Hodge theory on nearly Kahler manifolds

Let (M, I, omega, Omega) be a nearly Kahler 6-manifold, that is, an SU(3)-manifold with (3, 0)-form Omega and Hermitian form omega which satisfies d omega = 3 lambda Re Omega, dIm Omega = -2 lambda omega(2) for a nonzero real constant lambda. We develop an analogue of the Kahler relations on M, proving several useful identities for various intrinsic Laplacians on M. When M is compact, these identities give powerful results about cohomology of M. We show that harmonic forms on M admit a Hodge decomposition, and prove that H(p,q)(M) = 0 unless p = q or (p = 1, q = 2) or (p = 2, q = 1).