We develop a correspondence between deformations of the standard CR structure on $S^3$ and deformations of formal neighbourhoods of the hyperplane bundle over ${\mathbb{P}}^1$; this correspondence leads to a geometric description of obstructions to the embeddability of CR structures.

This note is an expanded version of the author’s lecture at the conference on Several Complex Variables held in Osaka, Dec. 1994. We consider deformations of complex spaces and their relationship to deformations of CR-structures. An invariant is introduced which measures the change in the algebra of CR-functions under a deformation. These issues are then considered in the context of the deformations of the total space of the tangent bundle of a Riemann surface. The last section contains problems in deformation theory.

In this paper we survey a number of results concerned with the deformations of quaternionic structures on classes of quaternionic manifolds and the deformation theory of Hermitian bundles with self-dual connections. The deformations in question are shown to correspond to the deformation theory of complex structures and holomorphic vector bundles over an associated complex manifold referred to as a twistor space. Results related to hypercomplex and hyperkähler manifolds are also discussed.

If the symbol $g_k$ of a SPDE $R_k$ is 2-acyclic, then the operations of prolongation and projection on $R_k$ commute \[ \rho^{k+l+2}_{k+l+1}((R_k)_{+l+2}) = \left(\rho^{k+l+1}_{k+l}((R_k)_{+l+1})\right)_{+1}.\] We apply this to study contact of three-dimensional CR-manifolds.