Hypersymplectic Structures Invariant Under an Effective Circle Action
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918124149800960 |
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| author | Fine, Joel He, Weiyong Yao, Chengjian |
| author_facet | Fine, Joel He, Weiyong Yao, Chengjian |
| contents | A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^1$-action. In particular we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_05272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypersymplectic Structures Invariant Under an Effective Circle Action Fine, Joel He, Weiyong Yao, Chengjian Symplectic Geometry Differential Geometry 53C26, 53D35 A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^1$-action. In particular we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^4$. |
| title | Hypersymplectic Structures Invariant Under an Effective Circle Action |
| topic | Symplectic Geometry Differential Geometry 53C26, 53D35 |
| url | https://arxiv.org/abs/2503.05272 |