Perverse filtration for generalized Kummer varieties of fibered surfaces

TL;DR

The paper proves that for a proper surjective morphism $f:A\to C$ from a connected smooth quasi-projective commutative group surface $A$ of dimension $2$ to a curve $C$, the perverse filtration on the cohomology associated with the induced map $h':A^{[[n]]}\to C^{((n))}$, where $A^{[[n]]}$ is a generalized Kummer variety, is multiplicative and even admits a strongly multiplicative splitting. This is established by a threefold approach: classifying all fibered group surfaces relevant to $(\dagger)$, deriving a cup-product formula for generalized Kummer varieties, and explicitly describing the perverse filtration for $A^{[[n]]}\to C^{((n))}$ (including behavior under partitions). The results extend multiplicativity properties known for Hitchin-type and Hilbert-scheme fibrations to a non-tautological, non-compact setting, suggesting broader multiplicativity phenomena in non-tautological geometric constructions and offering new insights related to the P=W paradigm. The work leverages the interplay between perverse filtrations, decomposition theorems, and base-change for symmetric products and Hilbert schemes to achieve a coherent multiplicativity framework for generalized Kummer geometries over fibered surfaces.

Abstract

Let $A\to C$ be a proper surjective morphism from a smooth connected quasi-projective commutative group scheme of dimension 2 to a smooth curve. The construction of generalized Kummer varieties gives a proper morphism $A^{[[n]]}\to C^{((n))}$. We show that the perverse filtration associated with this morphism is multiplicative.

Theorems & Definitions (38)

• Conjecture 1.1
• Theorem 1.2: Theorem \ref{['main']}
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5