Dual torus fibrations and homological mirror symmetry for $A_n$-singularities

Chan, Kwokwai; Ueda, Kanji

doi:10.4310/cntp.2013.v7.n2.a5

Cited by 14 publications

(19 citation statements)

References 19 publications

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“…Indeed, [Sym03] shows that there is at least a Lagrangian fibration when the singularity is factored. In the I k cases there are explicit formulas for special Lagrangian fibrations-see [Gro01], or see [CU13] which begins with a nice brief presentation of this. Further examples of cluster varieties are known to come from moduli of Higgs bundles, in which case the Hitchin fibration is a special Lagrangian fibration.…”

Section: When Lines Do Not Wrapmentioning

confidence: 99%

Classification of Rank 2 Cluster Varieties

Mandel

2019

SIGMA

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We classify rank 2 cluster varieties (those for which the span of the rows of the exchange matrix is 2-dimensional) according to the deformation type of a generic fiber U of their X -spaces, as defined by Fock and Goncharov. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call U positive if dim[Γ(U, O U )] = dim(U ) (which equals 2 in these rank 2 cases). This is the condition for the [GHK15b]-construction to produce an additive basis of theta functions on Γ(U, O U ). We find that U is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the monodromy of the tropicalization U trop of U is one of Kodaira's monodromies. In these cases we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is U trop . We also describe the action of the cluster modular group on U trop in the positive cases.

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Section: When Lines Do Not Wrapmentioning

confidence: 99%

Classification of Rank 2 Cluster Varieties

Mandel

2019

SIGMA

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“…The results of [29] can also be used to treat the case of the punctured A n Milnor fibers (in which the affine manifold has "parallel monodromy-invariant directions"). See [12] for a discussion of related cases.…”

Section: Wrapped Floer Cohomologymentioning

confidence: 99%

On the symplectic cohomology of log Calabi–Yau surfaces

Pascaleff

2019

Geom. Topol.

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This article studies the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symplectic cohomology, and compute enough differentials to identify a basis for the degree-zero part of the symplectic cohomology. This basis is indexed by integral points in a certain integral affine manifold, providing a relationship to the theta functions of Gross-Hacking-Keel. Included is a discussion of wrapped Floer cohomology of Lagrangian submanifolds and a description of the product structure in a special case. We also show that, after enhancing the coefficient ring, the degree-zero symplectic cohomology defines a family degenerating to a singular surface obtained by gluing together several affine planes.

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“…Hence we can define the SYZ transform of L as a holomorphic line bundle L overX. Moreover, in the overlap U C 01 ∩ U C 12 , we have∇ Similar techniques work for all mirrors of A n -resolutions (or toric Calabi-Yau 2-folds) [24] and also for the deformed conifold and mirrors of other small toric Calabi-Yau 3-folds [23]. Moreover, this SYZ transform, which is defined only on the object level, in fact induces HMS equivalences as predicted by Kontsevich; the readers may consult [24,23] for the precise statements and proofs.…”

Section: Local Mirror Symmetry By Syzmentioning

confidence: 99%

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This is a write-up of the author's talk in the conference Algebraic Geometry in East Asia 2016 held at the University of Tokyo in January 2016. We give a survey on the series of papers [16,24, 23,22] where the author and his collaborators Daniel Pomerleano and Kazushi Ueda show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.

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confidence: 99%