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
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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 [1] 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 [1] fields.
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brs L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne’s conjecture on $1$-motives, Ann. of Math. 158 (2003), no. 2, 557–597.
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be 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.
blr S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990.
ca J. A. Carlson, The one-motif of an algebraic surface, Compositio Math. 56 (1985), no. 3, 271–314.
ca2 ---, 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.
curt E. B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107–209.
dj A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51–93.
dlet P. Deligne, Letter to the author. 15 April 1998.
h1 ---, Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 425–430.
h2 ---, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57.
weil ---, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307.
h ---, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77.
hp ---, 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.
weil2 ---, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252.
pgal ---, 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.
dmot ---, 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.
dmos 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.
dg M. Demazure and P. Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland, Amsterdam, 1980.
ez F. El Zein, Introduction à la théorie de Hodge mixte, Hermann, Paris, 1991.
ev H. Esnault and E. Viehweg, Deligne-Beĭlinson cohomology, Beĭlinson’s conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 43–91.
gi H. Gillet, Universal cycle classes, Compositio Math. 49 (1983), no. 1, 3–49.
cime M. Green, J. Murre, and C. Voisin, Algebraic cycles and Hodge theory, Springer-Verlag, Berlin, 1994, Lecture Notes in Mathematics, Vol. 1594.
grocry A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 306–358.
gr ---, Le groupe de Brauer. III. Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88–188.
guillen 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.
hart R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 5–99.
illucie 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.
jannbson U. Jannsen, Deligne homology, Hodge-$\mathcal D$-conjecture, and motives, Beĭlinson’s conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 305–372.
jamm ---, Mixed motives and algebraic ${K}$-theory, Springer-Verlag, Berlin, 1990, Lecture Notes in Mathematics, Vol. 1400.
jamo ---, 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.
ja ---, 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.
katzdiff N. M. Katz, Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118.
ka3 ---, Wild ramification and some problems of “independence of $l$”, Amer. J. Math. 105 (1983), no. 1, 201–227.
ka ---, 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.
mame B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 370.
mes 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.
mi J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980.
milram 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.
oort F. Oort, Commutative group schemes, Springer-Verlag, Berlin, 1966.
banff 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.
ra ---, Duality of Albanese and Picard 1-motives, $K$-Theory 22 (2001), no. 3, 271–301.
sga4 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.
gaga J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42.
jp ---, Abelian $l$-adic representations and elliptic curves, second ed., Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
se2 ---, 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.
jp2 ---, Morphismes universels et différentiels de troisième espèce, Exposés de Séminaires (1950-1999), Documents Mathématiques 1, Soc. Math. France, Paris, 2001, pp. 161–168.
tate 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.
tsu N. Tsuzuki, Cohomological descent of rigid cohomology for proper coverings, Invent. Math. 151 (2003), 101–133.
vv V. Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238.
aweil A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.
Additional Information
Niranjan Ramachandran
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email:
atma@math.umd.edu
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
