Duality of Albanese and Picard 1-motives

Details Duality of Albanese and Picard 1-motives Duality of Albanese and Picard 1-motives

1998-04-07

arxiv.org/abs/math/9804042v3

Mathematics

Algebraic Geometry

29 pages, no figures, Latex. to appear in K-Theory Journal

Scientific paper

We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are t he homological 1-motive of Lichtenbaum and the motivic $H^1$ of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum's conjecture that the Albanese and the Picard 1-motives of a (simplicial) scheme are dual. This gives a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.

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