G-structures and Families of Isotropic Submanifolds in Complex Contact Manifolds

Abstract: We study a generalized twistor correspondence between irreducible G-structures (with torsion in general) on complex manifolds Z and moduli spaces M of deformations of isotropic homogeneous submanifolds X in complex contact manifolds Y.For any irreducible G-structure on a complex manifold M we present an explicit construction of a contact manifold (a generalized twistor space) Y with contact line bundle L and a family F of isotropic submanifolds X in Y having M as its moduli space. We study those special properties of this family which encode geometric invariants of the original G-structure.Conversely, given a contact manifold (Y,L) and an homogeneous isotropic submanifold X in Y satisfying certain properties, we show that the associated moduli space M of isotropic deformations of X inside Y has an induced G-structure, Gind, and then show how the invariant torsion of Gind can be read off from certain cohomology groups canonically associated with the holomorphic embedding data of X in Y.