Lagrangian Floer theory on compact toric manifolds, I

Fukaya, Kenji; Oh, Yong─Geun; Ohta, Hiroshi; Ono, Kaoru

doi:10.1215/00127094-2009-062

Cited by 186 publications

(620 citation statements)

References 48 publications

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“…Meanwhile (i) generalizes results that have been proven for a variety of toric Fano manifolds; in particular in Ostrover and Tyomkin [48] and Fukaya, Oh, Ohta and Ono [22], it is shown that if M is toric and Fano then QH.M; !/ 0 is semisimple for generic choices of the toric symplectic form ! .…”

Section: (Iii)supporting

confidence: 76%

“…This follows from the Batyrev-Givental formula for the quantum homology of such a manifold as re-expressed by eg Fukaya, Oh, Ohta and Ono [22] and Ostrover and Tyomkin [48]; in particular, in light of Proposition 7.11 above, we can simply read off this conclusion from [48,Theorem 4.1].…”

Section: 34mentioning

confidence: 88%

“…A reader who still prefers to work with undeformed (ie Á D 0) quantum homology may take solace in the following, which is somewhat reminiscent of Ostrover and Tyomkin [48,Theorem 4.1] and Fukaya, Oh, Ohta and Ono [22,Proposition 8.8]:…”

Section: Consider a General Elementmentioning

confidence: 99%

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Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher

2011

Geom. Topol.

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We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold .M; !/ which upon passing to homology yields ring isomorphisms with the big quantum homology of M . By studying the properties of the resulting deformed version of the OhSchwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero GromovWitten invariant with two point constraints, and we produce a new algebraic criterion for .M; !/ to admit a Calabi quasimorphism and a symplectic quasistate. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M ), and also whenever M is a point blowup of an arbitrary closed symplectic manifold. 53D40, 53D45

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Section: (Iii)supporting

confidence: 76%

Section: 34mentioning

confidence: 88%

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Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher

2011

Geom. Topol.

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“…The Gross-Siebert program [GS11] gave a sophisticated algebraic formulation of the SYZ construction. Moreover, the SYZ program was successfully carried out in various situations for Kähler manifolds [LYZ00, Leu05,Aur07,CO06,CL10,FOOO10,CLL12,AAK]. This paper explores the SYZ approach for non-Kähler Calabi-Yau manifolds, i.e.…”

Section: Introductionmentioning

confidence: 99%

Non-Kähler SYZ Mirror Symmetry

Lau

Tseng

Yau

2015

Commun. Math. Phys.

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Abstract. We study SYZ mirror symmetry in the context of non-Kähler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric SU (3) systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.

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“…It turns out that the Lagrangian Floer theory developed in [3] and [4,5] can be nicely applied to the various symplectic topological questions concerning polydisks D 2 (r 1 )×· · ·×D 2 (r n ). This is largely because the polydisks contain Lagrangian tori which can be embedded into the toric manifolds S 2 (a 1 ) × · · · × S 2 (a n ) or S 2 (a) × CP n−1 (λ) for suitable choices of a i 's or (a, λ).…”

Section: Introductionmentioning

confidence: 99%

Displacement of polydisks and Lagrangian Floer theory

Fukaya

Ohta

et al. 2013

Journal of Symplectic Geometry

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There are two purposes of the present article. One is to correct an error in the proof of Theorem 6.1.25 in [FOOO1], from which Theorem J [FOOO1] follows. In the course of doing so, we also obtain a new lower bound of the displacement energy of polydisks in general dimension. The results of the present article are motivated by the recent preprint of Hind [H] where the 4 dimensional case is studied. Our proof is different from Hind's even in the 4 dimensional case and provides stronger result, and relies on the study of torsion thresholds of Floer cohomology of Lagrangian torus fiber in simple toric manifolds associated to the polydisks.

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Lagrangian Floer theory on compact toric manifolds, I

Cited by 186 publications

References 48 publications

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Non-Kähler SYZ Mirror Symmetry

Displacement of polydisks and Lagrangian Floer theory

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