CR circle bundles and Mizohata structures

In this paper we study the relationship between CR-structures on 3-dimensional manifolds and Mizohata structures on 2-dimensional manifolds. Let ( X, H 0,1 ) be a 3-dimensional strictly pseudoconvex CR-manifold. Assume there is a free smooth S 1 -action on X , then X can be regarded as a principal S 1 -bundle π: X → M over a smooth 2-dimensional manifold M ; and assume the CR-structure H 0,1 is invariant under the circle action. First, we show that the projection π * ( H 0,1 ) of H 0,1 into C TM induces a Mizohata structure on M . If ( X, H 0,1 ) can be embedded into some C n by CR-functions, then the induced Mizohata structure on M is locally integrable. Moreover, we consider when a neighborhood of π -1 ( p ) can be embedded into C 2 by CR-functions. Second, for every Mizohata structure V on M we can construct a CR circle bundle ( X, H 0,1 ) via a singular curvature form by using the theory of singular forms and singular connections over circle bundles described here. This construction can be viewed as a generalization of geometric prequantization to a degenerate situation.