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
Keywords: Théorie de Hodge, motifs, cycles algébriques, singularités

@book{MSMF_2001_2_87__1_0,
     author = {Barbieri-Viale, Luca and Srinivas, Vasudevan},
     title = {Albanese and {Picard} $1$-motives},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     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/}
}

TY  - BOOK
AU  - Barbieri-Viale, Luca
AU  - Srinivas, Vasudevan
TI  - Albanese and Picard $1$-motives
T3  - Mémoires de la Société Mathématique de France
PY  - 2001
IS  - 87
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2001_2_87__1_0/
DO  - 10.24033/msmf.400
LA  - en
ID  - MSMF_2001_2_87__1_0
ER  -

%0 Book
%A Barbieri-Viale, Luca
%A Srinivas, Vasudevan
%T Albanese and Picard $1$-motives
%S Mémoires de la Société Mathématique de France
%D 2001
%N 87
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2001_2_87__1_0/
%R 10.24033/msmf.400
%G en
%F MSMF_2001_2_87__1_0

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/

References
Cited by

[1] L. Barbieri-Viale – On algebraic $1$ -motives related to Hodge cycles, preprint math.AG/0103179, 2001. | Zbl | MR

[2] L. Barbieri-Viale, C. Pedrini and C. Weibel – Roitman’s theorem for singular complex projective surfaces, Duke Math. J. 84, No. 1, (1996) 155–189. | Zbl | MR

[3] L. Barbieri-Viale, A. Rosenschon and M. Saito – Deligne’s conjecture on $1$ -motives, preprint math.AG/0102150, 2001. | Zbl | MR

[4] L. Barbieri-Viale and V. Srinivas – On the Néron-Severi group of a singular variety, J. Reine Ang. Math. (Crelles Journal) 435 (1993) 65–82. | Zbl | EuDML | MR

[5] L. Barbieri-Viale and V. Srinivas – The Néron-Severi group and the mixed Hodge structure on $H^{2}$ , J. Reine Ang. Math. (Crelles Journal) 450 (1994) 37–42. | Zbl | EuDML | MR

[6] L. Barbieri-Viale and V. Srinivas – Albanese and Picard $1$ -motives, C. R. Acad. Sci. Paris, t. 326, Série I (1998) 1397–1401. | Zbl | MR

[7] J. Biswas – Topics in Algebraic Cycles, PhD Thesis, TIFR and University of Mumbai, 1997.

[8] J. Biswas and V. Srinivas – Roitman’s theorem for singular projective varieties, Compositio Math. 119 (1999) 213–237. | Zbl | MR

[9] J. Biswas and V. Srinivas – A Lefschetz $(1, 1)$ -Theorem for normal projective complex varieties, Duke Math. J. 101 (2000) 427–458. | Zbl | MR

[10] J-F. Boutot – Schéma de Picard local, Springer LNM 632, 1978. | Zbl | MR

[11] S. Bosch, W. Lutkebohmert and M. Raynaud – Néron Models, Springer Ergebnisse der Math. 21 Heidelberg, 1990. | Zbl | MR

[12] J.A. Carlson – The one-motif of an algebraic surface, Compositio Math. 56 (1985) 271–314. | Numdam | Zbl | EuDML | MR

[13] C. Chevalley – Sur la théorie de la variété de Picard, American J. Math. 82 (1960) 435-490. | Zbl | MR

[14] P. Deligne – Equations différentielles à points singuliers réguliers, Springer LNM 163 Heidelberg, 1970. | Zbl | MR

[15] P. Deligne – Théorie de Hodge II, III Publ. Math. IHES 40 (1972) 5–57 and Publ. Math. IHES 44 (1974) 5–78. | EuDML | MR

[16] P. Deligne – A quoi servent les motifs ?, in “Motives” Proc. Symp. Pure Math AMS 55, Part 1, (1994) 143–161. | JFM | MR

[17] P. Deligne – Letter to H. Esnault, November 4, 1994.

[18] P. Deligne and L. Illusie – Rélèvements modulo $p^{2}$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987) 247–270. | Zbl | EuDML | MR

[19] H. Esnault, V. Srinivas and E. Viehweg – The universal regular quotient of the Chow group of points on projective varieties, Inv. Math. 135 (1999) 595–664. | Zbl | MR

[20] H. Esnault and E. Viehweg – Lectures on vanishing theorems, Birkäuser DMV Seminar 20 Basel, 1992. | MR

[21] W. Fulton – Intersection theory, Springer Ergebnisse der Math. 2 Heidelberg, 1984. | Zbl | MR

[22] H. Gillet – On the $K$ -theory of surfaces with multiple curves and a conjecture of Bloch, Duke Math. J. 51, No. 1, (1984) 195–233. | Zbl | MR

[23] A. Grothendieck – Technique de descente et théorèmes d’existence en géométrie algébrique I–VI, “Fondements de la Géométrie Algébrique” Extraits du Séminaire Bourbaki 1957/62. | Numdam | Zbl

[24] A. Grothendieck et al. – SGA2 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (1962), North-Holland, Amsterdam, 1968. | MR

[25] A. Grothendieck et al. – SGA4 - Théorie des topos et cohomologie étale des schémas (1963-64), Springer LNM 269 270 305, Heidelberg, 1972-73.

