Higher normal functions and Griffiths groups

Author:
Shuji Saito

Journal:
J. Algebraic Geom. **11** (2002), 161-201

DOI:
https://doi.org/10.1090/S1056-3911-01-00294-6

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 Abel-Jacobi maps originally developed by Griffiths. Another is the Bloch-Beilinson’s (conjectural) filtration on Chow groups arising from the theory of mixed motives. The outcome is the theory of higher normal functions and higher Abel-Jacobi maps, which we apply to the study of algebraic cycles on hypersurfaces in $\mathbb {P}^n$.

[AS]AS M. Asakura and S. Saito, *Generalized Jacobian rings and Beilinson’s Hodge and Tate conjectures*, preprint.
[Be]Be A. Beilinson, *Height pairings between algebraic cycles*, Lecture Notes in Math. **1289** (1987), 1-26.
[Bl]Bl S. Bloch, *Lectures on Algebraic Cycles*, vol. 4, Duke Univ. Math. Series, Durham, 1980.
[BO]BO S. Bloch and A. Ogus, *Gersten’s conjecture and the homology of schemes*, Ann. Ec. Norm. Sup. 4 serie **7, fasc. 2** (1974), 181–202.
[Bo]Bo C. Borcea, *Deforming varieties of $k$-planes of projective complete intersections*, Pacific Journal of Math. **143** (1990), 25–36.
[CDT]CDT D. Cox, R. Donagi and L. Tu, *Variational Torelli implies generic Torelli*, Invent. Math. **88** (1987), 439–446.
[Co]Co A. Collino, *The Abel-Jacobi isomorphism for the cubic fivefold*, Pacific Journal of Math. **122** (1986), 43-55.
[D1]D1 P. Deligne, *Théorème de Lefschetz et critères de dégénérescence de suites spectrales*, Publ. Math. IHES **35** (1968), 107–126.
[D2]D2 ---, *Théorie de Hodge II*, Publ. Math. IHES **40** (1972), 5–57.
[El]El F. El Zein, *Complexe dualisant et applications à la classe fondamentale d’un cycle*, Bull. Soc. Math. France, Mémoire **58** (1978).
[EV]EV H.Esnault and E.Viehweg, *Deligne-Beilinson cohomology*, in: Beilinson’s Conjectures on Special Values of $L$-Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
[G1]G1 M. Green, *Infinitesimal methods in Hodge theory*, Lecture Notes in Math. **1594** (1993), 1-92.
[G2]G2 ---, *Koszul cohomology and Geometry*, in Cornalba, Gomez-Mont, and Verjovsky, Lectures on Riemann surfaces, ICTP, Trieste, Italy, 177-200.
[G3]G3 ---, *Griffiths infinitesimal invariant and the Abel-Jacobi map*, J. Diff. Geom. **29** (1989), 545-555.
[G4]G4 ---, *Lectures at Banff in 1998*.
[Gri]Gri P. Griffiths, *Periods of certain rational integrals: I and II*, Ann. of Math. **90** (1969), 460-541.
[J1]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]J2 ---, *A letter from Jannsen to Gross on higher Abel-Jacobi maps*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), NATO Science Series Vol. 548, Kluwer Academic Publishers, 2000, pp. 261–275.
[J3]J3 ---, *Deligne homology, Hodge-$D$-conjecture, and motives*, in: Beilinson’s Conjectures on Special Values of $L$-Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
[Le1]Le1 J. Lewis, *Introductory lectures in transcendental algebraic geometry: A survey of the Hodge conjecture*, Les Publications CRM, 1991.
[Le2]Le2 ---, *Cylinder homomorphisms and Chow groups*, Math. Nachr. **160** (1993), 205–221.
[No]No M. V. Nori, *Algebraic cycles and Hodge theoretic connectivity*, Invent. of Math. (1993), 349–373.
[Pa]Pa K. Paranjape, *Cohomological and cycle-theoretic connectivity*, Ann. of Math. **140** (1994), 641–660.
[R]R M. Rosenlicht, *A remark on quotient spaces*, An. Acad. Bras. Cienc. **35** (1963), 487–489.
[Sa1]Sa1 S. Saito, *Motives and filtration on Chow groups*, Invent. Math. **125** (1996), 149–196.
[Sa2]Sa2 ---, *Motives and filtration on Chow groups, II*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), NATO Science Series Vol.548, Kluwer Academic Publishers, 2000, pp. 321–346.
[Sa3]Sa3 ---, *Motives, Filtration on Chow groups and Hodge theory*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), CRM Proceedings and Lecture Notes Vol. 24, American Mathematical Society, 2000, pp. 235–253.
[Sch]Sch C. Schoen, *On Hodge structures and non-representability of Chow groups*, Compositio Math. **88** (1993), 285–316.
[SGA5]SGA5 A. Grothendieck et al., *Cohomologie $l$-adique et Fonctions $L$*, Lecture Notes in Math. **589**, Springer-Verlag.
[V1]V1 C. Voisin, *Transcendental methods in the study of algebraic cycles*, Lecture Notes in Math., Springer **1594** (1993), 153-222.
[V2]V2 ---, *Variations de structure de Hodge et zéro-cycles sur les surfaces générals*, Math. Ann. **299** (1994), 77-103.

