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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Hodge theory on nearly Kähler manifolds

Misha Verbitsky

Geometry & Topology 15 (2011) 2111–2133
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).

Keywords
nearly Kähler, $G_2$–manifold, Hodge decomposition, Hodge structure, Calabi–Yau manifold, almost complex structure, holonomy
References
Publication
Received: 19 June 2008
Revised: 7 October 2010
Accepted: 12 June 2011
Published: 28 October 2011
Proposed: Gang Tian
Seconded: Simon Donaldson, Yasha Eliashberg
Authors
Misha Verbitsky
Laboratory of Algebraic Geometry
Faculty of Mathematics, NRU HSE
7 Vavilova Ul
Moscow 117312
Russia
http://verbit.ru/