Aspects of functoriality in homological mirror symmetry for toric varieties

Hanlon, Andrew D.; Hicks, Jeffrey Stephen

doi:10.1016/j.aim.2022.108317

Cited by 8 publications

(3 citation statements)

References 24 publications

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“…This procedure can be generalized to apply to conormals of subtori mirror to toric subvarieties. Moreover, the relevant collection of line bundles can be identified with Θ using [Abo09;HH22]. In fact, it is possible to prove Corollary D by itself using variations on [GPS18a, Proposition 5.22] and [HH22, Lemma 3.14].…”

Section: Mirror Symmetry Motivationmentioning

confidence: 99%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

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Given any toric subvariety Y of a smooth toric variety X of codimension k, we construct a length k resolution of O Y by line bundles on X. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of O X under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry.As a corollary, we obtain a virtual analogue of Hilbert's syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman, and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal's claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara's conjecture that this remains true for arbitrary line bundles.

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Section: Mirror Symmetry Motivationmentioning

confidence: 99%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

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“…When Y is smooth, the wrapping can be described in terms of a "superpotential" function on the torus [18]; more generally, the stop for the wrapping is a certain singular Legendrian introduced in [5,6]. There are several approaches to this mirror symmetry: the microlocal sheaf theoretic approach of [3,5,6,27] is completed, first by [21], and also by logically independent methods in [30] and [28]; meanwhile [1,2,16,17] provides a direct Floer theoretic approach. (Comparing sheaf and Floer results requires the foundational [13][14][15] together with the Lagrangian skeleton calculations of [12,31].…”

Section: Let G N Mmentioning

confidence: 99%

Toric mirror symmetry revisited

Shende¹

2022

Comptes Rendus. Mathématique

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“…In recent years, a variation in the form of toric hyperplane arrangements has been studied more prominently, from combinatorial, algebraic, and topological point of views; see [2,3] for examples. Some applications of the theory of toric hyperplane arrangements to toric varieties have been presented in [4,5].…”

Section: Introduction 1hyperplane Arrangementsmentioning

confidence: 99%

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

Bergerová

2024

PUMP J. Undergrad. Res.

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A toric hyperplane is the preimage of a point in a circle of a continuous surjective group homomorphism from the n-torus to the circle. A toric hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of toric hyperplane arrangements on n-tori which are spanning and in general position. Specifically, we describe the symmetry of f-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.

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Aspects of functoriality in homological mirror symmetry for toric varieties

Cited by 8 publications

References 24 publications

Resolutions of toric subvarieties by line bundles and applications

Resolutions of toric subvarieties by line bundles and applications

Toric mirror symmetry revisited

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

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