Nonvanishing of external products for higher Chow groups
Authors:
Andreas Rosenschon and Morihiko Saito
Journal:
J. Algebraic Geom. 13 (2004), 441-459
DOI:
https://doi.org/10.1090/S1056-3911-03-00361-8
Published electronically:
December 9, 2003
MathSciNet review:
2047676
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References |
Additional Information
Abstract: Consider an external product of a higher cycle and a usual cycle which is algebraically equivalent to zero. Assume there exists an algebraically closed subfield $k$ such that the higher cycle and its ambient variety are defined over $k$, but the image of the usual cycle by the Abel-Jacobi map is not. Then we prove that the external product is nonzero if the image of the higher cycle by the cycle map to the reduced Deligne cohomology does not vanish. We also give examples of indecomposable higher cycles on even-dimensional hypersurfaces of degree at least four in a projective space which satisfy the last condition.
1 Asakura, M., Motives and algebraic de Rham cohomology, in: The arithmetic and geometry of algebraic cycles (Banff), CRM Proc. Lect. Notes, 24, AMS, 2000, pp. 133–154.
2 Beilinson, A., Higher regulators and values of $L$-functions, J. Soviet Math. 30 (1985), 2036–2070.
3 ---, Notes on absolute Hodge cohomology, Contemporary Math. 55 (1986), 35–68.
4 ---, Height pairing between algebraic cycles, Lect. Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 1–26.
5 Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque, vol. 100, Soc. Math. France, Paris, 1982.
6 Bloch, S., Lectures on algebraic cycles, Duke University Mathematical Series, 4, Durham, 1980.
7 ---, Algebraic cycles and values of $L$-functions, J. Reine Angew. Math. 350 (1984), 94–108.
8 ---, Algebraic cycles and higher $K$-theory, Advances in Math., 61 (1986), 267–304.
9 ---, Algebraic cycles and the Beilinson conjectures, Contemporary Math. 58 (1) (1986), 65–79.
10 Carlson, J., Extensions of mixed Hodge structures, in: Journées de Géométrie Algébrique d’Angers 1979, Sijthoff-Noordhoff Alphen a/d Rijn, 1980, pp. 107–128.
11 Clemens, C.H., Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), 215–290.
12 Collino, A., Griffiths’ infinitesimal invariant and higher $K$-theory on hyperelliptic Jacobians, J. Alg. Geom. 6 (1997), 393–415.
13 ---, Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds, in Algebraic $K$-theory and Application (Eds. H. Bass et al.), World Scientific, 1999, pp. 370–402.
14 del Angel, P. and Müller-Stach, S., The transcendental part of the regulator map for $K_{1}$ on a mirror family of $K3$ surfaces, Duke Math. J. 112 (2002), 581–598.
15 Deligne, P., Equations différentielles à points singuliers réguliers, Lect. Notes in Math., Vol. 163. Springer, Berlin, 1970.
16 ---, Théorie de Hodge I, Actes Congrès Intern. Math., 1970, vol. 1, 425-430; II, Publ. Math. IHES 40 (1971), 5–57; III, ibid. 44 (1974), 5–77.
17 ---, Valeurs de fonctions $L$ et périodes d’intégrales, Proc. Symp. in Pure Math., 33 (1979) part 2, pp. 313–346.
18 Deligne, P., Milne, J., Ogus, A. and Shih, K., Hodge Cycles, Motives, and Shimura varieties, Lect. Notes in Math., vol 900, Springer, Berlin, 1982.
19 Deninger, C. and Scholl, A., The Beilinson conjectures, Proceedings Cambridge Math. Soc. (eds. Coats and Taylor) 153 (1992), 173–209.
20 El Zein, F. and Zucker, S., Extendability of normal functions associated to algebraic cycles, in: Topics in transcendental algebraic geometry, Ann. Math. Stud., 106, Princeton Univ. Press, Princeton, N.J., 1984, pp. 269–288.
21 Esnault, H. and Viehweg, E., Deligne-Beilinson cohomology, in: Beilinson’s conjectures on Special Values of L-functions, Academic Press, Boston, 1988, pp. 43–92.
22 Green, M., Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Diff. Geom. 29 (1989), 545–555.
23 Griffiths, P., On the period of certain rational integrals I, II, Ann. Math. 90 (1969), 460–541.
24 Jannsen, U., Deligne homology, Hodge-$D$-conjecture, and motives, in Beilinson’s conjectures on Special Values of $L$-functions, Academic Press, Boston, 1988, pp. 305–372.
25 ---, Mixed motives and algebraic $K$-theory, Lect. Notes in Math., vol. 1400, Springer, Berlin, 1990.
26 ---, Motivic sheaves and filtrations on Chow groups, Proc. Symp. Pure Math. 55 (1994), Part 1, pp. 245–302.
