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
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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.
[A]Atiyah M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
[AS]AsakuraSaito M. Asakura and S. Saito, Filtration on Chow groups and generalized normal functions, preprint, 1996.
[BC]BatyrevCox 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]BlochSrinivas S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Am. J. Math. 105 (1983), 1235–1253.
[BM]BraunSMS 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]CollinoGriffithsinfinitesimalinvariant A. Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic jacobians, J. Algebraic Geometry 6 (1997), 393–415.
[C2]Collinoregulator 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]CGGH J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variations of Hodge structures, Comp. Math. 50 (1983), 109–205.
[D]DeligneSGA7II 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]DeligneDimca 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]ENS H. Esnault, M.V. Nori and V. Srinivas, Hodge type of projective varieties of low degree, Math. Ann. 293 (1992), 1–6.
[FH]FultonHarris W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Math. 129, Springer–Verlag (1991).
[G1]Greenperiodmap M. Green, The period map for hypersurface sections of high degree of an arbitrary variety, Comp. Math. 55 (1984), 135–156.
[G2]GreenKoszul M. Green, Koszul cohomology and geometry, in: Lectures on Riemann surfaces, Trieste, Italy, World Scientific Press (1987), 177–200.
[G3]GreenexplicitNoetherLefschetz M. Green, A new proof of the explicit Noether–Lefschetz theorem, J. Diff. Geom. 27 (1988), 155–159.
[G4]GreenAbelJacobi M. Green, Griffiths’ infinitesimal invariant and the Abel–Jacobi map, J. Diff. Geom. 29 (1989), 545–555.
[G5]GreenCIME M. Green, Infinitesimal methods in Hodge theory, in: Algebraic cycles and Hodge theory, Lecture Notes in Math. 1594, Springer–Verlag (1994).
[Gr]Grothendieck A. Grothendieck, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 29 (1969), 460–495.
[GM1]GreenSMS M. Green and S. Müller–Stach, Algebraic cycles on a general complete intersection of high multi-degree, Comp. Math. 100 (1996), 305–309.
[GM2]GreenSMSmanuscript M. Green and S. Müller–Stach, Algebraic cycles on a general hypersurface section of high degree of a smooth projective variety, manuscript.
[K]Kleiman S. Kleiman, The enumerative theory of singularities, in: Real and complex singularities, Oslo 1976 (P. Holm, ed.), Slijthoff & Noordhoff.
[Mul1]SMSGriffithsgroup S. Müller–Stach, On the nontriviality of the Griffiths group, J. Reine Angew. Math. 427 (1992), 209–218.
[Mul2]SMSregulator S. Müller–Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geometry 6 (1997), 513–543.
[Mum]MumfordAV D. Mumford, Abelian varieties, TIFR Studies in Mathematics 5, London: Oxford University Press (1974).
[Na1]W9510 J. Nagel, The Abel–Jacobi map for complete intersections, Indag. Math. 8 (1997), 95–113.
[Na2]CRAS J. Nagel, Effective bounds for Nori’s connectivity theorem, Comptes Rendus Acad. Sci. Paris, t. 327, Série I, 189–192 (1998).
[No]Nori M.V. Nori, Algebraic cycles and Hodge–theoretic connectivity, Inv. Math. 111 (1993), 349–373.
[Pa]Paranjape K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. 140 (1994), 641–660.
[Pe]Perkinson D. Perkinson, Curves in Grassmannians, Trans. of the AMS 347 (1995), 3179–3246.
[R]Ravi M.S. Ravi, An effective version of Nori’s theorem, Math. Z. 214 (1993), 1–7.
[Sh]Shioda 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]Snow D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1987), 159–176.
[SSU]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]VoisinCIME 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]VoisinICM 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]Voisinlezing C. Voisin, Théorème de connexité de Nori et groupes de Chow supérieurs, lecture at Paris VI, October 2000.
[W]Welters G.E. Welters, Polarized abelian varieties and the heat equations, Comp. Math. 49 (1983), 173–194.
[A]Atiyah M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.
