On generalized Beauville decompositions

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-K\"ahler 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. Our argument relies on the recently established universal double ramification cycle relations. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears.