Perverse filtrations and Fourier transforms
Davesh Maulik, Junliang Shen, Qizheng Yin
TL;DR
The paper unifies the understanding of perverse filtrations under Fourier--Mukai dualities for dualizable abelian fibrations, proving that under Fourier vanishing the perverse filtration is multiplicative and satisfies Perverse $\supset$ Chern. It develops a motivic enhancement via relative Chow motives, enabling a motivic decomposition of the fibration and linking Tauberian-type tautological classes to perverse gradings. The authors apply this framework to compactified Jacobians of integral locally planar curves, proving a motivic decomposition, a new proof of $P=W$ for $GL_r$, and partial $P=C$ results for local $\mathbb{P}^2$, while extending the theory to twisted (gerbed) families. The work leverages Arinkin’s autoduality, Ngô’s support, Adams operations, and Corti–Hanamura motives to provide a cohesive, transportable toolkit across Hitchin-like systems, local curve families, and 1-dimensional sheaf moduli, with implications for motivic decompositions, refined BPS invariants, and Hodge-theoretic structures. Overall, the paper offers a broad, principled mechanism to derive deep structural properties of perverse filtrations in geometric representation theory and enumerative geometry, with concrete outcomes for several central conjectures and models.
Abstract
We study the interaction between Fourier-Mukai transforms and perverse filtrations for a certain class of dualizable abelian fibrations. Multiplicativity of the perverse filtration and the "Perverse $\supset$ Chern" phenomenon for these abelian fibrations are immediate consequences of our theory. We also show that our class of fibrations include families of compactified Jacobians of integral locally planar curves. Applications include the following: (a) we prove the motivic decomposition conjecture for this class (including compactified Jacobian fibrations), which generalizes Deninger-Murre's theorem for abelian schemes; (b) we provide a new proof of the P=W conjecture for $\mathrm{GL}_r$; (c) we prove half of the P=C conjecture concerning refined BPS invariants for the local $\mathbb{P}^2$; (d) we show that the perverse filtration for the compactified Jacobian associated with an integral locally planar curve is multiplicative, which generalizes a result of Oblomkov-Yun for homogeneous singularities. Our techniques combine Arinkin's autoduality for coherent categories, Ngô's support theorem for the decomposition theorem, Adams operations in operational K-theory, and Corti-Hanamura's theory of relative Chow motives.