Floer cohomologies of non-torus fibers of the Gelfand–Cetlin system

Nohara, Yuichi; Ueda, Kanji

doi:10.4310/jsg.2016.v14.n4.a9

Cited by 17 publications

(20 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 4.12, the GC fiber Φ −1 λ (0) over the origin is an S 3 -bundle over S 5 . Meanwhile, Proposition 2.7 in [NU16] implies that Φ −1 λ (0) is SU (3). It is however that SU (3) and S 5 ×S 3 are not homotopy equivalent.…”

Section: W -Shaped Blocks and M -Shaped Blocksmentioning

confidence: 99%

See 1 more Smart Citation

Lagrangian fibers of Gelfand-Cetlin systems

Cho,

Kim,

2019

Preprint

View full text Add to dashboard Cite

A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda [NNU10] on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to (S 1 ) k × N for some smooth manifold N . We also prove that such N 's are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram. Contents 1. Introduction 1 2. Gelfand-Cetlin systems 4 3. Ladder diagram and its face structure 9 4. Classification of Lagrangian fibers 12 5. Iterated bundle structures on Gelfand-Cetlin fibers 24 6. Degenerations of fibers to tori 34 References 43

show abstract

Section: W -Shaped Blocks and M -Shaped Blocksmentioning

confidence: 99%

“…Moreover, some non-torus Lagrangian fibers are indeed non-zero objects in the Fukaya category. Partly because of lack of understanding of Lagrangian fibers in higher dimensional partial flag varieties, Floer theory of non-torus Lagrangians in only limited cases is understood, see [NU16,EL19].…”

Section: Introductionmentioning

confidence: 99%

Lagrangian fibers of Gelfand-Cetlin systems

Cho,

Kim,

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The Grassmannian Gr(n, 2n) admits a U(n)-action with a free Lagrangian orbit [49,Proposition 2.7] with minimal Maslov number 2n. We will show that, when n is a power of 2, this Lagrangian split-generates the Fukaya category over a field of characteristic 2.…”

Section: Grassmanniansmentioning

confidence: 99%

Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

Evans

Lekili²

2018

J. Amer. Math. Soc.

View full text Add to dashboard Cite

Let G be a compact Lie group and k be a field of characteristic p ≥ 0 such that H * (G) has no p-torsion if p > 0. We show that a free Lagrangian orbit of a Hamiltonian G-action on a compact, monotone, symplectic manifold X split-generates an idempotent summand of the monotone Fukaya category F (X; k) if and only if it represents a non-zero object of that summand (slightly more general results are also provided). Our result is based on: an explicit understanding of the wrapped Fukaya category W(T * G; k) through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor D b W(T * G; k) → D b F (X − × X; k) canonically associated to the Hamiltonian G-action on X . We explore several examples which can be studied in a uniform manner including toric Fano varieties and certain Grassmannians.

show abstract

“…The GC system admits nontorus Lagrangian GC fibers at the lower-dimensional strata of the GC polytope, which makes Floer theory of the system more interesting and challenging. Using non-Abelian symmetry or discrete symmetry, particular fibers of limited cases of Grassmannians have been investigated in [8], [9], and [28].…”

Section: Introductionmentioning

confidence: 99%

“…Nohara and Ueda [28] calculated a Floer cohomology of the Lagrangian 3-sphere Φ −1 λ (0, 0, 0), which turns out to be zero over the Novikov field Λ; hence nondisplaceability of the fiber remained open. Theorem A resolves the question by showing that the fiber is nondisplaceable.…”

Section: Introductionmentioning

confidence: 99%

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Cho

Kim

2021

Kyoto J. Math.

View full text Add to dashboard Cite

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya-Oh-Ohta-Ono's bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n ≥ 3) with a monotone Kirillov-Kostant-Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S 3 -fiber in Fl( 3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

show abstract

Floer cohomologies of non-torus fibers of the Gelfand–Cetlin system

Abstract: The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundary strata of the moment polytope. We compute Floer cohomologies of such non-torus Lagrangian fibers in the cases of the 3-dimensional full flag manifold and the Grassmannian of 2-planes in a 4-space.

Cited by 17 publications

References 4 publications

Lagrangian fibers of Gelfand-Cetlin systems

Lagrangian fibers of Gelfand-Cetlin systems

Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Contact Info

Product

Resources

About