  On a symplectic manifold, there is a natural elliptic complex replacing the
de Rham complex. It can be coupled to a vector bundle with connection and, when
the curvature of this connection is constrained to be a multiple of the
symplectic form, we find a new complex. In particular, on complex projective
space with its Fubini-Study form and connection, we can build a series of
differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from
parabolic differential geometry.
