Donaldson’s a priori bounds conjecture for the Calabi–Yau equation on tamed almost complex 4-manifolds

Establish uniform C^{∞} a priori bounds for an almost Kähler form \tilde{\omega} solving the Calabi–Yau equation \tilde{\omega}^2 = \sigma on a compact 4-manifold M equipped with an almost complex structure J tamed by a symplectic form \Omega, where \sigma is a smooth volume form satisfying \int_M \sigma = \int_M \Omega^2 and [\tilde{\omega}] = [\Omega]. The bounds should depend only on \Omega, J, and M.

Background

The conjecture concerns the Calabi–Yau equation in the almost complex, non-integrable setting. Given a compact 4-manifold with an almost complex structure J tamed by a symplectic form \Omega, one asks for uniform a priori estimates for any almost Kähler representative \tilde{\omega} in the cohomology class [\Omega] solving \tilde{\omega}^2 = \sigma, with \sigma matching the total volume of \Omega^2.

In this paper, the authors solve a generalized Monge–Ampère equation on closed almost Kähler surfaces and obtain existence, uniqueness, and a priori estimates. As an application, they show that in the case h_J^- = b^+ - 1, their results yield the desired a priori bounds, giving a positive answer to Donaldson’s conjecture under this condition.