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Cohomology of compactified Jacobians for locally planar integral curves

Junliang Shen

TL;DR

This survey connects the cohomology of compactified Jacobians for locally planar integral curves with deep structures from perverse filtrations, derived categories, and link theory. It explains Ngô’s support theorem as a foundational tool that renders the cohomology of singular fibers determined by the smooth locus, and it develops a canonical perverse filtration on H^*(𝑱̄_C, Q) whose intrinsic nature is established via full-support results. The work then links these filtrations to Hilbert schemes through a precise perverse-graded correspondence and shows how Arinkin–Fourier–Mukai transforms provide a powerful mechanism to realize and study multiplicativity of the perverse truncations. Collectively, the article highlights the rich interplay between algebraic geometry, representation theory, and low-dimensional topology in understanding compactified Jacobians and their broader connections to Hitchin systems and enumerative invariants.

Abstract

This article surveys some recent developments on the cohomology of the compactified Jacobian associated with a locally planar integral curve. Topics discussed here include the Ngô support theorem, the perverse filtration, connections to the Hilbert schemes, and cohomological structures induced by the Arinkin-Fourier-Mukai transform.

Cohomology of compactified Jacobians for locally planar integral curves

TL;DR

This survey connects the cohomology of compactified Jacobians for locally planar integral curves with deep structures from perverse filtrations, derived categories, and link theory. It explains Ngô’s support theorem as a foundational tool that renders the cohomology of singular fibers determined by the smooth locus, and it develops a canonical perverse filtration on H^*(𝑱̄_C, Q) whose intrinsic nature is established via full-support results. The work then links these filtrations to Hilbert schemes through a precise perverse-graded correspondence and shows how Arinkin–Fourier–Mukai transforms provide a powerful mechanism to realize and study multiplicativity of the perverse truncations. Collectively, the article highlights the rich interplay between algebraic geometry, representation theory, and low-dimensional topology in understanding compactified Jacobians and their broader connections to Hitchin systems and enumerative invariants.

Abstract

This article surveys some recent developments on the cohomology of the compactified Jacobian associated with a locally planar integral curve. Topics discussed here include the Ngô support theorem, the perverse filtration, connections to the Hilbert schemes, and cohomological structures induced by the Arinkin-Fourier-Mukai transform.

Paper Structure

This paper contains 25 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Ngô support theorem
  3. Compactfied Jacobian fibrations
  4. Decomposition Theorem
  5. Ngô's support theorem
  6. Further remarks
  7. Perverse filtrations
  8. Definition
  9. Independence of the family
  10. A local P=W conjecture
  11. Story 1: the P=W conjecture for the Hitchin fibration
  12. Story 2: link invariants associated with planar singularities
  13. Shende's proposal
  14. Predictions and known results
  15. Further remarks
  16. ...and 10 more sections

Key Result

Theorem 1.1

We have

Theorems & Definitions (19)

  • Theorem 1.1
  • Proposition 1.2: Goresky--MacPherson inequality
  • Theorem 1.3: Ngô support theorem Ngo
  • proof : Proof sketch of Theorem \ref{['thm1.1']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Conjecture 2.4: Oblomkov--Rasmussen--Shende ORS
  • Theorem 2.5: Oblomkov--Yun OY2
  • Theorem 2.6: Maulik--Shen--Yin MSY
  • ...and 9 more