Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Complex varieties for which the Chow group mod $n$ is not finite

Author: Chad Schoen
Journal: J. Algebraic Geom. 11 (2002), 41-100
Published electronically: November 16, 2001
MathSciNet review: 1865914
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Abstract | References | Additional Information

Abstract: Using the recent work of S. Bloch and H. Esnault, we give examples of smooth projective varieties $W/\mathbb{Q} $ and integers $n\neq 0$ for which $CH^{2}(W_{\bar{\mathbb{Q} }}) /nCH^{2}(W_{\bar{\mathbb{Q} }})$ is not a finite group.

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

  • [Al-Kl] Altman, A. and Kleiman, S., Introduction to Grothendieck Duality Theory, Lect. Notes in Math. 146 (1970), Springer-Verlag, Heidelberg.
  • [Ar-SD] Artin, M. and Swinnerton-Dyer, H. P. F., The Shafarevich-Tate conjecture for Pencils of Elliptic Curves on K3 Surfaces, Invent. Math. 20 (1973), 249-266.
  • [Be] Beauville, A., Les familles stables de courbes elliptiques sur $\mathbb{P} ^{1}$ admettant quatre fibres singulières, C.R. Acad. Sci. Paris 294 (1982), 657-660.
  • [Be-Og] Berthelot, P. and Ogus, A., Notes on crystalline cohomology, Princeton University Press, Princeton, 1978.
  • [Bl1] Bloch, S., Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), 107-127.
  • [Bl2] Bloch, S., Algebraic cycles and values of $L$-functions, J. f. reine u. angew. Math. 350 (1984), 94-108.
  • [Bl3] Bloch, S., Lectures on Algebraic Cycles, Duke University Mathematics Department, Durham, 1980.
  • [Bl-Es] Bloch, S. and Esnault, H., The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann. 304 (1996), 303-314.
  • [Bl-Ka] Bloch, S. and Kato, K., $p$-adic étale cohomology, Publ. Math. IHES 63 (1986), 107-152.
  • [Bl-Og] Bloch, S. and Ogus, A., Gersten's conjecture and the homology of schemes, Ann. Scient. Éc. Norm. Sup. $4^{e}$ série, t. 7 (1974), 181-202.
  • [Ca-Eil] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, Princeton, 1956.
  • [Co] Colliot-Thélène, J.-L., Birational invariants, purity and the Gersten conjecture, Proc. Symp. Pure Math., vol. 58.1, 1995, pp. 1-64.
  • [Co-Sa-So] Colliot-Thélène, J.-L., Sansuc J.-J., and Soulé, C., Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), 763-801.
  • [De] Deligne, P., Formes modulaires et représentations $l$-adiques, Séminaire Bourbaki 355, Lect. Notes in Math. 179, Springer-Verlag, 1971, 139-172.
  • [De2] Deligne, P., Courbes Elliptiques: Formulaire (d'après J. Tate) in Modular Functions of One Variable IV, Lect. Notes in Math. 476 (1973), Springer-Verlag, New York, 1973.
  • [De-Ra] Deligne, P. and Rapoport, M., Les schémas de modules des courbes elliptiques, Modular Functions of One Variable II, p. 143-316, Lecture Notes in Math. 349, Springer-Verlag, New York, 1973.
  • [De-Se] Deligne, P. and Serre, J.-P., Formes modulaires de pois 1, Ann. Sci. Ec. Norm. Sup. 7 (1974), 507-530.
  • [Fu] Fulton, W., Intersection Theory, Springer-Verlag, New York, 1984.
  • [Gro] Grothendieck, A., Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), 119-221.
  • [Ha] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1977.
  • [Il] Illusie, L., Letter to the author, April 29, 1996.
  • [Ja] Jannsen, U., Mixed motives and algebraic $K$-theory, Lect. Notes in Math. 1400 Springer-Verlag, Berlin, Heidelberg, New York, 1990.
  • [Ka] Katz, N., $p$-adic properties of modular schemes and modular forms, Modular Functions of One Variable III, Lecture Notes in Math. 350, Springer-Verlag, New York, 1973, pp. 69-190.
  • [Ka-Me] Katz, N. and Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73-77.
  • [La] Lang, S., Introduction to Modular Forms, Grundlehren der Math. Wissen. 222, Springer-Verlag, New York, 1976.
  • [La2] Lang, S., Elliptic Functions, Addison-Wesley, Reading, MA, 1973.
  • [La3] Lang, S., Sur les séries L d'une variété algébrique, Bull. Soc. Math. France 84, 1956, 385-407.
  • [Le] Lecomte, F., Rigidité des groupes de Chow, Duke Math. J. 53, (1986), 405-426.
  • [Lig] Ligozat, G., Courbes modulaires de genre 1, Bull. Math. France 43 (1975).
  • [Me-Su] Merkurjev, A. and Suslin, A., $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk. USSR Ser. Mat. 46 (1982).
  • [Mi] Milne, J., Étale Cohomology, Princeton University Press, Princeton, 1980.
  • [Mir1] Miranda, R., The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, Dipartimento di Mathematica dell' Università di Pisa, ETS Editrice Pisa, 1989.
  • [Mir2] Miranda, R., Smooth models of elliptic threefolds, The Birational Geometry of Degenerations, Progr. Math. 29, Birkhäuser, Boston, 1983, pp. 85-133.
  • [Mir-P] Miranda, R. and Persson, U., On extremal rational elliptic surfaces, Math. Zeit. 193 (1986), 537-558.
  • [Mu-S] Mumford, D. and Suominen, K., Introduction to the theory of moduli, in ``Algebraic Geometry'', Proc. of the 5-th Nordic Summer School in Mathematics, Oslo, August 5-25, 1970; F. Oort, editor: Walters-Noordhoff Publishing (1972), 172-222.
  • [Sa] Saito, M-H, On the infinitesimal Torelli problem of elliptic surfaces, J. Math. Kyoto Univ. 23 (1983), 441-460.
  • [Sch] Schoen, C., On the computation of the cycle class map for nullhomologous cycles over the algebraic closure of a finite field, Ann. scient. Éc. Norm. Sup. $4^{e}$ série, t. 28 (1995), 1-50.
  • [Sch2] Schoen, C., On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988), 177-199.
  • [Sch3] Schoen, C., Complex multiplication cycles and the conjecture of Beilinson and Bloch, Trans. A.M.S. 339 (1993), 87-115.
  • [Sch4] Schoen, C., On the image of the $l$-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field, J. Amer. Math. Soc. 12 (1999), 795-838.
  • [Sch5] Schoen, C., Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 (1986), 771-794.
  • [Se-Ta] Serre, J.-P. and Tate, J., Good reduction of abelian varieties, Ann. Math. 88 (1968), 492-517.
  • [Sil] Silverman, J., The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
  • [Sil2] Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1994.
  • [So] Soulé, C., Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268 (1984), 317-345.
  • [Su] Suslin, A., On the $K$-theory of algebraically closed fields, Inv. Math. 73 (1983), 241-245.
  • [Tam] Tamme, G., Introduction to Étale Cohomology, Springer-Verlag, New York, 1994.
  • [Wa] Waterhouse, W., Abelian varieties over finite fields, Ann. scient. Éc. Norm. Sup. $4^{e}$ série, t. 2 (1969), 521-560.

Additional Information

Chad Schoen
Affiliation: Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708-0320

Received by editor(s): December 28, 1999
Published electronically: November 16, 2001
Additional Notes: Partial support by NSF and NSA and hospitality of T.I.F.R. and I.H.E.S. gratefully acknowledged

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
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