Nondegenerate Monge-Ampere structures in dimension 6

Abstract

We define a nondegenerate Monge-Amp\`ere structure on a 6-dimensional manifold as a pair $(\Omega,\omega)$, such that $\Omega$ is a symplectic form and $\omega$ is a 3-differential form which satisfies $\omega\wedge\Omega=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Amp\`ere operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Amp\`ere equation in the dimension 4. We study from this point of view the example of the Stenzel metric on the cotangent bundle of the sphere $S^3$.