Geometry & Topology 15 (2011) 2111–2133

DOI: 10.2140/gt.2011.15.2111

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 H^p,q(M) = 0 unless p = q or (p = 1, q = 2) or (p = 2, q = 1).

Keywords

nearly Kähler, G₂–manifold, Hodge decomposition, Hodge structure, Calabi–Yau manifold, almost complex structure, holonomy

Mathematical Subject Classification

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