On the Chow motive of an abelian scheme with non-trivial endomorphisms

Abstract

Abstract Let X be an abelian scheme over a base variety S, with endomorphism algebra D. We prove that the relative Chow motive R ( X / S ) $R(X/S)$ has a canonical decomposition as a direct sum of motives R ( ξ ) $R^{(\xi )}$ where ξ runs over an explicitly determined finite set of irreducible representations of the group D o p , * $D^{\textup {op},*}$ , such that R ( ξ ) $R^{(\xi )}$ , seen as a functor from Chow motives to D o p , * $D^{\textup {op},*}$ -representations, is ξ-isotypic. Our decomposition refines the motivic decomposition of Deninger and Murre, as well as Beauville's decomposition of the Chow group. The second main result is that we construct a canonical generalized motivic Lefschetz decomposition. Inspired by work of Looijenga and Lunts in [Invent. Math. 129 (1997), 361–412], the role of 𝔰𝔩 2 $\mathfrak {sl}_2$ in the classical theory is here replaced by a larger Lie algebra, generated by all Lefschetz and Lambda operators associated to non-degenerate line bundles. We construct an action of this larger Lie algebra on R ( X / S ) $R(X/S)$ and deduce from this a generalized Lefschetz decomposition. We also give a precise structure result for the Lefschetz components that arise. As an application of our techniques, we provide a positive answer to a question of Claire Voisin concerning the analogue for abelian varieties of a conjecture of Beauville.