Exact, graded, immersed Lagrangians and Floer theory

Alston, Garrett; Bao, Erkao

doi:10.4310/jsg.2018.v16.n2.a2

Cited by 14 publications

(33 citation statements)

References 17 publications

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“…∃ K 1/2 Li ) [12], [15], X = L 1 ∩ L 2 is orientable. 2 Hitchin showed that this already implies L is a J-holomorphic submanifold of M Other interesting works towards the categorification or quantization of complex Lagrangian intersections include works of Baranovsky and Ginzburg [4], Kapustin and Rozansky [40], Kashiwara and Schapira [41], etc.…”

Section: 3mentioning

confidence: 99%

“…Proof. We apply (2) to O C1 , O C2 , and then use the following Lemma 3.6 to determine Chern characters of Ψ(O Ci ), i = 1, 2.…”

Section: Mukai Flops and Plücker Type Formulasmentioning

confidence: 99%

“…It was observed by Donaldson-Thomas [24] (in fact first by Tyurin [64]) that the transpose of the above sequence with respect to Serre duality pairings on Y and S remains the same, and r(M Y ) will be a complex Lagrangian submanifold of M S provided that H 0,2 (Y, EndE) = 0. Let S be a (K3) generic hyperplane section of type (2,3) in Y = P 1 × P 2 , and L m = O Y (1, m) be a polarization. Then the moduli space M Y of rank 2, L m -stable bundles on Y with Chern classes c 1 = (ǫ 1 , ǫ 2 ), c 2 = (−1, 1) · (ǫ 1 + 1, ǫ 2 − 1) is isomorphic to P k , where (ǫ 1 , ǫ 2 ) = (0, 1) or (1, 0) or (1, 1), and k = (5 + 6ǫ 1…”

Section: Appendix On Examples Of Complex Lagrangians From Higher Dimementioning

confidence: 99%

“…Fukaya, Oh, Ohta and Ono introduced quantum corrections in terms of counting holomorphic disks bounding Lagrangians and defined their Floer theory [28]. It was later generalized by Akaho and Joyce [1] to immersed Lagrangian submanifolds (see also Alston and Bao [2] for some exact immersed cases).…”

Section: Introductionmentioning

confidence: 99%

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Mukai flops and Plücker-type formulas for hyper-Kähler manifolds

Cao

Leung

2020

Proc. Amer. Math. Soc.

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We study the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds. In particular, we study their behaviour under Mukai flops and give a rigorous proof of the Plücker type formula for Legendre dual subvarieties written down by the second author before. Then we apply the formula to study projective dual varieties in projective spaces. Intersection theory of complex Lagrangians in hyper-Kähler manifoldsIn this section, we recall some standard facts about the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds.2.1. Some basic notions from hyper-Kähler geometry. Let (M, g) be a hyper-Kähler manifold with complex structures I, J, K which satisfy the Hamilton relationThen the twistor family of complex structures on M can be expressed as S 2 = {aI + bJ + cK | a 2 + b 2 + c 2 = 1 and a, b, c ∈ R} Fixing a complex structure J, (M, g, J) is a Calabi-Yau manifold which corresponds to the embedding Sp(n) ⊆ SU (2n). We denote the Kähler form by ω J . The parallel form Ω J = ω I − iω K defines a holomorphic symplectic structure on (M, J). And the J-holomorphic volume form for the Calabi-Yau structure on (M, g, J) is just the top exterior power of Ω J .

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Section: 3mentioning

confidence: 99%

“…Proof. We apply (2) to O C1 , O C2 , and then use the following Lemma 3.6 to determine Chern characters of Ψ(O Ci ), i = 1, 2.…”

Section: Mukai Flops and Plücker Type Formulasmentioning

confidence: 99%

Section: Appendix On Examples Of Complex Lagrangians From Higher Dimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mukai flops and Plücker-type formulas for hyper-Kähler manifolds

Cao

Leung

2020

Proc. Amer. Math. Soc.

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“… also gains a special feature: in this case the associated sphere is equivalent to equipped with a non-trivial -local system in the Fukaya category (see [Dam12, AB14, She15]). Therefore, the iterated cone relation can be packaged directly into a long exact sequence without invoking the iterated cones.…”

Section: Introductionmentioning

confidence: 99%

Dehn twist exact sequences through Lagrangian cobordism

Mak¹,

Wu²

2018

Compositio Math.

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This paper introduces a new Lagrangian surgery construction that generalizes Lalonde–Sikorav and Polterovich’s well-known construction, and combines this with Biran and Cornea’s Lagrangian cobordism formalism. With these techniques, we build a framework which both recovers several known long exact sequences (Seidel’s exact sequence, including the fixed point version and Wehrheim and Woodward’s family version) in symplectic geometry in a uniform way, and yields a partial answer to a long-term open conjecture due to Huybrechts and Thomas; this also involved a new observation which relates projective twists with surgeries.

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Immersed Lagrangian Floer cohomology via pearly trajectories

Alston¹,

Bao

2021

Journal of Geometry and Physics

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Exact, graded, immersed Lagrangians and Floer theory

Cited by 14 publications

References 17 publications

Mukai flops and Plücker-type formulas for hyper-Kähler manifolds

Mukai flops and Plücker-type formulas for hyper-Kähler manifolds

Dehn twist exact sequences through Lagrangian cobordism

Immersed Lagrangian Floer cohomology via pearly trajectories

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