CR structures on the 3-sphere and the blow-up of the projective plane

Motivated by a problem of characterizing CR-structures on the 3-sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are describ ed by cohomology groups of the blow-up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subs paces. This stratification reflects algebro-geometric properties of X. It i s expected that our construction will clarify the complex geometric nature of the space of non-embeddable CR-structures on the 3-sphere.