[26] A. Grothendieck et al. – SGA6 - Théorie des intersections et théorème de Riemann-Roch (1966-67), Springer LNM 589, 1977.

[27] A. Grothendieck et al. – SGA7 - Groupes de monodromie en géométrie algébrique (1967-68), Springer LNM 288 340, 1972-73.

[28] R. Hartshorne – Ample subvarieties of algebraic varieties, Springer LNM 156, 1970. | Zbl | EuDML | MR

[29] R. Hartshorne – On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 45 (1976) 5–99. | Numdam | Zbl | EuDML | MR

[30] M. Levine and C. Weibel – Zero-cycles and complete intersections on singular varieties, J. Reine Ang. Math. (Crelles Journal) 359 (1985) 106–120. | Zbl | EuDML | MR

[31] S. Lichtenbaum – Suslin homology and Deligne $1$ -motives, in “Algebraic $K$ -theory and algebraic topology” NATO ASI Series, 1993, 189–197. | Zbl | MR

[32] B. Mazur and W. Messing – Universal extensions and one dimensional crystalline cohomology, Springer LNM 370, 1974. | Zbl | MR

[33] J. S. Milne – Étale cohomology, Princeton University Press, Princeton, New Jersey, 1980. | Zbl | MR

[34] D. Mumford – Biextensions of formal groups, International Colloquium on Algebraic Geometry (Bombay, 1968) Oxford University Press, 1969, 307–322. | MR

[35] D. Mumford – The topology of normal singularities on an algebraic surface and a criterion for simplicity, Publ. Math. IHES 9 (1961) 5-22. | Numdam | Zbl | EuDML | MR

[36] J. Murre – On contravariant functors from the category of preschemes over a field into the category of abelian groups (with application to the Picard functor), Publ. Math. IHES 23 (1964) 581–619. | Numdam | Zbl | EuDML | MR

[37] F. Oort – A construction of generalized Jacobian varieties by group extensions, Math. Annalen 147 (1962) 277–286. | Zbl | EuDML | MR

[38] F. Oort – Commutative group schemes, Springer LNM 15, 1966. | Zbl | MR

[39] N. Ramachandran – Albanese and Picard $1$ -motives, Ph D Thesis (preliminary version, 1995).

[40] N. Ramachandran – Albanese and Picard one-motives of schemes, preprint, math.AG/9804042, 1998. | MR

[41] N. Ramachandran – A conjecture of Deligne on one-motives, preprint, math.AG/9806117, 1998.

[42] M. Raynaud – Spécialisation du foncteur de Picard, Publ. Math. IHES 38 (1970) 27–76. | Numdam | Zbl | EuDML | MR

[43] M. Raynaud – $1$ -motifs et monodromie géométrique, Exposé VII, Astérisque 223 (1994) 295–319. | Numdam | Zbl | MR

[44] M. Saito – Mixed Hodge modules, Publ. RIMS Kyoto Univ. 26 (1990) 221–333. | Zbl | MR

[45] M. Saito – Introduction to mixed Hodge modules, Astérisque, 179-180 (1989) 145-162. | Numdam | Zbl | MR

[46] S. Saito – Torsion zero-cycles and étale homology of singular schemes, Duke Math. J. 64, No. 1, (1991) 71–83. | Zbl | MR

[47] J.-P. Serre – Morphismes universels et variétés d’Albanese, dans “Variétés de Picard” ENS Séminaire C. Chevalley 3e année: 1958/59. | Numdam | EuDML

[48] J.-P. Serre – Morphismes universels et différentielles de troisième espèce, dans “Variétés de Picard” ENS Séminaire C. Chevalley 3e année: 1958/59. | Numdam | Zbl | EuDML

[49] J.-P. Serre – Groupes algébriques et corps de classes, Hermann, Paris, 1959. | Zbl | MR

[50] E. H. Spanier – Algebraic Topology, McGraw-Hill, New York, 1966. | Zbl | MR

[51] J.H.M. Steenbrink – Cup products and mixed Hodge structures, preprint, math.AG/9902134, 1999. | Zbl

[52] A. Suslin and V. Voevodsky – Singular homology of abstract algebraic varieties, Inv. Math. 123 (1996) 61–94. | Zbl | EuDML | MR

[53] Bin Zhang – Sur les jacobiennes des courbes à singularités ordinaires, Manuscripta Math. 92 (1997) 1-12. | EuDML | MR

Cited by Sources: