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arXiv · 2603.07193

Lefschetz filtration and Perverse filtration on the compactified Jacobian

Yao Yuan
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Abstract

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.

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Yao Yuan. 2026-03-07. Lefschetz filtration and Perverse filtration on the compactified Jacobian. https://arxiv.org/abs/2603.07193

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