#### Volume 15, issue 4 (2011)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Hodge theory on nearly Kähler manifolds

### Misha Verbitsky

Geometry & Topology 15 (2011) 2111–2133
##### 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 $\left(p=1,\phantom{\rule{1em}{0ex}}q=2\right)$ or $\left(p=2,\phantom{\rule{1em}{0ex}}q=1\right)$.

##### Keywords
nearly Kähler, $G_2$–manifold, Hodge decomposition, Hodge structure, Calabi–Yau manifold, almost complex structure, holonomy