The Geometry of Three-Forms on Symplectic Six-Manifolds

In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We show that there are extremely rich geometric structures attached to certain unstable 3-forms arising naturally from degeneration of Calabi-Yau structures, which in turn provides us a new perspective towards the SYZ conjecture. We give concrete examples and demonstrate that the limiting behavior of the Type IIA flow can be used to detect canonical geometric structures on symplectic manifolds.