## Geometry & Topology

### Hodge theory on nearly Kähler manifolds

Misha Verbitsky

#### Abstract

Let $(M,I,ω,Ω)$ be a nearly Kähler $6$–manifold, that is, an $SU(3)$–manifold with $(3,0)$–form $Ω$ and Hermitian form $ω$ which satisfies $dω=3λReΩ$, $dImΩ=−2λω2$ for a nonzero real constant $λ$. 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 $Hp,q(M)=0$ unless $p=q$ or $(p=1,q=2)$ or $(p=2,q=1)$.

#### Article information

Source
Geom. Topol., Volume 15, Number 4 (2011), 2111-2133.

Dates
Revised: 7 October 2010
Accepted: 12 June 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732363

Digital Object Identifier
doi:10.2140/gt.2011.15.2111

Mathematical Reviews number (MathSciNet)
MR2860989

Zentralblatt MATH identifier
1246.58002

#### Citation

Verbitsky, Misha. Hodge theory on nearly Kähler manifolds. Geom. Topol. 15 (2011), no. 4, 2111--2133. doi:10.2140/gt.2011.15.2111. https://projecteuclid.org/euclid.gt/1513732363

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