Geometry & Topology

Hodge theory on nearly Kähler manifolds

Misha Verbitsky

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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
Received: 19 June 2008
Revised: 7 October 2010
Accepted: 12 June 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

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

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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