Lagrangian Floer theory on compact toric manifolds II: bulk deformations

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

doi:10.1007/s00029-011-0057-z

Cited by 104 publications

(368 citation statements)

References 25 publications

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“…In this discussion we use the C-coefficients as in [4,5] but one can also use the Q-coefficients as in [3]. As we explained at the beginning of Section 5, we assume that L (0) and L (1) are transversal.…”

Section: Corrected Proofs Of Theorem J and Theorem 6125 [3]mentioning

confidence: 99%

“…In this subsection, we generalize this inequality by involving bulk deformations. In [5] we developed bulk deformations of Lagrangian Floer theory of L(u). We briefly recall the result of Section 3 [5] in which we defined Floer cohomology…”

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

“…We also recall the following definition (Definition 3.17 [5]) that is a generalization of Definition 4.11 [4]. Definition 7.4.…”

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

“…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%

“…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%

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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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Section: Corrected Proofs Of Theorem J and Theorem 6125 [3]mentioning

confidence: 99%

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

“…We also recall the following definition (Definition 3.17 [5]) that is a generalization of Definition 4.11 [4]. Definition 7.4.…”

Section: Lemma 64 the Two Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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Enumerative Aspects of the Gross-Siebert Program

Garrel¹,

Peter²,

Ruddat³

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Michel van Garrel KIAS,discretizing Hitchin's Legendre duality.Enumerative aspects of the Gross-Siebert program 3 Kontsevich and Soibelman [32] demonstrated how one could reconstruct a K3 surface from an affine structure with singularities on S 2 . Using logarithmic geometry, Gross and Siebert were able to solve the reconstruction problem [20] in any dimension, obtaining a degenerating family of Calabi-Yau manifolds X → D over a holomorphic disk from the information of (B, P, ϕ) and a log structure. Furthermore, this family is parametrized by a canonical coordinate (in the usual sense in mirror symmetry). The construction features wall-crossings and scatterings, structures that encode enumerative information linking symplectic with complex geometry via tropical geometry. As will be hinted at in this exposition, Gromov-Witten theory [21] can also be incorporated in this framework. Toric conventionsWe assume familiarity with toric geometry. The interested reader is referred to the excellent exposition of Fulton [10]. As the following story is closely tied to toric geometry, it is convenient to begin by making a few conventions regarding notation.SetFor n ∈ N, set n, m to be the evaluation of n on m. Set a toric fan Σ in M R . Let Σ [n] signify the set of n dimensional cones of Σ . Let X Σ be the toric variety defined by Σ .Denote by T Σ the free abelian group generated by Σ [1] . For ρ ∈ Σ [1] , denote by v ρ the corresponding generator in T Σ . We will need the mapwhereρ is the integral vector generating ρ, that is ρ ∩ M = Z ≥0ρ .

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Introduction

Fukaya

Ohta

et al. 2020

Springer Monographs in Mathematics

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Lagrangian Floer theory on compact toric manifolds II: bulk deformations

Cited by 104 publications

References 25 publications

Displacement of polydisks and Lagrangian Floer theory

Displacement of polydisks and Lagrangian Floer theory

Enumerative Aspects of the Gross-Siebert Program

Introduction

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