Toric Degeneration and Nondisplaceable Lagrangian Tori in S2×S2

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

doi:10.1093/imrn/rnr128

Cited by 55 publications

(154 citation statements)

References 16 publications

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“…An interesting example to which Theorem 7.2 applies but the version without bulk deformations may not apply can be constructed using the idea of [7]. 7.2.…”

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

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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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“…An interesting example to which Theorem 7.2 applies but the version without bulk deformations may not apply can be constructed using the idea of [7]. 7.2.…”

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

“…In this way, we obtain various improvements and generalizations of the theorems concerning symplectic topology of polydisks proven in [8]. See [7] for other applications of Lagrangian Floer theory a la [3][4][5]. We give two high-dimensional generalizations of (1.1).…”

Section: Introductionmentioning

confidence: 99%

Displacement of polydisks and Lagrangian Floer theory

Fukaya

Ohta

et al. 2013

Journal of Symplectic Geometry

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“…Finally, we relate the ATBD of CP 1 ×CP 1 in Fig. 1 with the singular Lagrangian fibration given by Fukaya et al [13], as well as a similar ATBD of CP 2 with the singular Lagrangian fibration described in [38]. In [12,13] it was shown that there are a continuous of non-displaceable fibres in the monotone CP 1 ×CP 1 ; in CP 2 #2CP 2 for blowups of sizes (α,…”

Section: To Mutually Different Hamiltonian Isotopy Classesmentioning

confidence: 68%

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

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We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP 2 #CP 2 , CP 2 #2CP 2 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in CP 2 #kCP 2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP 2 #CP 2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP 2 #2CP 2 . Finally, the Lagrangian tori n 1 ,n 2 ,n 3 p,q,r

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“…This argument is also based upon a pseudoholomorphic disk count. We also refer to [19] for an alternative proof of the same statement.…”

Section: 23mentioning

confidence: 99%

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

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Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

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Toric Degeneration and Nondisplaceable Lagrangian Tori in S2×S2

Cited by 55 publications

References 16 publications

Displacement of polydisks and Lagrangian Floer theory

Displacement of polydisks and Lagrangian Floer theory

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

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