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Lefschetz filtration and Perverse filtration on the compactified Jacobian

Yao Yuan

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.

Lefschetz filtration and Perverse filtration on the compactified Jacobian

Abstract

Let be a complex integral curve with plannar singularities. Let be the compactified Jacobian of . There are two filtrations on the cohomology group . One is obtained by the nilpotent morphism defined by cupping a certain ample divisor on , which we call the Lefschetz filtration. To obtain the other filtration, we put into a family of curves so that can be embedded into a family , and we let be smooth. Then decomposes into a direct sum of its (shifted) perverse cohomologies. Restricting this decomposition to fibers, we get a filtration on called the perverse filtration. We show in this paper that these two filtrations are opposite to each other as conjectured by Maulik-Yun.
Paper Structure (18 sections, 19 theorems, 137 equations)

This paper contains 18 sections, 19 theorems, 137 equations.

Table of Contents

  1. Introduction.
  2. The Lefschetz filtration.
  3. The perverse filtrations.
  4. The main result.
  5. The plan of the paper.
  6. Notations, Conventions and Prelimilaries.
  7. Acknowledgements.
  8. Some properties on $J$ and $C^{[n]}$
  9. A decomposition on $H_*(J)$.
  10. An eigenvalue decomposition on $\mathcal{H}_n$.
  11. Pull-backs of the Poincaré sheaves to the flag Hilbert scheme.
  12. Proof of the main result.
  13. The Fourier transform.
  14. The main result.
  15. A review on some homology theory.
  16. ...and 3 more sections

Key Result

Theorem 1.3

$(e^{\vee}, [e^{\vee},f^{\vee}],f^{\vee})$ forms a $sl_2$-triple on $H^*(J)$, whose eigenvalue decomposition coincides with the decomposition (Ddemp)on $H^*(J)$. In particular, the Lefschetz filtration induced by cupping $\Theta$ on $H^*(J)$ is $W_{k}(H^*(J))=\bigoplus_{j\geq 2g-k}D_jH^*(J)$ and it

Theorems & Definitions (43)

  • Remark 1.1
  • Conjecture 1.2: Conjecture 2.17 in MY
  • Theorem 1.3: Corollary \ref{['maincoro']}
  • Remark 1.4
  • Theorem 2.1: Theorem 1.2 + Lemma 3.2 in Ren
  • Theorem 2.2: Theorem 1.3 in Ren
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 33 more