The Calabi–Yau equation on the Kodaira–Thurston manifold

We prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.