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Deforming hypersymplectic four-manifolds to hyperkähler triples
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Abstract
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.
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Possibly relevant literature (found by the agent, not checked)
- C. H. Taubes, Tamed to compatible: Symplectic forms via moduli space integration (arXiv:0910.5440, 2009). For compact four-manifolds with $b^+=1$, it produces compatible symplectic forms cohomologous to a given taming form for an open dense set of tamed almost-complex structures, using currents from pseudoholomorphic curves; this bears on the paper's tamed-to-compatible and prescribed-taming-class steps, which work with $b^+=3$.
- Q. Tan, H. Wang, J. Zhou, P. Zhu, On tamed almost complex four manifolds (arXiv:1712.02948, 2017). It proves existence of a compatible symplectic form on tamed closed almost-complex four-manifolds under a condition on $h^-_J$ and $b^+$, earlier partial work on the tamed-to-compatible question the paper uses in Sections 3–4.
- K. Zheng, The space of closed G₂-structures. II. Concavity and hypersymplectic structures (arXiv:2507.20456, 2025). It studies concavity of Hitchin's volume functional along geodesics of closed G₂-structures and constructs hypersymplectic examples, adding to the flow and G₂-reduction literature surveyed in the paper's introduction.
Digests and expositions
- Latent Seminar, Deforming positive triples to hyperkähler geometry (7 October 2026). This exposition outlines the manuscript's prescribed-class criterion, torus-fibration construction, auxiliary equal-square form, and two final Kähler steps; it describes itself as an explanation of selected mechanisms rather than an independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 341
Independent sites; their content is not reviewed here.
Cite
@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}
}Versions
- Earlier edition: September 23, 2026
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.
Family 341: Donaldson's hypersymplectic deformation conjecture
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.
This is the only paper in this family.