Graded rings and equivariant sheaves on toric varieties

Perling, Markus

doi:10.1002/mana.200310130

Cited by 53 publications

(173 citation statements)

References 19 publications

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“…We give Klyacho's description of T -equivariant torsion free sheaves on a smooth projective toric 3-fold X [Kly1, Kly2]. Klyachko's work was further developed by M. Perling [Per1] and we will mostly follow his notation. Most constructions of Sections 2.2-2.4 work for X of any dimension, but we stick to dimension 3 for notational convenience.…”

Section: Fixed Locimentioning

confidence: 99%

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Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Gholampour

Kool

Young

2017

Int Math Res Notices

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Abstract. Let M be the moduli space of rank 2 stable torsion free sheaves with Chern classes c i on a smooth 3-fold X. When X is toric with torus T , we describe the T -fixed locus of the moduli space. Connected components of M T with constant reflexive hulls are isomorphic to products of P 1 . We mainly consider such connected components, which typically arise for any c 1 , "low values" of c 2 , and arbitrary c 3 .In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function Z(q) of topological Euler characteristics of M (summing over all c 3 ). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for Z(q) involving the MacMahon function.In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the T -fixed locus is in general not equal to its signed Euler characteristic due to T -fixed obstructions. Nevertheless, the generating function of all invariants is given by Z(q) up to signs.

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Section: Fixed Locimentioning

confidence: 99%

“…Moreover F α is torsion free if and only if the multiplication maps x 1 , x 2 , x 3 are injective on all weight spaces F α (k 1 , k 2 , k 3 ) [Per1]. Therefore, the category of Tequivariant rank r torsion free sheaves F α on U α is equivalent to the category of collections of complex vector spaces…”

Section: Fixed Locimentioning

confidence: 99%

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Gholampour

Kool

Young

2017

Int Math Res Notices

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“…This section is a brief exposition of the main results of [Kly1,Kly2,Per,Koo]. We review Klyacho's and Perling's description of T -equivariant coherent, torsion free, and reflexive sheaves on toric varieties.…”

Section: Equivariant Sheaves On Toric Varietiesmentioning

confidence: 99%

Stable reflexive sheaves and localization

Gholampour¹,

Kool²

2017

Journal of Pure and Applied Algebra

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Abstract. We study moduli spaces N of rank 2 stable reflexive sheaves on P 3 . Fixing Chern classes c 1 , c 2 , and summing over c 3 , we consider the generating function Z refl (q) of Euler characteristics of such moduli spaces. The action of the torus T on P 3 lifts to N and we classify all sheaves in N T . This leads to an explicit expression for Z refl (q). Since c 3 is bounded below and above, Z refl (q) is a polynomial. We find a simple formula for its leading term when c 1 = −1.Next, we study moduli spaces of rank 2 stable torsion free sheaves on P 3 and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of Z refl (q) and Euler characteristics of Quot schemes of certain T -equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, Z refl (q) depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of P 2 × P 1 .

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“…An object in the categry C is called ∆-family [17]. Proposition 3.1 is a special case of more general statement about pure equivariant sheaves on a toric variety [13].…”

Section: Then This Correspondence Is An Equivalence Between the Catementioning

confidence: 99%

Genus Zero BPS Invariants for Local ℙ1

Choi

2012

International Mathematics Research Notices

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Abstract. We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P 1 whose determinant is O P 1 (−2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of stable sheaves of dimension one as proposed by Sheldon Katz [11]. We compute these invariants for low degrees by counting the torus fixed stable sheaves. The results agree with the prediction in local Gromov-Witten theory studied in [3].

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Graded rings and equivariant sheaves on toric varieties

Cited by 53 publications

References 19 publications

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Stable reflexive sheaves and localization

Genus Zero BPS Invariants for Local ℙ1

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