Albanese and Picard 1-motives

Details Albanese and Picard 1-motives Albanese and Picard 1-motives

1999-06-24

arxiv.org/abs/math/9906165v1

Memoire de la Soc. Math. de France 87, vi+104 pages, 2001

Mathematics

Algebraic Geometry

81 pages (428 K dvi file)

Scientific paper

We describe algebraically defined cohomological and homological Albanese and Picard 1-motives (or mixed motives) of any algebraic variety in characteristic zero, generalizing the classical Albanese and Picard varieties. We compute Hodge, l-adic and De Rham realizations proving Deligne's conjecture for the concerned mixed Hodge structures. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1-motives for normal schemes. For proper schemes, we obtain an Abel-Jacobi map from the (Levine-Weibel) Chow group of zero cycles to our cohomological Albanese 1-motive which is the universal regular homomorphism to semi-abelian varieties. By using this universal property we get 'motivic' Gysin maps for projective local complete intersection morphisms. This paper is an extended version of our preliminary Comptes Rendus Note, Academie des Sciences, Paris, Vol. 326, 1998.

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