[AS]AsakuraSaito M. Asakura and S. Saito, Filtration on Chow groups and generalized normal functions, preprint, 1996.
[BC]BatyrevCox 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]BlochSrinivas S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Am. J. Math. 105 (1983), 1235–1253.
[BM]BraunSMS 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]CollinoGriffithsinfinitesimalinvariant A. Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic jacobians, J. Algebraic Geometry 6 (1997), 393–415.
[C2]Collinoregulator 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]CGGH J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variations of Hodge structures, Comp. Math. 50 (1983), 109–205.
[D]DeligneSGA7II 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]DeligneDimca 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]ENS H. Esnault, M.V. Nori and V. Srinivas, Hodge type of projective varieties of low degree, Math. Ann. 293 (1992), 1–6.
[FH]FultonHarris W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Math. 129, Springer–Verlag (1991).
[G1]Greenperiodmap M. Green, The period map for hypersurface sections of high degree of an arbitrary variety, Comp. Math. 55 (1984), 135–156.
[G2]GreenKoszul M. Green, Koszul cohomology and geometry, in: Lectures on Riemann surfaces, Trieste, Italy, World Scientific Press (1987), 177–200.
[G3]GreenexplicitNoetherLefschetz M. Green, A new proof of the explicit Noether–Lefschetz theorem, J. Diff. Geom. 27 (1988), 155–159.
[G4]GreenAbelJacobi M. Green, Griffiths’ infinitesimal invariant and the Abel–Jacobi map, J. Diff. Geom. 29 (1989), 545–555.
[G5]GreenCIME M. Green, Infinitesimal methods in Hodge theory, in: Algebraic cycles and Hodge theory, Lecture Notes in Math. 1594, Springer–Verlag (1994).
[Gr]Grothendieck A. Grothendieck, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 29 (1969), 460–495.
[GM1]GreenSMS M. Green and S. Müller–Stach, Algebraic cycles on a general complete intersection of high multi-degree, Comp. Math. 100 (1996), 305–309.
[GM2]GreenSMSmanuscript M. Green and S. Müller–Stach, Algebraic cycles on a general hypersurface section of high degree of a smooth projective variety, manuscript.
[K]Kleiman S. Kleiman, The enumerative theory of singularities, in: Real and complex singularities, Oslo 1976 (P. Holm, ed.), Slijthoff & Noordhoff.
[Mul1]SMSGriffithsgroup S. Müller–Stach, On the nontriviality of the Griffiths group, J. Reine Angew. Math. 427 (1992), 209–218.
[Mul2]SMSregulator S. Müller–Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geometry 6 (1997), 513–543.
[Mum]MumfordAV D. Mumford, Abelian varieties, TIFR Studies in Mathematics 5, London: Oxford University Press (1974).
[Na1]W9510 J. Nagel, The Abel–Jacobi map for complete intersections, Indag. Math. 8 (1997), 95–113.
[Na2]CRAS J. Nagel, Effective bounds for Nori’s connectivity theorem, Comptes Rendus Acad. Sci. Paris, t. 327, Série I, 189–192 (1998).
[No]Nori M.V. Nori, Algebraic cycles and Hodge–theoretic connectivity, Inv. Math. 111 (1993), 349–373.
[Pa]Paranjape K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. 140 (1994), 641–660.
[Pe]Perkinson D. Perkinson, Curves in Grassmannians, Trans. of the AMS 347 (1995), 3179–3246.
[R]Ravi M.S. Ravi, An effective version of Nori’s theorem, Math. Z. 214 (1993), 1–7.
[Sh]Shioda 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]Snow D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1987), 159–176.
[SSU]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]VoisinCIME 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]VoisinICM 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]Voisinlezing C. Voisin, Théorème de connexité de Nori et groupes de Chow supérieurs, lecture at Paris VI, October 2000.
[W]Welters 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
MR Author ID:
626511
Email:
nagel@agat.univ-lille1.fr
Received by editor(s):
March 29, 1999
Published electronically:
November 16, 2001
