Dualizable abelian fibrations
Davesh Maulik, Junliang Shen, Qizheng Yin
TL;DR
The work develops a framework of dualizable abelian fibrations to generalize the Fourier–Mukai duality and the Beauville motivic/decomposition theory from abelian schemes to fibrations with singular fibers. It shows that, under Axioms A–D (and refinements like C+), one obtains motivic decompositions and multiplicative perverse truncations, with a dual fibration $f^\vee: M^\vee\to B$ whose convolution algebra matches the cup-product on the original side. The main result provides an explicit isomorphism of algebra objects between the cohomological summands on $M$ and those on $M^\vee$, mediated by the kernel of multiplication, yielding a robust duality that underpins applications to Hitchin systems, compactified Jacobians, and universal Jacobians. These results culminate in multiplicativity of the perverse filtration, the construction of intrinsic cohomology rings, and new evidence for $P=C$ and related conjectures in non-abelian Hodge theory, with Lagrangian fibrations showing particularly close alignment to the abelian-scheme case. The framework thereby unifies and extends key structural phenomena across a broad class of geometric fibrations with singular fibers, providing tools for future investigations into motivic and cohomological structures in algebraic geometry.
Abstract
In his proof of the fundamental lemma of the Langlands program, Ngô initiated the study of the decomposition theorem for abelian fibrations. When an abelian fibration admits a duality structure, the decomposition theorem and the perverse filtration on cohomology exhibit rich structures. The purpose of these notes is to describe a framework for dualizable abelian fibrations and to discuss some recent progress and applications.