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On the Chow motive of an abelian scheme. (English) Zbl 0823.14032

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 189-205 (1994).
For the Chow motive \(h(A)\) of an abelian variety \(A\), A. M. Shermenev [cf. Funct. Anal. Appl. 8, 47–53 (1974); translation from Funkts. Anal. Prilozh. 8, No. 1, 55–61 (1974; Zbl 0294.14003)] has constructed a (non-canonical) decomposition \(h(A) \cong \bigoplus_{i=0}^{2\dim (A)} h^ i (A)\) such that the realisation \(H^* (h^ i (A))\) is \(H^ i (A)\) for any Weil cohomology \(H^*\). This result was generalized to abelian schemes by C. Deninger and J. Murre [J. Reine Angew. Math. 422, 201–219 (1991; Zbl 0745.14003)], who also made a canonical functorial choice of the decomposition. – The paper under review contains an explicit closed formula for the projectors \(h(A)\to h^ i (A)\), using the Pontryagin \(*\)-product.
For the entire collection see [Zbl 0788.00053].

MSC:

14K05 Algebraic theory of abelian varieties
14A20 Generalizations (algebraic spaces, stacks)
14C25 Algebraic cycles
14C05 Parametrization (Chow and Hilbert schemes)
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