Algebraic cycles and infinitesimal invariants on Jacobian varieties
Author:
Atsushi Ikeda
Journal:
J. Algebraic Geom. 12 (2003), 573-603
DOI:
https://doi.org/10.1090/S1056-3911-03-00360-6
Published electronically:
March 11, 2003
MathSciNet review:
1966027
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Abstract |
References |
Additional Information
Abstract: We study the infinitesimal invariant for a family of algebraic cycles on Jacobian varieties, and prove the formula for calculating the infinitesimal invariant. Applying this formula to Jacobian varieties of plane curves, we detect a non-torsion element in the higher Griffiths group, which is a group of algebraic cycles modulo certain algebraic equivalence based on the theory of mixed motives.
as M. Asakura and S. Saito, Generalized Jacobian rings for open complete intersections, preprint.
b A. Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann. 273 (1986), 647-651.
be A. Beilinson, Height pairing between algebraic cycles, Lecture Notes in Math. 1289 (1987), 1-26.
cggh J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variation of Hodge structure (I), Compositio Math. 50 (1983), 109-205.
c G. Ceresa, $C$ is not algebraically equivalent to $C^-$ in its Jacobian, Ann. of Math. (2) 117 (1983), 285-291.
cp A. Collino and G. Pirola, The Griffiths infinitesimal invariant for curve in its Jacobian, Duke Math. J. 78 (1995), 59-88.
d1 P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 107-126.
d2 P. Deligne, Théorie de Hodge, II, III, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-58; Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-78.
dm C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201-219.
e F. El Zein, La classe fondamentale d’un cycle, Compositio Math. 29 (1974), 9-33.
e1 F. El Zein, Complexe dualisant et applications à la classe fondamentale d’un cycle, Bull. Soc. Math. France Mém. 58 (1978).
g M. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), 545-555.
gr0 P. Griffiths, On the periods of certain rational integrals: I, II, Ann. of Math. (2) 90 (1969), 460-541.
gr1 P. Griffiths, Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping), Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125-180.
gr2 P. Griffiths, A theorem concerning the differential equations satisfied by normal functions associated to algebraic cycles, Amer. J. Math. 101 (1979), 94-131.
gr3 P. Griffiths, Infinitesimal variations of Hodge structure (III): Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), 267-324.
h R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20 (1966).
m J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, Indag. Math. (N.S.) 4 (1993), 177-201.
s1 S. Saito, Motives and filtration on Chow groups, Invent. Math. 125 (1996), 149-196.
s2 S. Saito, Higher normal functions and Griffiths groups, J. Algebraic Geom. 11 (2002), 161-201.
v C. Voisin, Une approche infinitésimale du théorème de H. Clemens sur les cycles d’une quintique générale de $\mathbb {P}^4$, J. Algebraic Geom. 1 (1992), 157-174.
as M. Asakura and S. Saito, Generalized Jacobian rings for open complete intersections, preprint.
b A. Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann. 273 (1986), 647-651.
be A. Beilinson, Height pairing between algebraic cycles, Lecture Notes in Math. 1289 (1987), 1-26.
cggh J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variation of Hodge structure (I), Compositio Math. 50 (1983), 109-205.
c G. Ceresa, $C$ is not algebraically equivalent to $C^-$ in its Jacobian, Ann. of Math. (2) 117 (1983), 285-291.
cp A. Collino and G. Pirola, The Griffiths infinitesimal invariant for curve in its Jacobian, Duke Math. J. 78 (1995), 59-88.
d1 P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 107-126.
d2 P. Deligne, Théorie de Hodge, II, III, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-58; Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-78.
dm C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201-219.
e F. El Zein, La classe fondamentale d’un cycle, Compositio Math. 29 (1974), 9-33.
e1 F. El Zein, Complexe dualisant et applications à la classe fondamentale d’un cycle, Bull. Soc. Math. France Mém. 58 (1978).
g M. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), 545-555.
gr0 P. Griffiths, On the periods of certain rational integrals: I, II, Ann. of Math. (2) 90 (1969), 460-541.
gr1 P. Griffiths, Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping), Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125-180.
gr2 P. Griffiths, A theorem concerning the differential equations satisfied by normal functions associated to algebraic cycles, Amer. J. Math. 101 (1979), 94-131.
gr3 P. Griffiths, Infinitesimal variations of Hodge structure (III): Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), 267-324.
h R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20 (1966).
m J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, Indag. Math. (N.S.) 4 (1993), 177-201.
s1 S. Saito, Motives and filtration on Chow groups, Invent. Math. 125 (1996), 149-196.
s2 S. Saito, Higher normal functions and Griffiths groups, J. Algebraic Geom. 11 (2002), 161-201.
v C. Voisin, Une approche infinitésimale du théorème de H. Clemens sur les cycles d’une quintique générale de $\mathbb {P}^4$, J. Algebraic Geom. 1 (1992), 157-174.
Additional Information
Atsushi Ikeda
Affiliation:
Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan
Email:
atsushi@math.sci.osaka-u.ac.jp
Received by editor(s):
January 25, 2001
Published electronically:
March 11, 2003
