On generalized Beauville decompositions

TL;DR

The paper investigates generalized Beauville decompositions for compactified Jacobian fibrations. It proves a positive cohomological decomposition for Beauville–Mukai systems via a Fourier-stable, multiplicative perverse-splitting realized by a Lefschetz $\mathfrak{sl}_2$-triple within the LLV framework, and explores a motivic lifting program with connections to Beauville–Voisin. It also constructs genus $g$ families with nodal fibers for which a multiplicative splitting does not exist, illustrating the limits of extending Beauville-type decompositions. Additionally, the work develops motivic conjectures and verifies them in the elliptic $K3$ setting, tying in universal double ramification relations and BV-type phenomena to the broader structure of the cohomology and Chow rings of compact hyper-Kähler varieties.

Abstract

Motivated by the Beauville decomposition of an abelian scheme and the "Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. On the one hand, we prove for the Beauville-Mukai system associated with an irreducible curve class on a K3 surface the existence of a Fourier-stable multiplicative splitting of the perverse filtration, which extends the Beauville decomposition for the nonsingular fibers. Our approach is to construct a Lefschetz decomposition associated with a Fourier-conjugate $\mathfrak{sl}_2$-triple, which relies heavily on recent work concerning the interaction between derived equivalences and LLV algebras for hyper-Kähler varieties. Motivic lifting and connections to the Beauville-Voisin conjectures are also discussed. On the other hand, we construct for any $g\geq 2$ a compactified Jacobian fibration of genus g curves such that each curve is integral with at worst simple nodes and the (multiplicative) perverse filtration does not admit a multiplicative splitting. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears.

Theorems & Definitions (57)

• Theorem 1
• Theorem 3
• Theorem 4
• Theorem 5
• Remark 1.1
• Remark 1.3
• Proposition 2.1
• Example 2.2: The symplectic $\mathfrak{sl}_2$-triple
• Proposition 2.3: Verbitsky relations Ver95
• Proposition 2.4: c.f. SY