27 Lang, S., Abelian varieties, Interscience Publishers, New York, 1959.
28 Levine, M., Localization on singular varieties, Inv. Math. 91 (1988), 423–464.
29 Mildenhall, S.J.M., Cycles in a product of elliptic curves, and a group analogous to the class group, Duke Math. J. 67 (1992), 387–406.
30 Milnor, J., Singular points of complex hypersurfaces, Ann. Math. Stud. vol. 61, Princeton Univ. Press, 1969.
31 Müller-Stach, S., Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Alg. Geom. 6 (1997), 513–543.
32 Rosenschon, A. and Saito, M., Cycle map for strictly decomposable cycles, Amer. J. Math. 125 (2003), 773–790.
33 Saito, M., Mixed Hodge Modules, Publ. RIMS, Kyoto Univ., 26 (1990), 221–333.
34 ---, Extension of mixed Hodge Modules, Compos. Math. 74 (1990), 209–234.
35 ---, Admissible normal functions, J. Alg. Geom. 5 (1996), 235–276.
36 ---, Bloch’s conjecture, Deligne cohomology and higher Chow groups, preprint RIMS–1284 (or math.AG/9910113).
37 ---, Arithmetic mixed sheaves, Inv. Math. 144 (2001), 533–569.
38 ---, Refined cycle maps, Algebraic Geometry 2000, Azumino (Hotaka) Adv. Stud. Pure Math. 36 Math. Soc. Japan, Tokyo (2002), 115–143.
39 Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Inv. Math. 22 (1973), 211–319.
40 Schoen, C., Zero cycles modulo rational equivalence for some varieties over fields of transcendence degree one, Proc. Symp. Pure Math. 46 (1987), part 2, pp. 463–473.
41 ---, On certain exterior product maps of Chow groups, Math. Res. Let. 7 (2000), 177–194,
42 Steenbrink, J.H.M., Limits of Hodge structures, Inv. Math. 31 (1975/76), 229–257.
43 Voisin, C., Variations de structures de Hodge et zéro-cycles sur les surfaces générales, Math. Ann. 299 (1994), 77–103.
44 ---, Transcendental methods in the study of algebraic cycles, Lect. Notes in Math. vol. 1594, pp. 153–222.
45 ---, Nori’s connectivity theorem and higher Chow groups, J. Inst. Math. Jussieu 1 (2002), 307–329.
46 Zucker, S., Hodge theory with degenerating coefficients, $L_{2}$-cohomology in the Poincaré metric, Ann. Math., 109 (1979), 415–476.
1 Asakura, M., Motives and algebraic de Rham cohomology, in: The arithmetic and geometry of algebraic cycles (Banff), CRM Proc. Lect. Notes, 24, AMS, 2000, pp. 133–154.
2 Beilinson, A., Higher regulators and values of $L$-functions, J. Soviet Math. 30 (1985), 2036–2070.
3 ---, Notes on absolute Hodge cohomology, Contemporary Math. 55 (1986), 35–68.
4 ---, Height pairing between algebraic cycles, Lect. Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 1–26.
5 Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque, vol. 100, Soc. Math. France, Paris, 1982.
6 Bloch, S., Lectures on algebraic cycles, Duke University Mathematical Series, 4, Durham, 1980.
7 ---, Algebraic cycles and values of $L$-functions, J. Reine Angew. Math. 350 (1984), 94–108.
8 ---, Algebraic cycles and higher $K$-theory, Advances in Math., 61 (1986), 267–304.
9 ---, Algebraic cycles and the Beilinson conjectures, Contemporary Math. 58 (1) (1986), 65–79.
10 Carlson, J., Extensions of mixed Hodge structures, in: Journées de Géométrie Algébrique d’Angers 1979, Sijthoff-Noordhoff Alphen a/d Rijn, 1980, pp. 107–128.
11 Clemens, C.H., Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), 215–290.
12 Collino, A., Griffiths’ infinitesimal invariant and higher $K$-theory on hyperelliptic Jacobians, J. Alg. Geom. 6 (1997), 393–415.
13 ---, Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds, in Algebraic $K$-theory and Application (Eds. H. Bass et al.), World Scientific, 1999, pp. 370–402.
14 del Angel, P. and Müller-Stach, S., The transcendental part of the regulator map for $K_{1}$ on a mirror family of $K3$ surfaces, Duke Math. J. 112 (2002), 581–598.
15 Deligne, P., Equations différentielles à points singuliers réguliers, Lect. Notes in Math., Vol. 163. Springer, Berlin, 1970.
16 ---, Théorie de Hodge I, Actes Congrès Intern. Math., 1970, vol. 1, 425-430; II, Publ. Math. IHES 40 (1971), 5–57; III, ibid. 44 (1974), 5–77.
