Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911830872424448 |
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| author | Lee, Sukjoo |
| author_facet | Lee, Sukjoo |
| contents | The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski, predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit 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. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of $D$. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09217 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence Lee, Sukjoo Algebraic Geometry Symplectic Geometry 11F23 (Primary) 14J33, 53D37 (Secondary) The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski, predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit 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. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of $D$. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it. |
| title | Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence |
| topic | Algebraic Geometry Symplectic Geometry 11F23 (Primary) 14J33, 53D37 (Secondary) |
| url | https://arxiv.org/abs/2206.09217 |