For a family of smooth projective curves, the relative Jacobian is a family of principally polarized abelian varieties. Such abelianization gives a powerful tool to study the moduli space of smooth curves. When curves degenerate into singular curves, the relative Jacobian is no longer compact, and one can consider compactified Jacobians. By Arinkin, Fourier-Mukai transformation for abelian schemes extends to compactified Jacobian for a family of integral local planar curves.
We discuss two directions on the structure of cohomology/Chow group of compactified Jacobians related to the Fourier transformation. First, I will explain the Pixton’s formula computing the class of Abel-Jacobi section over the moduli space of nodal curves. This formula is a natural extension of the class of the unit section for abelian schemes. Pixton’s formula leads to compute intersection numbers of compactified Jacobians. Second, we consider the perverse filtration on the rational cohomology of the relative compactified Jacobian and ask when the filtration has a multiplicative splitting. If the relative compactified Jacobian arises from the moduli space of stable one-dimensional shaves on K3 surfaces, we get a Fourier-stable multiplicative splitting of the perverse filtration. We give a connection to Beauville-Voisin’s conjecture on the rational Chow group of hyperkahler varieties.