Higher normal functions and Griffiths groups
Author:
Shuji Saito
Journal:
J. Algebraic Geom. 11 (2002), 161201
Published electronically:
November 16, 2001
MathSciNet review:
1865917
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Abstract 
References 
Additional Information
Abstract: In this paper we present some results on the problem of identifying algebraic cycles by means of periods of integrals. The key idea is to combine the two main streams in the study of algebraic cycles. One is the theory of normal functions and AbelJacobi maps originally developed by Griffiths. Another is the BlochBeilinson's (conjectural) filtration on Chow groups arising from the theory of mixed motives. The outcome is the theory of higher normal functions and higher AbelJacobi maps, which we apply to the study of algebraic cycles on hypersurfaces in .
 [AS]
M. Asakura and S. Saito, Generalized Jacobian rings and Beilinson's Hodge and Tate conjectures, preprint.
 [Be]
A. Beilinson, Height pairings between algebraic cycles, Lecture Notes in Math. 1289 (1987), 126.
 [Bl]
S. Bloch, Lectures on Algebraic Cycles, vol. 4, Duke Univ. Math. Series, Durham, 1980.
 [BO]
S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes, Ann. Ec. Norm. Sup. 4 serie 7, fasc. 2 (1974), 181202.
 [Bo]
C. Borcea, Deforming varieties of planes of projective complete intersections, Pacific Journal of Math. 143 (1990), 2536.
 [CDT]
D. Cox, R. Donagi and L. Tu, Variational Torelli implies generic Torelli, Invent. Math. 88 (1987), 439446.
 [Co]
A. Collino, The AbelJacobi isomorphism for the cubic fivefold, Pacific Journal of Math. 122 (1986), 4355.
 [D1]
P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. IHES 35 (1968), 107126.
 [D2]
, Théorie de Hodge II, Publ. Math. IHES 40 (1972), 557.
 [El]
F. El Zein, Complexe dualisant et applications à la classe fondamentale d'un cycle, Bull. Soc. Math. France, Mémoire 58 (1978).
 [EV]
H.Esnault and E.Viehweg, DeligneBeilinson cohomology, in: Beilinson's Conjectures on Special Values of Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
 [G1]
M. Green, Infinitesimal methods in Hodge theory, Lecture Notes in Math. 1594 (1993), 192.
 [G2]
, Koszul cohomology and Geometry, in Cornalba, GomezMont, and Verjovsky, Lectures on Riemann surfaces, ICTP, Trieste, Italy, 177200.
 [G3]
, Griffiths infinitesimal invariant and the AbelJacobi map, J. Diff. Geom. 29 (1989), 545555.
 [G4]
, Lectures at Banff in 1998.
 [Gri]
P. Griffiths, Periods of certain rational integrals: I and II, Ann. of Math. 90 (1969), 460541.
 [J1]
U.Jannsen, Mixed sheaves and filtrations on Chow groups, in: Motives, ed. U. Jannsen, S. Kleiman, J.P. Serre, Proceedings of Symposia in Pure Math., AMS 55, Part 1 (1991).
 [J2]
, A letter from Jannsen to Gross on higher AbelJacobi maps, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), NATO Science Series Vol. 548, Kluwer Academic Publishers, 2000, pp. 261275.
 [J3]
, Deligne homology, Hodgeconjecture, and motives, in: Beilinson's Conjectures on Special Values of Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
 [Le1]
J. Lewis, Introductory lectures in transcendental algebraic geometry: A survey of the Hodge conjecture, Les Publications CRM, 1991.
 [Le2]
, Cylinder homomorphisms and Chow groups, Math. Nachr. 160 (1993), 205221.
 [No]
M. V. Nori, Algebraic cycles and Hodge theoretic connectivity, Invent. of Math. (1993), 349373.
 [Pa]
K. Paranjape, Cohomological and cycletheoretic connectivity, Ann. of Math. 140 (1994), 641660.
 [R]
M. Rosenlicht, A remark on quotient spaces, An. Acad. Bras. Cienc. 35 (1963), 487489.
 [Sa1]
S. Saito, Motives and filtration on Chow groups, Invent. Math. 125 (1996), 149196.
 [Sa2]
, Motives and filtration on Chow groups, II, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), NATO Science Series Vol.548, Kluwer Academic Publishers, 2000, pp. 321346.
 [Sa3]
, Motives, Filtration on Chow groups and Hodge theory, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), CRM Proceedings and Lecture Notes Vol. 24, American Mathematical Society, 2000, pp. 235253.
 [Sch]
C. Schoen, On Hodge structures and nonrepresentability of Chow groups, Compositio Math. 88 (1993), 285316.
 [SGA5]
A. Grothendieck et al., Cohomologie adique et Fonctions , Lecture Notes in Math. 589, SpringerVerlag.
 [V1]
C. Voisin, Transcendental methods in the study of algebraic cycles, Lecture Notes in Math., Springer 1594 (1993), 153222.
