The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves
Younghan Bae, Davesh Maulik, Junliang Shen, Qizheng Yin
TL;DR
This workstudies the cohomology of universal (fine) compactified Jacobians over the moduli of stable curves, revealing that while the actual cohomology rings depend on the stability condition, the associated graded ring with respect to the perverse filtration is intrinsic and independent of both degree and stability. The authors develop a framework based on dualizable abelian fibrations and a Fourier–Mukai transform, showing that the perverse filtration is multiplicative and that a reduced convolution product governs the intrinsic ring structure, independent of degree. They prove degree-independence in the integral locally planar setting and extend to families of reduced locally planar curves, including the universal curve over $\overline{\mathcal{M}}_{g,n}$, via universal double ramification cycle relations. The results connect Fourier-analytic methods with perverse-filtered degenerations and offer a canonical, stability-insensitive invariant that augments our understanding of Jacobian degenerations and tautological structures in moduli theory.
Abstract
The purpose of this paper is to study the cohomology rings of universal compactified Jacobians. 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.