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.