Symplectic Manifolds with no Kähler Structure

Tralle, Aleksy; Oprea, John

doi:10.1007/bfb0092608

Cited by 123 publications

(85 citation statements)

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“…This is the case for compact non-toral nilmanifolds (see [15] and the table of the classification of 6 -dimensional compact nilmanifolds in Section 4).…”

Section: Corollary 26 If (M 2m ω) Is a Symplectic Manifold That Imentioning

confidence: 96%

“…(See [9,15] for the definition of the Poisson bracket.) This (as well as any other) Poisson bracket yields a covariant skew-symmetric tensor field of type (2,0)…”

Section: Symplectically Harmonic Formsmentioning

confidence: 99%

“…Three important facts in the study of compact nilmanifolds are (see [15]): 1. Let g be a nilpotent Lie algebra with structural constants c ij k with respect to some basis, and let {α 1 , .…”

Section: Symplectically Harmonic Forms In Homogeneous Spacesmentioning

confidence: 99%

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On symplectically harmonic forms on six-dimensional nilmanifolds

Ibáñez

Rudyak

Tralle

et al. 2001

Comment. Math. Helv.

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Abstract. In the present paper we study the variation of the dimensions h k of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms ω t for which numbers h k (M, ωt) vary with respect to t . This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.

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“…This is the case for compact non-toral nilmanifolds (see [15] and the table of the classification of 6 -dimensional compact nilmanifolds in Section 4).…”

Section: Corollary 26 If (M 2m ω) Is a Symplectic Manifold That Imentioning

confidence: 96%

“…(See [9,15] for the definition of the Poisson bracket.) This (as well as any other) Poisson bracket yields a covariant skew-symmetric tensor field of type (2,0)…”

Section: Symplectically Harmonic Formsmentioning

confidence: 99%

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On symplectically harmonic forms on six-dimensional nilmanifolds

Ibáñez

Rudyak

Tralle

et al. 2001

Comment. Math. Helv.

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“…We say that M satisfies the c-splitting condition for a topological monoid [24]. The following result is proved in Section 4.2.…”

Section: Higher Homotopy Groupsmentioning

confidence: 98%

Homotopy properties of Hamiltonian group actions

Kędra

McDuff

2004

Geom. Topol.

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Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M, ω) and let G be a subgroup of the diffeomorphism group Diff M . We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BG → BG are injective. For example, we extend Reznikov's result for complex projective space CP n to show that both in this case and the case of generalized flag manifolds the natural map H * (BSU (n + 1)) → H * (BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if λ is a Hamiltonian circle action that contracts in G := Ham(M, ω) then there is an associated nonzero element in π 3 (G) that deloops to a nonzero element of H 4 (BG). This result (as well as many others) extends to c-symplectic manifolds (M, a), ie, 2n-manifolds with a class a ∈ H 2 (M ) such that a n = 0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

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“…Let us recall its definition (see [8], [22] for more details). Let X be a simply connected smooth manifold and consider its algebra of differential forms (…”

Section: Nonformality Of the Constructed Manifoldmentioning

confidence: 99%