The Hill-Penrose-Sparling CR manifolds

This paper gives a simple formulation of the CR structures of Hill, Penrose, and Sparling. An elementary power series argument shows that they cannot be realized as hypersurfaces in an ambient complex manifold. Suppose M is a smooth three-dimensional real manifold equipped with a nowhere vanishing complex vector field X and a smooth complex-valued function g such that, even locally, the equation Xf = g has no solutions. One may regard X as defining a CR structure on M and then g represents a nowhere vanishing class in the abcohomology H0" (M). As described in [2], this class may be exponentiated to a CR line bundle over M with total space T. This example T of a CR manifold is due to Hill, Penrose, and Sparling who have shown (see [2]) that it cannot be locally realized as a hypersurface in C3. (An alternative proof of this fact is given by Jacobowitz [1].) The manifold T may be defined as M x C with CR structure induced by the vector fields X+gza/lz and 9/lz, where z is the usual coordinate on C. To show that this is nonrealizable, first notice that if a smooth function a satisfies Xca = nga, where n is a nonzero constant, then a must be identically zero for, otherwise, any local choice of n-1 log a would provide a solution of Xf = g. Suppose that h is CR holomorphic on T. In other words, Xh + gzah/lz = 0 = ahla2. The second equation says that this function is holomorphic in the variable z and so may be expanded as a Taylor series h = E hj (x)zi with coefficients hj depending smoothly on x E M. One may apply X term by term and the second equation yields Xhj + jghj = 0 for all j. As noticed above, the forces hj _ 0 for j > 1. Thus, h is independent of z: h = h(x) for h(x) a CR function on M. Hence, the CR functions fail to separate points as they would do if T were a hypersurface in C3. As noted by Jacobowitz [1] the CR manifold M need not be restricted to have dimension three. The only requirement is that H0 (M) be nonzero and the above argument easily generalizes to cover this case. For example, M can be taken to be a hypersurface of Levi type (1, k). The manifold T = M x C is always Levi flat in the C direction. Received by the editors July 3, 1986 and, in revised form, December 9, 1986. 1980 Mathematics Subject Classification (1985 Revi3ion). Primary 32C10; Secondary 58A30, 58G30. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page