The Calabi-Yau equation for T\(^2\)-bundles over T\(^2\): the non-Lagrangian case (Q2898488)

We study the Calabi-Yau equation on \(T^2\)-bundles over \(T^2\) endowed with an invariant non-Lagrangian almost-Kähler structure showing that for \(T^2\)-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space \(M^4\) of an orientable \(T^2\)-bundle over \(T^2\) endowed with an invariant almost-Kähler structure the Calabi-Yau problem has a solution for every normalized \(T^2\)-invariant volume form.