Donaldson's hypersymplectic deformation conjecture

1 paper

Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by \(\int\omega_i\wedge\omega_j=\delta_{ij}\), deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.

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Papers 1

Deforming hypersymplectic four-manifolds to hyperkähler triples

OpenAI · October 7, 2026

Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.

This version clarifies the prescribed cohomology-class criterion used in the deformation argument, removes an unnecessary cone-comparison citation, and updates the companion-paper citation; see the previous version.

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@misc{OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026,
  author = {{OpenAI}},
  title = {{Deforming hypersymplectic four-manifolds to hyperk\"ahler triples}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf}{OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026}},
  year = {2026}
}

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