Closed 3-forms in five dimensions and embedding problems

We consider the question if a five-dimensional manifold can be embedded into a Calabi–Yau manifold of complex dimension 3 such that the real part of the holomorphic volume form induces a given closed 3-form on the 5-manifold. We define an open set of 3-forms in dimension five which we call strongly pseudoconvex, and show that for closed strongly pseudoconvex 3-forms, the perturbative version of this embedding problem can be solved if a finite-dimensional vector space of obstructions vanishes.