On the topology of real Lagrangians in toric symplectic manifolds

Brendel, Joé; Kim, Joon‐Tae; Moon, Joo-Hee

doi:10.48550/arxiv.1912.10470

Cited by 4 publications

(10 citation statements)

References 16 publications

(27 reference statements)

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“…Together with J. Kim and J. Moon, we show in [7] that central symmetry of the moment polytope is a sufficient condition for the central fibre T 0 to be real. Under property F S, Theorem 1.2 is therefore an equivalence.…”

Section: Introductionsupporting

confidence: 56%

“…Proof. It is proved in [7] that the central fibre T 0 is real whenever ∆ = −∆. Hence we can take an anti-symplectic involution σ of M such that Fix σ = T 0 .…”

Section: Now Suppose That θ Nmentioning

confidence: 99%

“…In collaboration with J. Kim and J. Moon [7], we construct many real Lagrangians in toric symplectic manifolds by lifting symmetries of the moment polytope.…”

Section: Introductionmentioning

confidence: 99%

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Real Lagrangian Tori and Versal Deformations

Brendel

2020

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Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called real. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not real, extending Kim's result [16] for S 2 × S 2 . Inside products of S 2 , we show that all products of Chekanov tori are pairwise distinct and not real either. These results indicate that real tori are rare.

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Section: Introductionsupporting

confidence: 56%

“…Proof. It is proved in [7] that the central fibre T 0 is real whenever ∆ = −∆. Hence we can take an anti-symplectic involution σ of M such that Fix σ = T 0 .…”

Section: Now Suppose That θ Nmentioning

confidence: 99%

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Real Lagrangian Tori and Versal Deformations

Brendel

2020

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“…Together with J. Kim and J. Moon, we show in [BKM19] that central symmetry of the moment polytope is a sufficient condition for the central fibre T 0 to be real. Under property F S, Theorem 4.1.2 is therefore an equivalence.…”

Section: Introductionsupporting

confidence: 56%

On the topology of Lagrangian submanifolds in toric symplectic manifolds

Brendel¹

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This thesis consists of four main chapters. Chapter 2, An introduction to the Chekanov torus, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, On the topology of real Lagrangians in toric symplectic manifolds, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, Real Lagrangian tori and versal deformations, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, Squeezing via degenerations of the complex projective plane, is an appendix to the paper On certain quantifications of Gromov’s non-squeezing theorem by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.

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“…For topological reasons, there is no closed real Lagrangian in R 2n at all. Known monotone symplectic 4-manifolds containing real Lagrangian tori are S 2 ˆS2 and the three-fold monotone blow-up of CP 2 , see [6,Theorem D]. In [28], it was proved that the Chekanov-Schlenk torus in S 2 ˆS2 is not real, from which we believed that our main result holds.…”

Section: Introductionmentioning

confidence: 76%