Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

Przyjalkowski, Victor; Shramov, Constantin

doi:10.1134/s0081543815060097

Cited by 15 publications

(16 citation statements)

References 19 publications

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“…In addition to the threefold case discussed in the introduction, the existence of Laurent polynomials satisfying period condition for smooth toric varieties is shown in [Gi97], for complete intersections in Grassmannians is shown in [PSh14a], [PSh14b], [PSh15b], for some complete intersections in some toric varieties is shown in [CKP14] and [DH15]. In [DH15] the toric condition for some complete intersections in toric varieties and partial flag varieties is also checked.…”

Section: Definitionmentioning

confidence: 99%

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Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

Przyjalkowski¹

2017

Sb. Math.

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Section: Definitionmentioning

confidence: 99%

“…For a variety X i−j construct its Givental's type Landau-Ginzburg models, see [Gi97], for a compact explanation see [PSh14a]. Then present it by Laurent polynomial f i−j , see, for instance, [Prz11], and [Prz13].…”

Section: Minkowski Toric Landau-ginzburg Modelsmentioning

confidence: 99%

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

Przyjalkowski¹

2017

Sb. Math.

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“…№14.641.31.0001. V. Lunts is partially supported by the NSA grant H98230-15-1-0255; V. Przyjalkowski is partially supported by the Program of the Presidium This conjecture for del Pezzo surfaces follows from the construction of compactifications of toric Landau-Ginzburg models; it is proven for Fano threefolds of rank one (see [Prz13]) and for complete intersections (see [PSh15b]).…”

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confidence: 99%

Landau–Ginzburg Hodge numbers for mirrors of del Pezzo surfaces

Lunts

Przyjalkowski

2018

Advances in Mathematics

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“…This conjecture holds for smooth Fano threefolds (see [Prz13] and [ILP13] for the Picard rank one case and [CCG + ], [IKKPS], [DHKLP], and [Prz16] for the general case) and complete intersections (see [Prz13], [ILP13], and also [PSh15a]). The existence of Laurent polynomials satisfying the period condition for smooth toric varieties is shown in [Gi97], for complete intersections in Grassmannians it is shown in [PSh14a], [PSh14b] [PSh15b], for some complete intersections in some toric varieties it is shown in [CKP14] and [DH15]. In [DH15] the toric condition for some complete intersections in toric varieties and partial flag manifolds is also checked.…”

Section: Period Conditionmentioning

confidence: 99%

On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

Przyjalkowski

2018

Math Notes

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Toric Landau-Ginzburg models of Givental's type for Fano complete intersections are known to have Calabi-Yau compactifications. We give an alternative proof of this fact. As an output of our proof we get a description of fibers over infinity for compactified toric Landau-Ginzburg models. IntroductionIn [Gi97] (see also [HV00]) Givental presented Landau-Ginzburg models for complete intersections with non-positive canonical classes in smooth toric varieties. They are certain linear sections in complex tori equipped by complex-valued functions called superpotentials. He proved that the regularized I-series, that are generating series for genus 0 one-pointed Gromov-Witten invariants with descendants, are solutions of Picard-Fuchs equations for families of fibers of superpotentials (see more details below). In some cases, such as for Fano complete intersections in projective spaces, Givental's models can be birationally presented as complex tori with complex valued functions on them. These functions are (in some bases) given by Laurent polynomials. Such Laurent polynomials are toric Landau-Ginzburg models, which, besides the I-series-periods correspondence (see [Gi97] and [Prz13]), means that there is a correspondence between Laurent polynomials and toric degenerations of the complete intersections (see [ILP13]), and that they admit Calabi-Yau compactifications (see [Prz13]), see details in Definition 5. (These Calabi-Yau compactifications are described in details in [PSh15a].)More precise, a Landau-Ginzburg model from Homological Mirror Symmetry point of view is a smooth quasiprojective variety with a complex-valued function (superpotential). This variety and the superpotential satisfy a number of properties, algebraic and symplectic. In particular, fibers of the superpotential should be Calabi-Yau varieties. Compactification Principle (see [Prz13, Principle 32]) states that "correct" toric Landau-Ginzburg model, considered as a family of fibers of the superpotential, admits a fiberwise compactification to a Landau-Ginzburg model satisfying Homological Mirror Symmetry conjecture. In particular this compactification is a family of compact Calabi-Yau varieties with smooth total space.A Landau-Ginzburg model by definition is a family of varieties over A 1 . However sometimes a compactification to a family over P 1 is important, see, for instance, [AKO06], [KKP17], [LP16]. In [Prz16] such compactifications of toric Landau-Ginzburg models for smooth Fano threefolds are studied in a systematic way. That is, for any Fano threefold one can choose a "good" toric Landau-Ginzburg model f such that the following holds. Let ∆ be a Newton polytope for f and let ∇ be a polytope dual to ∆. The polytope ∆ is reflexive, that is ∇ is integral, and thus it defines a toric Fano variety T . Fibers of the map determined by f are given by Laurent polynomials supported in ∆. This means that they correspond to elements of an anticanonical linear

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Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

Cited by 15 publications

References 19 publications

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

Landau–Ginzburg Hodge numbers for mirrors of del Pezzo surfaces

On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

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