Hodge theory on nearly Kähler manifolds

Verbitsky, Misha

doi:10.2140/gt.2011.15.2111

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Hodge theory on nearly Kähler manifolds

Misha Verbitsky

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 $d ω = 3 λ Re Ω$ , $d Im Ω = - 2 λ ω^{2}$ 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, $G_2$–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/

Volume 15, issue 4 (2011)