Closed 3-forms in five dimensions and embedding problems

TL;DR

This paper investigates the conditions under which a five-dimensional manifold with a specific type of closed 3-form can be embedded into a Calabi-Yau threefold, focusing on strongly pseudoconvex forms and obstruction theory.

Contribution

It introduces the concept of strongly pseudoconvex 3-forms in five dimensions and provides conditions for embedding these forms into Calabi-Yau manifolds, addressing a geometric embedding problem.

Findings

Defined strongly pseudoconvex 3-forms in five dimensions.

Established conditions for embedding based on vanishing obstructions.

Provided a perturbative solution framework for the embedding problem.

Abstract

We consider the question if a five dimensional manifold can be embedded into a Calabi-Yau manifold of complex dimension three 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.