A Lefschetz decomposition for Chow motives of abelian schemes
Let \(A\) be an abelian variety of dimension \(g\) over a field \(k\) and let \(h(A) = \bigoplus^{2g}_{i = 0} h^ i(A)\) be the Chow motive of \(A\) constructed by \textit{C. Deninger} and \textit{J.-P. Murre} [J. Reine Angew. Math. 422, 201-219 (1991; Zbl 0745.14003)]. The author proves the motivic analog of the hard Lefschetz theorem: \(L^{g-i} : h^ i(A) @>\sim>> h^{2g-i} (A)(g-i)\) and of the Lefschetz decomposition theorem \[ h^ i(A) = \bigoplus_ k L^ kP^{i-2k} (A). \] Here \(L\) is the Lefschetz operator associated to a symmetric polarization of \(A\), and \(P^ i(A)\) is a motivic analog of the primitive part in cohomology. The decomposition isomorphism was independently proven by \textit{A. J. Scholl} [in Motives, Proc. Res. Conf. Motives, Seattle 1991, Proc. Symp. Pure Math. 55, Part 1, 163-187 (1994)]. All these results can be also applied to abelian schemes. As an application the author reproves an earlier result of \textit{C. Soulé} [Math. Ann. 268, 317-345 (1984; Zbl 0573.14001)] on isomorphism of the Chow groups \(CH^ p (A,\mathbb{Q}) \to CH^{g-p} (A,\mathbb{Q})\), \(2p \leq g\), for any abelian variety \(A\) over a finite field \(k\).
scientific article; zbMATH DE number 549682