Mixed Hodge Structures

Elzein, Fouad; Tráng, Lê Dũng

doi:10.48550/arxiv.1302.5811

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“…Then we get 14) where γ i is the collection of canonical Gysin maps we introduced in Section 2.1. Similarly, with putting the same signs, the corresponding A-side diagram will be 15) where β i 's are the collection of the connecting homomorphisms with the chosen sign. The gluing property ensures that each Y I,sm is topologically the union ∪ i / ∈I (Y I ∩ Y i ).…”

Section: 22mentioning

confidence: 99%

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

Lee¹

2022

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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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Section: 22mentioning

confidence: 99%

“…We recollect some amounts of the Deligne's Hodge theory [12][13] that are used in the main part of the paper. We mainly follows the logistics in [15]. We only deal with the coefficient Q and C for simplicity.…”

Section: The Perverse Filtration Let (Y Hmentioning

confidence: 99%

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

Lee¹

2022

Preprint

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Mixed Hodge Structures

Cited by 1 publication

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

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

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