Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

Effective bounds for Hodge-theoretic connectivity


Author: J. Nagel
Journal: J. Algebraic Geom. 11 (2002), 1-32
DOI: https://doi.org/10.1090/S1056-3911-01-00302-2
Published electronically: November 16, 2001
MathSciNet review: 1865913
Full-text PDF

Abstract | References | Additional Information

Abstract: We prove an effective version of Nori's connectivity theorem using Koszul cohomology computations. We apply this result to study the cycle class, Abel-Jacobi and regulator maps and the nonvanishing of certain Griffiths groups for complete intersections in projective spaces, abelian varieties and quadrics.


References [Enhancements On Off] (What's this?)

  • [A] M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207.
  • [AS] M. Asakura and S. Saito, Filtration on Chow groups and generalized normal functions, preprint, 1996.
  • [BC] V.V. Batyrev and D.A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293-338.
  • [BS] S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Am. J. Math. 105 (1983), 1235-1253.
  • [BM] R. Braun and S. Müller-Stach, Effective bounds for Nori's connectivity theorem, in: Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, Walter de Gruyter (1996), 83-88.
  • [C1] A. Collino, Griffiths' infinitesimal invariant and higher K-theory on hyperelliptic jacobians, J. Algebraic Geometry 6 (1997), 393-415.
  • [C2] A. Collino, Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds, in: Algebraic K-theory and its applications (Trieste, 1997), 370-402, World Sci. Publishing, River Edge, NJ, 1999.
  • [CGGH] J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variations of Hodge structures, Comp. Math. 50 (1983), 109-205.
  • [D] P. Deligne, Le théorème de Noether, Exposé XIX dans: P. Deligne et N. Katz, Groupes de Monodromie en Géométrie Algébrique, SGA 7II, Lecture Notes in Math. 340, Springer-Verlag (1973).
  • [DD] P. Deligne and A. Dimca, Filtrations de Hodge et par l'ordre du pôle pour les hypersurfaces singulières, Ann. Sci. ENS 23 (1990), 645-656.
  • [ENS] H. Esnault, M.V. Nori and V. Srinivas, Hodge type of projective varieties of low degree, Math. Ann. 293 (1992), 1-6.
  • [FH] W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Math. 129, Springer-Verlag (1991).
  • [G1] M. Green, The period map for hypersurface sections of high degree of an arbitrary variety, Comp. Math. 55 (1984), 135-156.
  • [G2] M. Green, Koszul cohomology and geometry, in: Lectures on Riemann surfaces, Trieste, Italy, World Scientific Press (1987), 177-200.
  • [G3] M. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Diff. Geom. 27 (1988), 155-159.
  • [G4] M. Green, Griffiths' infinitesimal invariant and the Abel-Jacobi map, J. Diff. Geom. 29 (1989), 545-555.
  • [G5] M. Green, Infinitesimal methods in Hodge theory, in: Algebraic cycles and Hodge theory, Lecture Notes in Math. 1594, Springer-Verlag (1994).
  • [Gr] A. Grothendieck, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 29 (1969), 460-495.
  • [GM1] M. Green and S. Müller-Stach, Algebraic cycles on a general complete intersection of high multi-degree, Comp. Math. 100 (1996), 305-309.
  • [GM2] M. Green and S. Müller-Stach, Algebraic cycles on a general hypersurface section of high degree of a smooth projective variety, manuscript.
  • [K] S. Kleiman, The enumerative theory of singularities, in: Real and complex singularities, Oslo 1976 (P. Holm, ed.), Slijthoff & Noordhoff.
  • [Mul1] S. Müller-Stach, On the nontriviality of the Griffiths group, J. Reine Angew. Math. 427 (1992), 209-218.
  • [Mul2] S. Müller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geometry 6 (1997), 513-543.
  • [Mum] D. Mumford, Abelian varieties, TIFR Studies in Mathematics 5, London: Oxford University Press (1974).
  • [Na1] J. Nagel, The Abel-Jacobi map for complete intersections, Indag. Math. 8 (1997), 95-113.
  • [Na2] J. Nagel, Effective bounds for Nori's connectivity theorem, Comptes Rendus Acad. Sci. Paris, t. 327, Série I, 189-192 (1998).
  • [No] M.V. Nori, Algebraic cycles and Hodge-theoretic connectivity, Inv. Math. 111 (1993), 349-373.
  • [Pa] K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. 140 (1994), 641-660.
  • [Pe] D. Perkinson, Curves in Grassmannians, Trans. of the AMS 347 (1995), 3179-3246.
  • [R] M.S. Ravi, An effective version of Nori's theorem, Math. Z. 214 (1993), 1-7.
  • [Sh] T. Shioda, Algebraic cycles on hypersurfaces in $\mathbb P^n$, Adv. Studies in Pure Math. 10 (1987), Algebraic Geometry Sendai 1985, North Holland Publ. (1987), 717-732.
  • [Sn] D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1987), 159-176.
  • [SSU] M.-H. Saito, Y. Shimizu and S. Usui, Variation of mixed Hodge structure and Torelli problem, in: Advanced Studies in Pure Math. 10, Algebraic Geometry Sendai 1985, North Holland Publ. (1987), 649-693.
  • [V1] C. Voisin, Transcendental methods in the study of algebraic cycles, in: Algebraic cycles and Hodge theory, Lecture Notes in Math. 1594, Springer-Verlag (1994).
  • [V2] C. Voisin, Variations of Hodge structure and algebraic cycles, in: Proceedings of the ICM Zürich 1994 (S.D. Chatterji, ed.), Basel: Birkhäuser (1995), vol. I, 706-715.
  • [V3] C. Voisin, Théorème de connexité de Nori et groupes de Chow supérieurs, lecture at Paris VI, October 2000.
  • [W] G.E. Welters, Polarized abelian varieties and the heat equations, Comp. Math. 49 (1983), 173-194.


Additional Information

J. Nagel
Affiliation: Université Lille 1, Mathématiques - Bât. M2, F-59655 Villeneuve d’Ascq Cedex, France
Email: nagel@agat.univ-lille1.fr

DOI: https://doi.org/10.1090/S1056-3911-01-00302-2
Received by editor(s): March 29, 1999
Published electronically: November 16, 2001

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website