Hodge theory on nearly Kähler manifolds

Abstract

Let .M; I; !; / be a nearly Khler 6-manifold, that is, an SU.3/-manifold with .3; 0/-form and Hermitian form ! which satisfies d! D 3 Re , d Im D 2 ! 2 for a nonzero real constant . We develop an analogue of the Khler 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 / D 0 unless p D q or .p D 1; q D 2/ or .p D 2; q D 1/.