 [V2]
, Variations de structure de Hodge et zérocycles sur les surfaces générals, Math. Ann. 299 (1994), 77103.
 [AS]
 M. Asakura and S. Saito, Generalized Jacobian rings and Beilinson's Hodge and Tate conjectures, preprint.
 [Be]
 A. Beilinson, Height pairings between algebraic cycles, Lecture Notes in Math. 1289 (1987), 126.
 [Bl]
 S. Bloch, Lectures on Algebraic Cycles, vol. 4, Duke Univ. Math. Series, Durham, 1980.
 [BO]
 S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes, Ann. Ec. Norm. Sup. 4 serie 7, fasc. 2 (1974), 181202.
 [Bo]
 C. Borcea, Deforming varieties of planes of projective complete intersections, Pacific Journal of Math. 143 (1990), 2536.
 [CDT]
 D. Cox, R. Donagi and L. Tu, Variational Torelli implies generic Torelli, Invent. Math. 88 (1987), 439446.
 [Co]
 A. Collino, The AbelJacobi isomorphism for the cubic fivefold, Pacific Journal of Math. 122 (1986), 4355.
 [D1]
 P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. IHES 35 (1968), 107126.
 [D2]
 , Théorie de Hodge II, Publ. Math. IHES 40 (1972), 557.
 [El]
 F. El Zein, Complexe dualisant et applications à la classe fondamentale d'un cycle, Bull. Soc. Math. France, Mémoire 58 (1978).
 [EV]
 H.Esnault and E.Viehweg, DeligneBeilinson cohomology, in: Beilinson's Conjectures on Special Values of Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
 [G1]
 M. Green, Infinitesimal methods in Hodge theory, Lecture Notes in Math. 1594 (1993), 192.
 [G2]
 , Koszul cohomology and Geometry, in Cornalba, GomezMont, and Verjovsky, Lectures on Riemann surfaces, ICTP, Trieste, Italy, 177200.
 [G3]
 , Griffiths infinitesimal invariant and the AbelJacobi map, J. Diff. Geom. 29 (1989), 545555.
 [G4]
 , Lectures at Banff in 1998.
 [Gri]
 P. Griffiths, Periods of certain rational integrals: I and II, Ann. of Math. 90 (1969), 460541.
 [J1]
 U.Jannsen, Mixed sheaves and filtrations on Chow groups, in: Motives, ed. U. Jannsen, S. Kleiman, J.P. Serre, Proceedings of Symposia in Pure Math., AMS 55, Part 1 (1991).
 [J2]
 , A letter from Jannsen to Gross on higher AbelJacobi maps, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), NATO Science Series Vol. 548, Kluwer Academic Publishers, 2000, pp. 261275.
 [J3]
 , Deligne homology, Hodgeconjecture, and motives, in: Beilinson's Conjectures on Special Values of Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
 [Le1]
 J. Lewis, Introductory lectures in transcendental algebraic geometry: A survey of the Hodge conjecture, Les Publications CRM, 1991.
 [Le2]
 , Cylinder homomorphisms and Chow groups, Math. Nachr. 160 (1993), 205221.
 [No]
 M. V. Nori, Algebraic cycles and Hodge theoretic connectivity, Invent. of Math. (1993), 349373.
 [Pa]
 K. Paranjape, Cohomological and cycletheoretic connectivity, Ann. of Math. 140 (1994), 641660.
 [R]
 M. Rosenlicht, A remark on quotient spaces, An. Acad. Bras. Cienc. 35 (1963), 487489.
 [Sa1]
 S. Saito, Motives and filtration on Chow groups, Invent. Math. 125 (1996), 149196.
 [Sa2]
 , Motives and filtration on Chow groups, II, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), NATO Science Series Vol.548, Kluwer Academic Publishers, 2000, pp. 321346.
 [Sa3]
 , Motives, Filtration on Chow groups and Hodge theory, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui, eds.), CRM Proceedings and Lecture Notes Vol. 24, American Mathematical Society, 2000, pp. 235253.
 [Sch]
 C. Schoen, On Hodge structures and nonrepresentability of Chow groups, Compositio Math. 88 (1993), 285316.
 [SGA5]
 A. Grothendieck et al., Cohomologie adique et Fonctions , Lecture Notes in Math. 589, SpringerVerlag.
 [V1]
 C. Voisin, Transcendental methods in the study of algebraic cycles, Lecture Notes in Math., Springer 1594 (1993), 153222.
 [V2]
 , Variations de structure de Hodge et zérocycles sur les surfaces générals, Math. Ann. 299 (1994), 77103.
Additional Information
Shuji Saito
Affiliation:
Graduate School of Mathematics, Nagoya University Chikusaku, NAGOYA, 4648602, Japan
Email:
sshuji@msb.biglobe.ne.jp
DOI:
http://dx.doi.org/10.1090/S1056391101002946
PII:
S 10563911(01)002946
Received by editor(s):
January 28, 2000
Received by editor(s) in revised form:
May 4, 2000
Published electronically:
November 16, 2001
