A categorification of Morelli’s theorem

Fang, Bohan; Liu, Chiu Chu Melissa; Treumann, David; Zaslow, Eric

doi:10.1007/s00222-011-0315-x

Cited by 59 publications

(126 citation statements)

References 19 publications

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“…The pair ((C * ) n , Λ) then has two flavors of Fukaya category, called partially wrapped and infinitesimally wrapped, which are meant to be equivalent to the dg-categories Coh(X) and P erf(X), respectively. The equivalence proceeds by combining the coherentconstructible correspondence [10] and the Nadler-Zaslow correspondence [25]. This approach is explored by Fang-Liu-Treumann-Zaslow in [11].…”

Section: Mirror Symmetrymentioning

confidence: 99%

On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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We define a new class of symplectic objects called 'stops', which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an A∞-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates these partially wrapped Fukaya categories to the wrapped Fukaya category of the underlying Liouville domain. This operation is mirror to removing a divisor. 373is the open-closed map. Here, the action filtration on Hochschild homology is induced from the action filtration on Floer cochain groups; for a precise definition, see Section 4.3.In particular, any punctured Riemann surface other than C and any cotangent bundle is strongly nondegenerate.Remark 1.1. A nondegenerate Liouville domain F is one for which some element e ∈ HH * (W(F )) satisfies OC(e) = 1. This condition is Abouzaid's generation criterion [2], which says that such an F admits a finite collection of Lagrangians which split-generate W(F ). Ganatra further proved that in this case, OC is an isomorphism [15], which in particular implies that e is unique.An approximate version of Theorem 4.21 can be stated as follows: Theorem 1.2. Let M be a Liouville domain, and let σ be a strongly nondegenerate stop in M . Let B ⊂ W σ (M ) be the full subcategory of objects supported near σ. Then the inclusion W σ (M ) → W(M ) induces a fully faithful functorwhere the quotient is a quotient of an A ∞ category by a full subcategory in the sense of .At the most basic level, Theorem 1.2 shows that the partially wrapped Fukaya category, together with B, really knows at least as much as the wrapped Fukaya category. That is, restricting to the zero-filtered part but remembering the stop does not lose information. At an intuitive level, this happens because the 'nice' presentation of W(M ) giving rise to the stop filtration uses a contact form with a large number of canceling Reeb chords. These chords live at different levels in the filtration, so passing to the zero-filtered part results in them no longer canceling. In a more minimal presentation, these chords could be eliminated geometrically.The key ingredient of the proof of Theorem 1.2 is an auxiliary filtration on W(M ), presented as the trivial quotient A = W(M )/B. The benefit of this quotient presentation is that it naturally contains the category A 0 = W σ (M )/B(σ) as the minimally filtered part, which makes it possible to build a homotopy which retracts A onto A 0 . The homotopy itself requires a filtered version of the annulus trick, which was introduced in [2] and extended in a forthcoming paper by Abouzaid and Auroux. Specifically, one factors the identity operation as a composition of a product and a coproduct, where ...

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Section: Mirror Symmetrymentioning

confidence: 99%

On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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“…The authors in [FLTZ1,FLTZ2] conjecture that a similar statement holds when replacing the equivariant category with ordinary perfect complexes, Perf(X Σ ). For toric varieties partial results in this direction were obtained by Treumann in the preprint [Tr].…”

Section: Introductionmentioning

confidence: 83%

“…We start by reviewing the statement of the equivariant coherent-constructible (CC) correspondence of [FLTZ1,FLTZ2]. Next we extend to toric orbifolds partial results of Treumann [Tr] on the non-equivariant CC-correspondence.…”

Section: The Coherent-constructible Correspondencementioning

confidence: 99%

“…Let Σ = (N, Σ, β) be a stacky fan (see Section 3 for a definition), and let M = Hom(N, Z) and M R = M ⊗ Z R. Denote X Σ the corresponding toric orbifold. In [FLTZ1,FLTZ2] the authors define a conical LagrangianΛ Σ ⊂ T * M R , and construct an equivalence between the torus equivariant category of perfect complexes Perf T (X Σ ), and the category of compactly supported constructible sheaves on M R with singular support inΛ Σ , Sh c (M R ,Λ Σ ). The notion of singular support is due to Kashiwara and Schapira [KS], informally it is an invariant that measures the way sections of constructible sheaves propagate.…”

Section: Introductionmentioning

confidence: 99%

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The non-equivariant coherent-constructible correspondence and a conjecture of King

Scherotzke

Sibilla

2015

Sel. Math. New Ser.

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Abstract. The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we prove the equivariant coherent-constructible correspondence for a class of toric varieties including weighted projective space. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.

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“…between the dg categories of perfect complexes and constructible sheaves is referred to as the coherent-constructible correspondence (or CCC) [FLTZ11], proved in the stated generality in [Kuw16]. Local systems on the graph Γ give rise to Lagrangian branes in T * T 2 via the construction of [STWZ15]: up to Hamiltonian isotopy there is a canonical embedded exact Lagrangian L Γ in T * T 2 which deformation retracts onto Γ.…”

Section: Introductionmentioning

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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A categorification of Morelli’s theorem

Cited by 59 publications

References 19 publications

On partially wrapped Fukaya categories

On partially wrapped Fukaya categories

The non-equivariant coherent-constructible correspondence and a conjecture of King

Kasteleyn operators from mirror symmetry

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