Cohomology of compactified Jacobians for locally planar integral curves

TL;DR

This paper surveys recent advances in understanding the cohomology of compactified Jacobians for locally planar integral curves, highlighting key theorems, filtrations, and connections to other geometric structures.

Contribution

It provides a comprehensive overview of recent developments, including the Ngô support theorem, perverse filtrations, and Fourier-Mukai transforms in this context.

Findings

Ngô support theorem applied to compactified Jacobians

Perverse filtration elucidates cohomological structure

Connections established with Hilbert schemes and Fourier-Mukai transform

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 Ng\^o support theorem, the perverse filtration, connections to the Hilbert schemes, and cohomological structures induced by the Arinkin-Fourier-Mukai transform.