The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves

Abstract: The purpose of this paper is to study the cohomology rings of universal compactified Jacobians. Over the moduli space $\mathcal{M}g$ of nonsingular curves, we show that the cohomology ring of the universal Jacobian is independent of the degree. Over the moduli space $\overline{\mathcal{M}}{g,n}$ of Deligne-Mumford stable marked curves with $n\geq 1$, on the one hand we show that the cohomology ring of a universal fine compactified Jacobian is sensitive to the choice of a nondegenerate stability condition which answers a question of Pandharipande; on the other hand, we prove that the cohomology ring admits a degeneration via the perverse filtration which is independent of the (nondegenerate) stability condition. The latter defines the intrinsic cohomology ring of the universal compactified Jacobian which only relies on $g,n$. Our main tools include the support theorems, the recently developed Fourier theory for dualizable abelian fibrations, and the universal double ramification cycle relations associated with the universal Picard stack.