Hodge theory on nearly Kaehler manifolds

Abstract

Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies $dω=3λ\ReΩ, d\ImΩ=-2λω^2$, for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. When M is compact, these identities bring powerful results about cohomology of M. We show that harmonic forms on M admit the Hodge decomposition, and prove that H^{p,q}(M)=0 unless p=q or (p=1, q=2) or (p=2, q=1).