Hodge theory on nearly Kähler manifolds

Abstract

Let $\left(M,I,\omega ,\Omega \right)$ be a nearly Kähler $6$–manifold, that is, an $SU\left(3\right)$–manifold with $\left(3,0\right)$–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 Kähler 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 $H^{p,q}\left(M\right)=0$ unless $p=q$ or $\left(p=1,q=2\right)$ or $\left(p=2,q=1\right)$.