Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T16:15:47.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher Abel-Jacobi maps for 0-cycles

Published online by Cambridge University Press:  07 January 2008

Matt Kerr
Matt Kerr
Affiliation:
[email protected] of Mathematics, University of Durham, Durham, DH1 1JLEngland
Get access

Abstract

Starting from the candidate Bloch-Beilinson filtration on constructed in [L2], we develop and describe geometrically a series of Hodgetheoretic invariants Ψi defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the Ψi, with applications to 0-cycles on products of curves.

Keywords

Chow grouphigher Abel-Jacobi mapMilnor K-theoryregulatordifferential character
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Issue 1 , August 2008 , pp. 41 - 101
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

B1.Bloch, S., ‘Lectures on algebraic cycles’, Duke Math Series IV, 1980Google Scholar
B2.Bloch, S., Some elementary theorems about algebraic cycles on Abelian varieties, Invent.Math. 37 (1976), no. 3, pp. 215–228CrossRefGoogle Scholar
BKL.Bloch, S., Kas, A. and Lieberman, D., Zero cycles on surfaces with pg = 0, Compositio Math. 33 (1976), no. 2, 135–145Google Scholar
Ca.Carlson, J., Extensions of mixed Hodge structures, Journées de Géométrie Algébrique d'Angers 1979, Sijthof and Nordhof (1980), pp. 107–127Google Scholar
De1.Deligne, P., Theorie de Hodge II, Publ. Math. IHES 40 (1971), pp. 5–58CrossRefGoogle Scholar
De2.Deligne, P., Theorie de Hodge III, Publ. Math. IHES 44 (1974), pp. 5–77Google Scholar
EP.Esnault, H. and Paranjape, K., Remarks on absolute de Rham and absolute Hodge cycles, C. R. Acad. Sci. Paris Ser. I Math 319 (1994), no. 1, pp. 67–72Google Scholar
EV.Esnault, H. and Viehweg, E., Deligne-Beilinson cohomology, Beilinson's Conjectures on Special Values of L-functions, Academic Press, Boston, 1988, pp. 43–92Google Scholar
Gr1.Green, M., Higher Abel-Jacobi maps, Proceedings of the ICM (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, pp. 267–276CrossRefGoogle Scholar
Gr2.Green, M., Infinitesimal methods in Hodge theory, Algebraic cycles and Hodge theory (Torino, 1993), Lecture Notes in math. 159, Springer, 1994, pp. 1–92CrossRefGoogle Scholar
Gr3.Green, M., Notes from the Banff conference, June 1998Google Scholar
GG1.Green, M. and Griffiths, P., On the tangent space to the space of algebraic cycles on a smooth projective variety, preprint, 2001Google Scholar
GG2.Green, M. and Griffiths, P., An interesting 0-cycle, Duke Math. J. 119 (2003), no. 2, 261–313Google Scholar
GG3.Green, M. and Griffiths, P., Hodge-theoretic invariants for algebraic cycles, Intl. Math. Res. Not. 2003, no. 9, pp. 477–510Google Scholar
G.Griffiths, P., ‘Introduction to algebraic curves’, American Math. Society, Providence, 1989CrossRefGoogle Scholar
GH.Griffiths, P. and Harris, J., ‘Principles of Algebraic Geometry’, John Wiley & Sons, Inc., 1978Google Scholar
GS.Griffiths, P. and Schmid, W., Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applications to moduli (Bombay 1973), Oxford Univ. Press, 1975, pp. 31–127Google Scholar
H.Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), pp. 109–326CrossRefGoogle Scholar
Ja1.Jannsen, U., Deligne homology, Hodge D-conjecture, and motives, Beilinson's conjectures on special values of L-functions, Academic Press, Boston, 1988, pp. 305–372CrossRefGoogle Scholar
Ja2.Jannsen, U., ‘Mixed motives and algebraic K-theory’, Lecture Notes in Math. 1400, Springer-Verlag, Berlin, 1990Google Scholar
Ke1.Kerr, M., A regulator formula for Milnor K-groups, K-Theory 29 (2003), pp. 175–210Google Scholar
Ke2.Kerr, M., An elementary proof of Suslin reciprocity, Can. Math. Bull. 48 v.2 (2005) pp. 221–236CrossRefGoogle Scholar
Ke3.Kerr, M., Geometric construction of regulator currents with applications to algebraic cycles, Princeton thesis, 2003Google Scholar
Ke4.Kerr, M., A survey of transcendental methods in the study of Chow groups of zero-cycles, in ‘Mirror symmetry V’ (Banff 2003), AMS/IP Stud. Adv. Math. 38 (2006), pp. 295–349Google Scholar
Ke5.Kerr, M., Exterior products of zero-cycles, J. Reine Angew. Math. 600 (2006), pp. 1–23CrossRefGoogle Scholar
KL.Kerr, M. and Lewis, J., The Abel-Jacobi map for higher Chow groups, II, to appear in Inventiones MathGoogle Scholar
KLM.Kerr, M., Lewis, J. and Muller-Stach, S., The Abel-Jacobi map for higher Chow groups, Compositio Math. 142 (2006), no. 2, pp. 374–396Google Scholar
K1.Kleiman, S., Algebraic cycles and the Weil conjectures, in ‘Dix exposés sur la cohomologie des schémas’, North-Holland, Amsterdam, 1968, pp. 359–386Google Scholar
L1.Lewis, J., ‘A survey of the Hodge conjecture,’ (2nd edition), Amer. Math. Soc., Providence, 1999Google Scholar
L2.Lewis, J., A filtration on the Chow groups of a complex projective variety, Compositio Math. 128 (2001), no. 3, pp. 299–322Google Scholar
Mu.Mumford, D., Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), pp. 195–204Google Scholar
M.Murre, J., On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990), pp. 190–204Google Scholar
PS.Piatetski-Shapiro, I. and Shafarevich, I., A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR, Vol. 5 (1971), No. 3, pp. 547–588Google Scholar
PL.Peters, C. and Looijenga, E., Torelli theorems for Kahler K3 surfaces, Compositio Math. 42 (1980/1981), no. 2, pp. 145–186Google Scholar
Ro.Roitman, A., Rational equivalence of zero-dimensional cycles (Russian), Mat. Zametki 28(1), (1980), pp. 85–90, 169Google Scholar
R.Rosenberg, J., ‘Algebraic K-theory and its applications,’ Graduate Texts in Math. 147, Springer, 1994Google Scholar
RS.Rosenschon, A. and Saito, M., Cycle map for strictly decomposable cycles, preprint, 2001Google Scholar
mS.Saito, M., Filtrations on Chow groups and transcendence degree, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, pp. 933–948CrossRefGoogle Scholar
sS.Saito, S., Hodge-theoretic approach to generalization of Abel's theorem, notes from Morelia conference on algebraic cyclesGoogle Scholar
Sc.Schneider, T., ‘Einführung in die transzendenten Zahlen’ (German), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957Google Scholar
T.Totaro, B., Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6 (1992), pp. 177–189Google Scholar
W.Waldschmidt, M., Variations on the six exponentials theorem, Algebra and Number Theory: Proceedings of the Silver Jubilee Conference(Hyderabad,2003),Hindustan Book Agency, 2005Google Scholar