Hodge–Tate Conditions for Landau–Ginzburg Models

Shamoto, Yota

doi:10.4171/prims/54-3-2

Cited by 8 publications

(5 citation statements)

References 37 publications

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“…Using this result, in [CP18] it is proven that Equality (3) holds for Fano threefolds. The second conjecture from [KKP17] for Fano threefolds (Equality (5)) is given by the following result, which is an extension of the main result of Shamoto in [Sh17].…”

Section: Connections To the Conjectures Of [Kkp17]mentioning

confidence: 80%

P=W Phenomena

Harder¹,

Katzarkov²,

Przyjalkowski³

2019

Preprint

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In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi-Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi-Yau manifold, taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi-Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [CHM12], and de Cataldo and Migliorini [CM10]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror P=W conjecture provides a possible bridge between the mirror P=W conjecture and the well-known P=W conjecture in nonabelian Hodge theory.

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Section: Connections To the Conjectures Of [Kkp17]mentioning

confidence: 80%

P=W Phenomena

Harder¹,

Katzarkov²,

Przyjalkowski³

2019

Preprint

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“…Corollary 3.8. If H k (Y ) admits a Deligne's mixed Hodge structure of the Hodge-Tate type for all k ≥ 0, then the conjectural relative HMS (3.7) implies the mirror P=W conjecture [42] (Conjecture 2. 16).…”

Section: Proposition 36 By Taking Hochschild Homology Hh a (−) On The...mentioning

confidence: 99%

“…Remark 13. In [42], the rational structure is constructed to promote the morphism ν to the morphism of Q-mixed Hodge complexes 6 . Since the shifted cone of the morphism of Q-mixed Hodge complexes is again Q-mixed Hodge complex, we have the limiting Q-mixed Hodge structure on H a (Y, Y sm ).…”

Section: Landau-ginzburg Hodge Numbersmentioning

confidence: 99%

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Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

Lee¹

2022

Preprint

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The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [29], predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we offer to reexamine this conjecture through the lens of mirror symmetry for a Fano pair (X, D) where X is a smooth Fano variety, and D is a simple normal crossing divisor. A mirror object is a multi-potential version of the Landau-Ginzburg (LG) model, called hybrid LG model, which encodes the mirrors of all irreducible components of D. We explain how the mirror P=W conjecture is induced by the relative version of homological mirror symmetry for such pairs. The key idea is to find a combinatorial description of the perverse filtration. The requisite combinatorics naturally appears in the course of studying Hodge numbers of hybrid LG models. We discuss this topic in the second half of the paper and use it to extend the work of to the hybrid LG setting. In application, we discover an interesting upper bound of the multiplicativity of the perverse filtration, which is an independent interest. Contents 1. Introduction 1.1. Motivation: the mirror P=W conjecture 1.2. Extended Fano/LG correspondence 1.3. Landau-Ginzburg Hodge numbers 1.4. Acknowledgement 2. The mirror P=W conjecture 2.1. The weight filtration 2.2. The perverse filtration 2.3. The mirror P=W conjecture 3. Fano/LG correspondence 3.1. B-side categories 3.2. A-side categories 3.3. From HMS to mirror P=W 4. Extended Fano/LG Correspondence 4.1. The case of two components 4.2. The general case 5. Landau-Ginzburg Hodge numbers 5.1. The smooth case 5.2. The case of two components 5.3. The general case 6. Appendix 6.1. Hodge theory of simplicial varieties 6.2. The combinatorial perverse filtration References

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“…Notice also that the paper [KKP17] gives three definitions of so-called Landau-Ginzburg Hodge numbers associated to a family f : X → P 1 , one of them being dim H p (Ω q f ). Conjecturally these three definitions coincide, but this seems to require some more assumptions (see [Sha18] and [Sab18b] for some partial results). Although these Hodge numbers have only integer indices, they are closely related to the dimensions of the graded parts of the filtration from [ESY17].…”

Section: Introductionmentioning

confidence: 99%

Irregular Hodge Filtration of Some Confluent Hypergeometric Systems

Domínguez¹,

Sevenheck²

2019

J. Inst. Math. Jussieu

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We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

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Hodge–Tate Conditions for Landau–Ginzburg Models

Cited by 8 publications

References 37 publications

P=W Phenomena

P=W Phenomena

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

Irregular Hodge Filtration of Some Confluent Hypergeometric Systems

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