Donaldson's hypersymplectic deformation conjecture

The explanation

Take the four-torus, a cube in four dimensions whose opposite faces are glued. Use coordinates x1 to x4 and write dxi dxj for a small area element in the xi, xj plane. Form three two-forms: ω1 = dx1 dx2 + dx3 dx4, ω2 = dx1 dx3 + dx4 dx2 and ω3 = dx1 dx4 + dx2 dx3. Multiplying two of them, with the wedge product, gives a multiple of the volume element. Each form times itself gives 2. Each pair of different forms gives 0. So the table of products is 2 times the identity, the same at every point. This is a hyperkähler triple, the most symmetric kind.

Now tilt the third form to ω3 + ω1/2. The table becomes 2, 0, 1 in the first row, 0, 2, 0 in the second and 1, 0, 2.5 in the third. Its eigenvalues are about 3.28, 2 and 1.22, all positive. A triple of closed two-forms whose table is positive like this is hypersymplecticSource 1. On a general space the table can change from point to point. A cohomology class is a kind of fixed label for each closed form. The question is whether the forms can move to a hyperkähler triple with their classes keptSource 1.

Donaldson asked whether every compact four-manifold with such a triple carries a hyperkähler structureSource 2. Fine and Yao stated the deformation version, with all three classes fixed, as a conjectureSource 3. The main tool so far was the hypersymplectic flow, an equation that moves the forms over timeSource 4. Full convergence results needed symmetry or other extra conditionsSource 5.

The paper claims the deformation always existsSource 6. The endpoint forms are parallel, meaning constant in the right sense, and form a hyperkähler tripleSource 7. As a result, the space must be smoothly a K3 surface or the four-torusSource 8. The obstacle was keeping the third class while deforming. The paper works with two of the forms at a time. Those two forms define an almost-complex structure J, a rule for rotating directions by a right angle. A form tames J if it is positive on each direction paired with its rotationSource 9. The deformation works as long as the third class still contains a taming form. To keep one, the paper studies positive currents, which are generalized surfaces that could block the classSource 10. It bounds their mass on small balls. It builds on a companion paper that shows taming implies compatibility on four-manifoldsSource 11.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof is long and uses delicate estimates on currents. No specialist review of the paper has been published.

What it doesn't show. This does not claim the hypersymplectic flow converges; the deformation is built another way. It has no Lean check.