Calabi quasi-morphisms for some non-monotone symplectic manifolds

Ostrover, Yaron

doi:10.2140/agt.2006.6.405

Cited by 45 publications

(64 citation statements)

References 31 publications

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“…For CP 2 semi-simplicity is elementary (see e.g. [12]), and for S 2 × S 2 and CP 2 blown up at one point it was established by Y.Ostrover in [35]. The cases of the blow-ups of CP 2 at 2 or 3 points are new.…”

Section: Application To Symplectic Toric Fano 4-manifoldsmentioning

confidence: 99%

Symplectic quasi-states and semi-simplicity of quantum homology

Entov¹,

Polterovich²

2008

Toric Topology

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We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D. McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.

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Section: Application To Symplectic Toric Fano 4-manifoldsmentioning

confidence: 99%

Symplectic quasi-states and semi-simplicity of quantum homology

Entov¹,

Polterovich²

2008

Toric Topology

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“…The following theorem has been originally proven in the case of monotone symplectic manifolds in [16] (using a slightly different setting), then generalized by the first named author in [40] to the class of rational strongly semipositive symplectic manifolds that satisfy a technical condition which was eventually removed in [17].…”

Section: Introductionmentioning

confidence: 99%

On the quantum homology algebra of toric Fano manifolds

Ostrover¹,

Tyomkin²

2009

Sel. math., New ser.

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We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique. (2000). Primary 53D45; Secondary 14M25. Mathematics Subject Classification

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“…The invariantĉ ∞ (H) is closely related to Calabi quasi-morphisms; see [EP,McD10,Os,U11]. For us, however,ĉ ∞ is of interest because of its role in the proofs of the main theorems.…”

Section: Local Floer Homologymentioning

confidence: 99%

Conley Conjecture Revisited

Ginzburg

Gürel

2017

International Mathematics Research Notices

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Abstract. We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class).The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the "largest" and the "smallest" spectral invariants remains bounded and, as a consequence, establish some new cases of the C ∞ -generic existence of infinitely many simple periodic orbits.

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Calabi quasi-morphisms for some non-monotone symplectic manifolds

Cited by 45 publications

References 31 publications

Symplectic quasi-states and semi-simplicity of quantum homology

Symplectic quasi-states and semi-simplicity of quantum homology

On the quantum homology algebra of toric Fano manifolds

Conley Conjecture Revisited

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