Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology

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Donaldson, Two-forms on four-manifolds and elliptic equations (arXiv:math/0607083) evidence for this event

Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology parent of this event

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Fine and Yao, Hypersymplectic 4-manifolds and the G2-Laplacian flow (arXiv:1704.07620) evidence for this event

VibeMathed record: Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology evidence for this event

Earlier edition (September 23, 2026) evidence for this event

Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology evidence for this event

Theorem 1.1: for a closed connected oriented smooth four-manifold $X$ and a smooth hypersymplectic triple with $\int_X\omega_i\wedge\omega_j=\delta_{ij}$, there is a smooth path $\omega(t)$ of hypersymplectic triples with $\omega(0)=\omega$, $[\omega_i(t)]=[\omega_i]$, ending at forms with $\omega_i(1)\wedge\omega_j(1)=2\delta_{ij}\mu_1$, parallel and self-dual for a hyperkahler metric. An arbitrary positive triple is reduced to this case by a constant linear change. Corollaries: every nonzero combination $\sum c_i\omega_i$ is Kahler for some hyperkahler metric, and any closed four-manifold with a positive closed triple is diffeomorphic to the K3 manifold or $T^4$. Not shown: long-time existence or convergence of the hypersymplectic flow, or any statement in higher dimensions. parent of this event