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 S¹-action on X, then X can be regarded as a principal S¹-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 CTM induces a Mizohata structure on M. If (X, H^0,1) can be embedded into some Cⁿ 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 C² 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.