Projecting Fanos in the mirror

Kasprzyk, Alexander; Katzarkov, Ludmil; Przyjalkowski, Victor; Sakovics, Dmitrijs

doi:10.48550/arxiv.1904.02194

Cited by 3 publications

(3 citation statements)

References 42 publications

(87 reference statements)

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“…This conjecture has been studied and is supported in many cases, such as: del Pezzo surfaces, Fano threefolds, and complete intersections [21][22][23]. See also the beautiful three-dimensional example by Petracci [24].…”

Section: Example 8 Consider the Laurent Polynomialsmentioning

confidence: 90%

Laurent polynomials in Mirror Symmetry: why and how?

Kasprzyk

Przyjalkowski

2022

Proyecciones (Antofagasta)

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Section: Example 8 Consider the Laurent Polynomialsmentioning

confidence: 90%

Laurent polynomials in Mirror Symmetry: why and how?

Kasprzyk

Przyjalkowski

2022

Proyecciones (Antofagasta)

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“…This conjecture has been studied and is supported in many cases: for example, del Pezzo surfaces, Fano threefolds, and complete intersections [2,31,32]. See also the beautiful threedimensional example by Petracci [47].…”

Section: Example 8 Consider the Laurent Polynomialsmentioning

confidence: 92%

Laurent polynomials in Mirror Symmetry: why and how?

Kasprzyk,

Przyjalkowski

2021

Preprint

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“…(The latter condition is called the toric condition.) Toric Landau-Ginzburg models exist for del Pezzo surfaces and Fano threefolds ([Prz13], [ILP13], [CCGK16], [KKPS19]) complete intersections ([CCG + ], [ILP13], [PSh15a], [Prz18b]); some partial results are known for Grassmannians ([PSh14], [PSh15b]) and complete intersections in toric varieties ([Gi97], [DH15]). In [Prz13,Conjecture 36] the existence of toric Landau-Ginzburg model for any smooth Fano variety is declared.…”

Section: Introduction and Setupmentioning

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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Projecting Fanos in the mirror

Cited by 3 publications

References 42 publications

Laurent polynomials in Mirror Symmetry: why and how?

Laurent polynomials in Mirror Symmetry: why and how?

Laurent polynomials in Mirror Symmetry: why and how?

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

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