On the Calabi-Yau equation in the Kodaira-Thurston manifold

We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kahler structure and assuming the volume form T-2-invariant. In particular, we observe that under some restrictions the problem is reduced to a Monge-Ampere equation by using the ansatz (omega) over tilde= Omega - dJdu + da, where u is a T-2-invariant function and a is a 1-form depending on u. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some basic cases and we finally take into account a generalization to higher dimensions.