Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915438412169216 |
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| author | Greb, Daniel Schwald, Martin |
| author_facet | Greb, Daniel Schwald, Martin |
| contents | We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperkähler metrics on members of the families via Penrose's Non-linear Graviton construction. On the way to proving this result, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. Over an open subset containing all twistor cycles we construct a family of such families, which is a universal small deformation for every twistor family. Whenever possible, we extend the results to higher-dimensional irreducible holomorphic-symplectic manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13420 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles Greb, Daniel Schwald, Martin Algebraic Geometry Complex Variables Differential Geometry 14J42, 14J28, 32G20, 14C05, 14C30, 32G07, 32G13, 32L25, 53C28 We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperkähler metrics on members of the families via Penrose's Non-linear Graviton construction. On the way to proving this result, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. Over an open subset containing all twistor cycles we construct a family of such families, which is a universal small deformation for every twistor family. Whenever possible, we extend the results to higher-dimensional irreducible holomorphic-symplectic manifolds. |
| title | Moduli of K3 families over $\mathbb{P}^1$ and complex-hyperkähler metrics induced by deformed twistor cycles |
| topic | Algebraic Geometry Complex Variables Differential Geometry 14J42, 14J28, 32G20, 14C05, 14C30, 32G07, 32G13, 32L25, 53C28 |
| url | https://arxiv.org/abs/2311.13420 |