Albanese and Picard $1$-motives - Volume (2001) no. 87

Albanese and Picard

1

-motives

Barbieri-Viale, Luca; Srinivas, Vasudevan

Mémoires de la Société Mathématique de France, no. 87 (2001) , 110 p.

Abstract
Résumé

Let $X$ be an $n$ -dimensional algebraic variety over a field of characteristic zero. We describe algebraically defined Deligne $1$ -motives ${Alb}^{+} (X)$ , ${Alb}^{-} (X)$ , ${Pic}^{+} (X)$ and ${Pic}^{-} (X)$ which generalize the classical Albanese and Picard varieties of a smooth projective variety. We compute Hodge, $ℓ$ -adic and De Rham realizations proving Deligne’s conjecture for $H^{2 n - 1}$ , $H_{2 n - 1}$ , $H^{1}$ and $H_{1}$ . 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.

Soit $X$ une variété algébrique de dimension $n$ sur un corps de caractéristique $0$ . Nous décrivons les $1$ -motifs de Deligne ${Alb}^{+} (X)$ , ${Alb}^{-} (X)$ , ${Pic}^{+} (X)$ et ${Pic}^{-} (X)$ définis algébriquement, qui généralisent les variétés d’Albanese et de Picard classiques d’une variété projective lisse. Nous calculons les réalisations de Hodge, $ℓ$ -adique et de De Rham, montrant ainsi la conjecture de Deligne pour $H^{2 n - 1}$ , $H_{2 n - 1}$ , $H^{1}$ et $H_{1}$ . Nous étudions la fonctorialité, l’universalité, l’invariance par homotopie et l’invariance par formation de fibrés projectifs. Nous comparons nos $1$ -motifs homologiques et cohomologiques pour les schémas normaux. Pour des schémas propres, nous obtenons une application d’Abel-Jacobi du groupe de (Levine-Weibel) Chow des zéro-cycles vers notre $1$ -motif cohomologique d’Albanese, qui est l’homomorphisme universel régulier vers les variétés semi-abéliennes. En utilisant cette propriété universelle, nous obtenons des applications de Gysin « motiviques » pour les morphismes projectifs localement intersection complète.

MR: 1891270 | Zbl: 1085.14011 | 2 citations in Numdam

DOI: 10.24033/msmf.400

Classification: 14F42, 14C30, 32S35, 19E15
Keywords: Hodge theory, motives, algebraic cycles, singularities

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     author = {Barbieri-Viale, Luca and Srinivas, Vasudevan},
     title = {Albanese and {Picard} $1$-motives},
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     publisher = {Soci\'et\'e math\'ematique de France},
     number = {87},
     year = {2001},
     doi = {10.24033/msmf.400},
     zbl = {1085.14011},
     mrnumber = {1891270},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2001_2_87__1_0/}
}

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Barbieri-Viale, Luca; Srinivas, Vasudevan. Albanese and Picard $1$-motives. Mémoires de la Société Mathématique de France, Serie 2, , no. 87 (2001), 110 p. doi : 10.24033/msmf.400. http://numdam.org/item/MSMF_2001_2_87__1_0/

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