Donaldson's tamed-to-compatible conjecture for almost complex four-manifolds

Theorem 1.1: if a smooth almost complex structure $J$ on a closed connected smooth four-manifold is tamed by a symplectic form, there is a smooth symplectic form compatible with the same $J$. The October 6 edition corrects the cone corollary of the first edition, which had claimed $\mathcal K_J^t=\mathcal K_J^c+H_J^-$ in general via Li-Zhang: it now characterizes the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone, proves that equality only when $h_J^-=b_2^+-1$ (so when $b_2^+=1$ every taming class still contains a compatible form), and gives a four-torus example where the inclusion is strict. Not shown: that an arbitrary taming class contains a compatible form, anything in dimensions above four, or Donaldson's separate a priori estimate conjecture for the prescribed-volume equation.

Theorem 1.1: if a smooth almost complex structure $J$ on a closed connected smooth four-manifold is tamed by a symplectic form, there is a smooth symplectic form compatible with the same $J$. The October 6 edition corrects the cone corollary of the first edition, which had claimed $\mathcal K_J^t=\mathcal K_J^c+H_J^-$ in general via Li-Zhang: it now characterizes the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone, proves that equality only when $h_J^-=b_2^+-1$ (so when $b_2^+=1$ every taming class still contains a compatible form), and gives a four-torus example where the inclusion is strict. Not shown: that an arbitrary taming class contains a compatible form, anything in dimensions above four, or Donaldson's separate a priori estimate conjecture for the prescribed-volume equation.