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arXiv · 2503.05272

Hypersymplectic Structures Invariant Under an Effective Circle Action

Joel Fine·Weiyong He·Chengjian Yao
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Abstract

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 hyperk\"ahler 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 $\mathbb{T}^4$.

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Joel Fine, Weiyong He, Chengjian Yao. 2025-03-07. Hypersymplectic Structures Invariant Under an Effective Circle Action. https://doi.org/10.1093/imrn%2Frnaf252

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