | He, Wenxin Xu, Bin |
| contents | In this paper, we study the complex structures of complete hyperkähler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. We show that for almost all complex structures in the hyperkähler family, the manifold is biholomorphic to a hypersurface in $\mathbb{C}^3$ defined by an explicit entire function. For the remaining complex structures, we further prove that the manifold is biholomorphic to the minimal resolution of a singular surface in $\mathbb{C}^3$ under certain conditions. Thus, we partially extend LeBrun's celebrated work to the context of countably many punctures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complex structures of the Gibbons-Hawking ansatz with infinite topological type He, Wenxin Xu, Bin Differential Geometry In this paper, we study the complex structures of complete hyperkähler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. We show that for almost all complex structures in the hyperkähler family, the manifold is biholomorphic to a hypersurface in $\mathbb{C}^3$ defined by an explicit entire function. For the remaining complex structures, we further prove that the manifold is biholomorphic to the minimal resolution of a singular surface in $\mathbb{C}^3$ under certain conditions. Thus, we partially extend LeBrun's celebrated work to the context of countably many punctures. |
| title | Complex structures of the Gibbons-Hawking ansatz with infinite topological type |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.18836 |