Generalized Calabi-Yau metric and generalized Monge-Ampère equation

In the neighborhood of a regular point, generalized Kähler geometry admits a description in terms of a single real function, the generalized Kähler potential. We study the local conditions for a generalized Kähler manifold to be a generalized Calabi-Yau manifold and we derive a non-linear PDE that the generalized Kähler potential has to satisfy for this to be true. This non-linear PDE can be understood as a generalization of the complex Monge-Ampère equation and its solutions give supergravity solutions with metric, dilaton and H-field.