Transversely Projective Structures on a Transversely Holomorphic Foliation, Ii

Given a transversely projective foliation F on a C∞ manifold M and a nonnegative integer k, a transversal differential operator DF (2k + 1) of order 2k + 1 from N⊗k to N⊗(−k−1) is constructed, where N denotes the normal bundle for the foliation. There is a natural homomorphism from the space of all infinitesimal deformations of the transversely projective foliation F to the first cohomology of the locally constant sheaf over M defined by the kernel of the operator DF (3). On the other hand, from this first cohomology there is a homomorphism to the first cohomology of the sheaf of holomorphic sections of N . The composition of these two homomorphisms coincide with the infinitesimal version of the forgetful map that sends a transversely projective foliation to the underlying transversely holomorphic foliation.