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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.07193 |
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| _version_ | 1866912951535927296 |
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| author | Yuan, Yao |
| author_facet | Yuan, Yao |
| contents | Let $C$ be a complex integral curve with plannar singularities. Let $J$ be the compactified Jacobian of $C$. There are two filtrations on the cohomology group $H^*(J)$. One is obtained by the nilpotent morphism defined by cupping a certain ample divisor on $J$, which we call the Lefschetz filtration. To obtain the other filtration, we put $C$ into a family of curves $\mathcal{C}\rightarrow B$ so that $J$ can be embedded into a family $f:\mathcal{J}\rightarrow B$, and we let $B, \mathcal{C},\mathcal{J}$ be smooth. Then $Rf_*(\mathbb{Q}_{\mathcal{J}})$ decomposes into a direct sum of its (shifted) perverse cohomologies. Restricting this decomposition to fibers, we get a filtration on $H^*(J)$ called the perverse filtration. We show in this paper that these two filtrations are opposite to each other as conjectured by Maulik-Yun. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lefschetz filtration and Perverse filtration on the compactified Jacobian Yuan, Yao Algebraic Geometry 14D22, 14J26 Let $C$ be a complex integral curve with plannar singularities. Let $J$ be the compactified Jacobian of $C$. There are two filtrations on the cohomology group $H^*(J)$. One is obtained by the nilpotent morphism defined by cupping a certain ample divisor on $J$, which we call the Lefschetz filtration. To obtain the other filtration, we put $C$ into a family of curves $\mathcal{C}\rightarrow B$ so that $J$ can be embedded into a family $f:\mathcal{J}\rightarrow B$, and we let $B, \mathcal{C},\mathcal{J}$ be smooth. Then $Rf_*(\mathbb{Q}_{\mathcal{J}})$ decomposes into a direct sum of its (shifted) perverse cohomologies. Restricting this decomposition to fibers, we get a filtration on $H^*(J)$ called the perverse filtration. We show in this paper that these two filtrations are opposite to each other as conjectured by Maulik-Yun. |
| title | Lefschetz filtration and Perverse filtration on the compactified Jacobian |
| topic | Algebraic Geometry 14D22, 14J26 |
| url | https://arxiv.org/abs/2603.07193 |