Transversely projective structures on a transversely holomorphic foliation, II

Given a transversely projective foliation F \mathcal F on a C ∞ C^\infty manifold M M and a nonnegative integer k k , a transversal differential operator D F ( 2 k + 1 ) {\mathcal D}_{\mathcal F}(2k+1) of order 2 k + 1 2k+1 from N ⊗ k N^{\otimes k} to N ⊗ ( − k − 1 ) N^{\otimes (-k-1)} is constructed, where N 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 \mathcal F to the first cohomology of the locally constant sheaf over M M defined by the kernel of the operator D F ( 3 ) {\mathcal D}_{\mathcal 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 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.