Duality of Albanese and Picard 1-motives
Let \(k\) be a perfect field. If \(X\) is a smooth projective variety over \(k\), then it is well-known that the associated Albanese variety \(\text{Alb} (X)\) and the Picard variety \(\text{Pic} (X)\) are naturally dual to each other. This result is extended in this paper to arbitrary schemes by the introduction of Albanese and Picard 1-motives for smooth simplicial schemes over \(k\). For smooth proper schemes these are the classical Albanese and Picard varieties, and for a curve they are the 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, which has also been proved independently by Barbieri-Viale and Srinivas. It also contains a purely algebraic proof of Lichtenbaum's conjecture that the Albanese and Picard 1-motives of a simplicial scheme are dual to each other. This provides a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.
scientific article; zbMATH DE number 1624882