Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems

Guillermou, Stéphane; Kashiwara, Masaki; Schapira, Pierre

doi:10.1215/00127094-1507367

Cited by 81 publications

(188 citation statements)

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“…Guillermou-Kashiwara-Schapira [GKS12] constructed sheaf quantizations of Hamiltonian isotopies. Since the microsupports of sheaves are conic subsets of cotangent bundles, the microlocal sheaf theory is related to the exact (homogeneous) symplectic structures rather than the symplectic structures of cotangent bundles.…”

Section: Sheaf Quantization Of Hamiltonian Isotopies ([Gks12])mentioning

confidence: 99%

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Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

Ike

2019

Publ. Res. Inst. Math. Sci.

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We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf quantizations of the Lagrangians in Tamarkin's category. Our sheaf-theoretic method can also deal with clean and degenerate Lagrangian intersections. k∈Z dim H k (M ; L) (1.4) for any rank 1 locally constant sheaf L on M over any field k. In particular, #(L 1 ∩ L 2 ) ≥ k∈Z dim H k (M ; k).

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Section: Sheaf Quantization Of Hamiltonian Isotopies ([Gks12])mentioning

confidence: 99%

“…Although φ does not satisfy[GKS12, (3.3)] in general, K| M ×R×M ×R×J is bounded for any relatively compact subinterval J of I. The author learned the detailed proof from S. Guillermou.…”

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confidence: 99%

Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

Ike

2019

Publ. Res. Inst. Math. Sci.

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“…Provided Γ satisfies a certain consistency condition, Λ Γ is Legendrian isotopic to the Legendrian link Λ Σ • associated to the rays of Σ. Following the results of [GKS12], such an isotopy quantizes to an equivalence Sh c Λ Γ (T 2 ) ∼ − → Sh c Λ Σ • (T 2 ). On the other hand, Λ Σ • is a subset of the singular Legendrian Λ Σ , hence there is a fully faithful inclusion Sh c Λ Σ • (T 2 ) ֒→ Sh c Λ Σ (T 2 ).…”

Section: Introductionmentioning

confidence: 94%

“…To understand this ambiguity we consider the following elementary autoisotopies of Λ Σ • : each ray ρ of Σ determines a collection of pairwise isotopic components with parallel front projections in T 2 , and we let σ ρ be the autoisotopy which moves these in their normal direction until they become cyclically permuted (see Figure 7). These act by autoequivalences on Sh c Λ Σ • (T 2 ) following [GKS12]. On the other hand, also associated to ρ is a line bundle L ρ on X Σ -when X Σ is a variety this is just the line bundle O(−D ρ ) defined by the toric divisor D ρ , and is a root of O(−D ρ ) when D ρ has nontrivial stabilizers.…”

Section: Isotopies and Integrabilitymentioning

confidence: 99%

Kasteleyn operators from mirror symmetry

Treumann

Williams

Zaslow

2019

Sel. Math. New Ser.

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Given a consistent bipartite graph Γ in T 2 with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (Γ, E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C × ) 2 . The second is to form the conjugate Lagrangian L ⊂ T * T 2 of Γ, equip it with a brane structure prescribed by E, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of (C × ) 2 determined by the Legendrian link which lifts the zig-zag paths of Γ (and to which the noncompact Lagrangian L is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of L.

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“…Consider the contact embedding of In [30,Theorem 4.1], the results of [18] are applied to establish that up to quasi-equivalence…”

Section: Singular Support and The Category Shmentioning

confidence: 99%

Generating families and augmentations for Legendrian surfaces

Rutherford

Sullivan

2018

Algebr. Geom. Topol.

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Given an augmentation for a Legendrian surface in a 1-jet space, Λ ⊂ J 1 (M ), we explicitly construct an object, F ∈ Sh • Λ (M × R, K), of the (derived) category from [30] of constructible sheaves on M × R with singular support determined by Λ. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on [25,26,27], is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of [30] for 1dimensional Legendrian knots to obtain a combinatorial model for sheaves in Sh • Λ (M × R, K) in the 2-dimensional case.

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Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems

Abstract: Let I be an open interval containing zero, let M be a real manifold, let P T M be its cotangent bundle with the zero-section removed, and letˆD ¹' t º t 2I be a homogeneous Hamiltonian isotopy of

Cited by 81 publications

References 23 publications

Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

Kasteleyn operators from mirror symmetry

Generating families and augmentations for Legendrian surfaces

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