Duality of Albanese and Picard 1-motives

Abstract

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 This gives a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.

arXiv comment29 pages, no figures, Latex. to appear in K-Theory Journal

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• Albanese kernels and Griffiths groups
• The Voevodsky motive of a rank one semiabelian variety
• On the Brauer-Manin obstruction for zero-cycles on curves
• Zero-cycles on normal varieties
• Motivic interpretation of Albanese varieties of smooth varieties
• Local-global principles for 1-motives
• On the Grothendieck ring of varieties in positive characteristic