Transversely projective structures on a transversely holomorphic foliation, II

Abstract

Given a transversely projective foliation F on a C^∞ manifold M and a nonnegative integer k, a transversal differential operator D_F(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 D_F(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.