[AS]AS M. Asakura and S. Saito, *Generalized Jacobian rings and Beilinson’s Hodge and Tate conjectures*, preprint.
[Be]Be A. Beilinson, *Height pairings between algebraic cycles*, Lecture Notes in Math. **1289** (1987), 1-26.
[Bl]Bl S. Bloch, *Lectures on Algebraic Cycles*, vol. 4, Duke Univ. Math. Series, Durham, 1980.
[BO]BO S. Bloch and A. Ogus, *Gersten’s conjecture and the homology of schemes*, Ann. Ec. Norm. Sup. 4 serie **7, fasc. 2** (1974), 181–202.
[Bo]Bo C. Borcea, *Deforming varieties of $k$-planes of projective complete intersections*, Pacific Journal of Math. **143** (1990), 25–36.
[CDT]CDT D. Cox, R. Donagi and L. Tu, *Variational Torelli implies generic Torelli*, Invent. Math. **88** (1987), 439–446.
[Co]Co A. Collino, *The Abel-Jacobi isomorphism for the cubic fivefold*, Pacific Journal of Math. **122** (1986), 43-55.
[D1]D1 P. Deligne, *Théorème de Lefschetz et critères de dégénérescence de suites spectrales*, Publ. Math. IHES **35** (1968), 107–126.
[D2]D2 ---, *Théorie de Hodge II*, Publ. Math. IHES **40** (1972), 5–57.
[El]El F. El Zein, *Complexe dualisant et applications à la classe fondamentale d’un cycle*, Bull. Soc. Math. France, Mémoire **58** (1978).
[EV]EV H.Esnault and E.Viehweg, *Deligne-Beilinson cohomology*, in: Beilinson’s Conjectures on Special Values of $L$-Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
[G1]G1 M. Green, *Infinitesimal methods in Hodge theory*, Lecture Notes in Math. **1594** (1993), 1-92.
[G2]G2 ---, *Koszul cohomology and Geometry*, in Cornalba, Gomez-Mont, and Verjovsky, Lectures on Riemann surfaces, ICTP, Trieste, Italy, 177-200.
[G3]G3 ---, *Griffiths infinitesimal invariant and the Abel-Jacobi map*, J. Diff. Geom. **29** (1989), 545-555.
[G4]G4 ---, *Lectures at Banff in 1998*.
[Gri]Gri P. Griffiths, *Periods of certain rational integrals: I and II*, Ann. of Math. **90** (1969), 460-541.
[J1]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]J2 ---, *A letter from Jannsen to Gross on higher Abel-Jacobi maps*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), NATO Science Series Vol. 548, Kluwer Academic Publishers, 2000, pp. 261–275.
[J3]J3 ---, *Deligne homology, Hodge-$D$-conjecture, and motives*, in: Beilinson’s Conjectures on Special Values of $L$-Functions (M.Rapoport, M.Schappacher and P.Schneider, eds.), Perspectives in Math., Academic Press.
[Le1]Le1 J. Lewis, *Introductory lectures in transcendental algebraic geometry: A survey of the Hodge conjecture*, Les Publications CRM, 1991.
[Le2]Le2 ---, *Cylinder homomorphisms and Chow groups*, Math. Nachr. **160** (1993), 205–221.
[No]No M. V. Nori, *Algebraic cycles and Hodge theoretic connectivity*, Invent. of Math. (1993), 349–373.
[Pa]Pa K. Paranjape, *Cohomological and cycle-theoretic connectivity*, Ann. of Math. **140** (1994), 641–660.
[R]R M. Rosenlicht, *A remark on quotient spaces*, An. Acad. Bras. Cienc. **35** (1963), 487–489.
[Sa1]Sa1 S. Saito, *Motives and filtration on Chow groups*, Invent. Math. **125** (1996), 149–196.
[Sa2]Sa2 ---, *Motives and filtration on Chow groups, II*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), NATO Science Series Vol.548, Kluwer Academic Publishers, 2000, pp. 321–346.
[Sa3]Sa3 ---, *Motives, Filtration on Chow groups and Hodge theory*, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7-19, 1998, Banff, Alberta, Canada (B. Gordon, J. Lewis, S. Müller-Stach, S. Saito and N. Yui, eds.), CRM Proceedings and Lecture Notes Vol. 24, American Mathematical Society, 2000, pp. 235–253.
[Sch]Sch C. Schoen, *On Hodge structures and non-representability of Chow groups*, Compositio Math. **88** (1993), 285–316.
[SGA5]SGA5 A. Grothendieck et al., *Cohomologie $l$-adique et Fonctions $L$*, Lecture Notes in Math. **589**, Springer-Verlag.
[V1]V1 C. Voisin, *Transcendental methods in the study of algebraic cycles*, Lecture Notes in Math., Springer **1594** (1993), 153-222.
[V2]V2 ---, *Variations de structure de Hodge et zéro-cycles sur les surfaces générals*, Math. Ann. **299** (1994), 77-103.

Additional Information

**Shuji Saito**

Affiliation:
Graduate School of Mathematics, Nagoya University Chikusa-ku, NAGOYA, 464-8602, Japan

Email:
sshuji@msb.biglobe.ne.jp

Received by editor(s):
January 28, 2000

Received by editor(s) in revised form:
May 4, 2000

Published electronically:
November 16, 2001