Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

One-motives and a conjecture of Deligne


Author: Niranjan Ramachandran
Journal: J. Algebraic Geom. 13 (2004), 29-80
DOI: https://doi.org/10.1090/S1056-3911-03-00370-9
Published electronically: September 22, 2003
MathSciNet review: 2008715
Full-text PDF

Abstract | References | Additional Information

Abstract: We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Other applications include some cases of conjectures of Serre, Katz, and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.


References [Enhancements On Off] (What's this?)

  • 1. M. Artin, Théorème de changement de base par un morphisme lisse, et applications, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305, pp. 206-249. MR 50:7132
  • 2. L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne's conjecture on $1$-motives, Ann. of Math. 158 (2003), no. 2, 557-597.
  • 3. L. Barbieri-Viale and V. Srinivas, Albanese and Picard 1-motives, Mém. Soc. Math. Fr. (N.S.) (2001), no. 87, vi+104.
  • 4. P. Berthelot, Altérations de variétés algébriques (d'après A. J. de Jong), Astérisque (1997), no. 241, Exp. No. 815, 5, 273-311, Séminaire Bourbaki, Vol. 1995/96. MR 98m:14021
  • 5. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990. MR 91i:14034
  • 6. J. A. Carlson, The one-motif of an algebraic surface, Compositio Math. 56 (1985), no. 3, 271-314. MR 87g:14005
  • 7. -, The geometry of the extension class of a mixed Hodge structure, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 46, Part 2, Bowdoin, 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 199-222. MR 89b:32036
  • 8. E. B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107-209. MR 43:5529
  • 9. A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51-93. MR 98e:14011
  • 10. P. Deligne, Letter to the author. 15 April 1998.
  • 11. -, Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 425-430. MR 56:356
  • 12. -, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-57. MR 58:16653a
  • 13. -, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273-307. MR 49:5013
  • 14. -, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5-77. MR 58:16653b
  • 15. -, Poids dans la cohomologie des variétés algébriques, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, Canad. Math. Congress, Montréal, 1975, pp. 79-85. MR 55:5634
  • 16. -, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137-252. MR 83c:14017
  • 17. -, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\mathbf{Q}$ (Berkeley, CA, 1987), Springer, New York, 1989, pp. 79-297. MR 90m:14016
  • 18. -, À quoi servent les motifs?, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 143-161. MR 95c:14013
  • 19. P. Deligne, J. S. Milne, A. Ogus, and K-y. Shih, Hodge cycles, motives, and Shimura varieties, Springer-Verlag, Berlin, 1982, Lecture Notes in Mathematics, Vol. 900. MR 84m:14046
  • 20. M. Demazure and P. Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland, Amsterdam, 1980. MR 82e:14001
  • 21. F. El Zein, Introduction à la théorie de Hodge mixte, Hermann, Paris, 1991. MR 93c:14013
  • 22. H. Esnault and E. Viehweg, Deligne-Beilinson cohomology, Be{\u{\i}}\kern.15emlinson's conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 43-91. MR 89k:14008
  • 23. H. Gillet, Universal cycle classes, Compositio Math. 49 (1983), no. 1, 3-49. MR 84j:14016
  • 24. M. Green, J. Murre, and C. Voisin, Algebraic cycles and Hodge theory, Springer-Verlag, Berlin, 1994, Lecture Notes in Mathematics, Vol. 1594. MR 96c:14001
  • 25. A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 306-358. MR 42:4558
  • 26. -, Le groupe de Brauer. III. Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88-188. MR 39:5586c
  • 27. F. Guillén, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Springer-Verlag, Berlin, 1988, Lecture Notes in Mathematics, Vol. 1335. MR 90a:14024
  • 28. R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 5-99. MR 55:5633
  • 29. L. Illusie, Report on crystalline cohomology, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 459-478. MR 52:13845
  • 30. U. Jannsen, Deligne homology, Hodge- $\mathcal D$-conjecture, and motives, Beilinson's conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 305-372. MR 89h:14016
  • 31. -, Mixed motives and algebraic ${K}$-theory, Springer-Verlag, Berlin, 1990, Lecture Notes in Mathematics, Vol. 1400. MR 91g:14008
  • 32. -, Motivic sheaves and filtrations on Chow groups, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 245-302. MR 95c:14006
  • 33. -, Mixed motives, motivic cohomology, and Ext-groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 667-679. MR 97j:14022
  • 34. N. M. Katz, Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1-118. MR 49:2728
  • 35. -, Wild ramification and some problems of ``independence of $l$'', Amer. J. Math. 105 (1983), no. 1, 201-227. MR 85f:14021
  • 36. -, Review of $l$-adic cohomology, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 21-30. MR 95j:14021
  • 37. B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 370. MR 51:10350
  • 38. W. Messing, The universal extension of an abelian variety by a vector group, Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), Academic Press, London, 1973, pp. 359-372. MR 49:2760
  • 39. J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980. MR 81j:14002
  • 40. J. S. Milne and N. Ramachandran, Integral motives and special values of zeta functions, (2002), Preprint. Available at http://xxx.lanl.gov/abs/math.NT/0204065.
  • 41. F. Oort, Commutative group schemes, Springer-Verlag, Berlin, 1966. MR 35:4229
  • 42. N. Ramachandran, From Jacobians to one-motives: exposition of a conjecture of Deligne, The arithmetic and geometry of algebraic cycles (CRM Proc. Lecture Notes, vol. 24, Banff, AB, 1998), Amer. Math. Soc., Providence, RI, 2000, pp. 215-234. MR 2001k:14022
  • 43. -, Duality of Albanese and Picard 1-motives, $K$-Theory 22 (2001), no. 3, 271-301. MR 2002e:14013
  • 44. B. Saint-Donat, Techniques de descente cohomologique, Théorie des topos et cohomologie étale des schémas. Tome 2, Springer-Verlag, Berlin, 1972 (SGA 4), Lecture Notes in Mathematics, Vol. 270, pp. 83-162. MR 50:7131
  • 45. J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955-1956), 1-42. MR 18:511a
  • 46. -, Abelian $l$-adic representations and elliptic curves, second ed., Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 91b:11071
  • 47. -, Propriétés conjecturales des groupes de Galois motiviques et des représentations $l$-adiques, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 377-400. MR 95m:11059
  • 48. -, Morphismes universels et différentiels de troisième espèce, Exposés de Séminaires (1950-1999), Documents Mathématiques 1, Soc. Math. France, Paris, 2001, pp. 161-168.
  • 49. J. Tate, Conjectures on algebraic cycles in $l$-adic cohomology, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 71-83. MR 95a:14010
  • 50. N. Tsuzuki, Cohomological descent of rigid cohomology for proper coverings, Invent. Math. 151 (2003), 101-133.
  • 51. V. Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188-238.
  • 52. A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948. MR 10:621d


Additional Information

Niranjan Ramachandran
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: atma@math.umd.edu

DOI: https://doi.org/10.1090/S1056-3911-03-00370-9
Received by editor(s): May 7, 2001
Received by editor(s) in revised form: September 20, 2002
Published electronically: September 22, 2003
Additional Notes: Funded in part by grants from Hewlett-Packard, Graduate Research Board (UMD), and MPIM (Bonn)
Dedicated: Dedicated to my parents

American Mathematical Society