Donaldson’s hypersymplectic deformation conjecture

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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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One current paper; one replaced by a later edition.

The paper states

Theorem 1.1

In Deforming hypersymplectic four-manifolds to hyperkähler triples, 7 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Corollary 1.2 (Individual Kähler realizations) in the same paper () and Corollary 1.3 (Smooth type) in the same paper ().

Let X be a closed connected oriented smooth four-manifold, and let \omega=(\omega_1,\omega_2,\omega_3) be a smooth hypersymplectic structure normalized by

\int_X\omega_i\wedge\omega_j=\delta_{ij}.

There is a smooth family of hypersymplectic structures \omega(t), 0\le t\le1, such that

\omega(0)=\omega,\qquad [\omega_i(t)]=[\omega_i] \quad (i=1,2,3),

and a positive volume form \mu_1 with

\omega_i(1)\wedge\omega_j(1)=2\delta_{ij}\mu_1.

The endpoint forms are parallel and self-dual for a hyperkähler metric.

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Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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Deforming hypersymplectic four-manifolds to hyperkähler triples

7 October 2026, current edition.

Read it here PDF

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.

Earlier editions: 23 September 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.

Previous editions

Deforming hypersymplectic four-manifolds to hyperkähler triples previous edition, 23 September 2026 PDF

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Papers elsewhere in the release that this family’s papers cite

Taming implies compatibility on four-manifoldsCurrent edition; no accompanying Lean statement Family Donaldson’s tamed-to-compatible conjectureCited by Deforming hypersymplectic four-manifolds to hyperkähler triples.

Papers elsewhere in the release that cite this family’s papers

Taming implies compatibility on four-manifoldsCurrent edition; no accompanying Lean statement Family Donaldson’s tamed-to-compatible conjectureCites Deforming hypersymplectic four-manifolds to hyperkähler triples.

References outside the release

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57 references in Deforming hypersymplectic four-manifolds to hyperkähler triples

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