17 ---, Valeurs de fonctions $L$ et périodes d’intégrales, Proc. Symp. in Pure Math., 33 (1979) part 2, pp. 313–346.
18 Deligne, P., Milne, J., Ogus, A. and Shih, K., Hodge Cycles, Motives, and Shimura varieties, Lect. Notes in Math., vol 900, Springer, Berlin, 1982.
19 Deninger, C. and Scholl, A., The Beilinson conjectures, Proceedings Cambridge Math. Soc. (eds. Coats and Taylor) 153 (1992), 173–209.
20 El Zein, F. and Zucker, S., Extendability of normal functions associated to algebraic cycles, in: Topics in transcendental algebraic geometry, Ann. Math. Stud., 106, Princeton Univ. Press, Princeton, N.J., 1984, pp. 269–288.
21 Esnault, H. and Viehweg, E., Deligne-Beilinson cohomology, in: Beilinson’s conjectures on Special Values of L-functions, Academic Press, Boston, 1988, pp. 43–92.
22 Green, M., Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Diff. Geom. 29 (1989), 545–555.
23 Griffiths, P., On the period of certain rational integrals I, II, Ann. Math. 90 (1969), 460–541.
24 Jannsen, U., Deligne homology, Hodge-$D$-conjecture, and motives, in Beilinson’s conjectures on Special Values of $L$-functions, Academic Press, Boston, 1988, pp. 305–372.
25 ---, Mixed motives and algebraic $K$-theory, Lect. Notes in Math., vol. 1400, Springer, Berlin, 1990.
26 ---, Motivic sheaves and filtrations on Chow groups, Proc. Symp. Pure Math. 55 (1994), Part 1, pp. 245–302.
27 Lang, S., Abelian varieties, Interscience Publishers, New York, 1959.
28 Levine, M., Localization on singular varieties, Inv. Math. 91 (1988), 423–464.
29 Mildenhall, S.J.M., Cycles in a product of elliptic curves, and a group analogous to the class group, Duke Math. J. 67 (1992), 387–406.
30 Milnor, J., Singular points of complex hypersurfaces, Ann. Math. Stud. vol. 61, Princeton Univ. Press, 1969.
31 Müller-Stach, S., Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Alg. Geom. 6 (1997), 513–543.
32 Rosenschon, A. and Saito, M., Cycle map for strictly decomposable cycles, Amer. J. Math. 125 (2003), 773–790.
33 Saito, M., Mixed Hodge Modules, Publ. RIMS, Kyoto Univ., 26 (1990), 221–333.
34 ---, Extension of mixed Hodge Modules, Compos. Math. 74 (1990), 209–234.
35 ---, Admissible normal functions, J. Alg. Geom. 5 (1996), 235–276.
36 ---, Bloch’s conjecture, Deligne cohomology and higher Chow groups, preprint RIMS–1284 (or math.AG/9910113).
37 ---, Arithmetic mixed sheaves, Inv. Math. 144 (2001), 533–569.
38 ---, Refined cycle maps, Algebraic Geometry 2000, Azumino (Hotaka) Adv. Stud. Pure Math. 36 Math. Soc. Japan, Tokyo (2002), 115–143.
39 Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Inv. Math. 22 (1973), 211–319.
40 Schoen, C., Zero cycles modulo rational equivalence for some varieties over fields of transcendence degree one, Proc. Symp. Pure Math. 46 (1987), part 2, pp. 463–473.
41 ---, On certain exterior product maps of Chow groups, Math. Res. Let. 7 (2000), 177–194,
42 Steenbrink, J.H.M., Limits of Hodge structures, Inv. Math. 31 (1975/76), 229–257.
43 Voisin, C., Variations de structures de Hodge et zéro-cycles sur les surfaces générales, Math. Ann. 299 (1994), 77–103.
44 ---, Transcendental methods in the study of algebraic cycles, Lect. Notes in Math. vol. 1594, pp. 153–222.
45 ---, Nori’s connectivity theorem and higher Chow groups, J. Inst. Math. Jussieu 1 (2002), 307–329.
46 Zucker, S., Hodge theory with degenerating coefficients, $L_{2}$-cohomology in the Poincaré metric, Ann. Math., 109 (1979), 415–476.
Additional Information
Andreas Rosenschon
Affiliation:
Department of Mathematics, University at Buffalo, SUNY, Buffalo, NY 14260
Email:
axr@buffalo.edu
Morihiko Saito
Affiliation:
RIMS Kyoto University, Kyoto 606–8502, Japan
Email:
msaito@kurims.kyoto-u.ac.jp
Received by editor(s):
January 11, 2002
Published electronically:
December 9, 2003
