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

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.

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.