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Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties

  • James D. Lewis
Chapter
Part of the Fields Institute Communications book series (FIC, volume 67)

Abstract

We review the transcendental aspects of algebraic cycles, and explain how this relates to Calabi–Yau varieties. More precisely, after presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne cohomology, as well as the generalized higher cycles due to Bloch that are connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi–Yau situation, and explain some recent developments in the field.

Key words

Calabi–Yau variety Algebraic cycle Abel–Jacobi map Regulator Deligne cohomology Chow group 

Mathematics Subject Classifications (2010)

Primary 14C25 Secondary 14C30 14C35 

Notes

Acknowledgements

Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

  1. 1.
    M. Asakura, On dlog Image of K 2 of Elliptic Surface Minus Singular Fibers, preprint (2006) [arXiv:math/0511190v4]Google Scholar
  2. 2.
    H. Bass, J. Tate, in The Milnor Ring of a Global Field, in Algebraic K-Theory II. Lecture Notes in Mathematics, vol. 342 (Springer, New York, 1972), pp. 349–446Google Scholar
  3. 3.
    A. Beilinson, Notes on absolute Hodge cohomology, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory. Contemporary Mathematics, vol. 55, Part 1 (AMS, Providence 1986), pp. 35–68Google Scholar
  4. 4.
    A. Beilinson, Higher regulators of modular curves, in Applications of K-Theory to Algebraic Geometry and Number Theory, Boulder, CO, 1983. Contemporary Mathematics, vol. 55 (AMS, Providence, 1986), pp. 1–34Google Scholar
  5. 5.
    S. Bloch, Lectures on Algebraic Cycles. Duke University Mathematics Series, vol. IV (Duke University, Durham, 1980)Google Scholar
  6. 6.
    S. Bloch, Algebraic cycles and higher K-theory. Adv. Math. 61, 267–304 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles. Am. J. Math. 105, 1235–1253 (1983)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    J. Carlson, Extension of mixed Hodge structures, in Journées de Géométrie Algébrique d’Angers 1979 (Sijthoff and Nordhoff, The Netherlands, 1980), pp. 107–127Google Scholar
  9. 9.
    X. Chen, J.D. Lewis, Noether-Lefschetz for K 1 of a certain class of surfaces. Bol. Soc. Mat. Mexicana (3) 10(1), 29–41 (2004)Google Scholar
  10. 10.
    X. Chen, J.D. Lewis, The Hodge\(-\mathcal{D}-\)conjecture for K3 and Abelian surfaces. J. Algebr. Geom. 14(2), 213–240 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    X. Chen, J.D. Lewis, Density of rational curves on K3 surfaces. Math. Ann. [arXiv:1004.5167] (2011)Google Scholar
  12. 12.
    X. Chen, C. Doran, M. Kerr, J.D. Lewis, Higher normal functions, derivatives of normal functions, and elliptic fibrations (2011) (submitted) [arXiv:1108.2223]Google Scholar
  13. 13.
    C.H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated. Publ. I.H.E.S. 58, 19–38 (1983)Google Scholar
  14. 14.
    A. Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic jacobians. J. Algebr. Geom. 6, 393–415 (1997)MathSciNetMATHGoogle Scholar
  15. 15.
    R. de Jeu, J.D. Lewis, Beilinson’s Hodge conjecture for smooth varieties, J. of K-Theor. [arXiv:1104.4364] (2011)Google Scholar
  16. 16.
    P. Deligne, Théorie de Hodge, II, III. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971); 44, 5–77 (1974)Google Scholar
  17. 17.
    P. Elbaz-Vincent, S. Müller-Stach, Milnor K-theory of rings, higher Chow groups and applications. Invent. Math. 148, 177–206 (2002)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H. Esnault, K.H. Paranjape, Remarks on absolute de Rham and absolute Hodge cycles. C. R. Acad. Sci. Paris t 319, Serie I, 67–72 (1994)Google Scholar
  19. 19.
    H. Esnault, E. Viehweg, Deligne-Beilinson cohomology, in Beilinson’s Conjectures on Special Values of L-Functions, ed. by Rapoport, Schappacher, Schneider. Perspectives in Mathematics, vol. 4 (Academic, New York, 1988), pp. 43–91Google Scholar
  20. 20.
    R. Friedman, R. Laza, Semi-algebraic horizontal subvarieties of Calabi-Yau type, preprint 2011 [arXive:1109.5632v1]Google Scholar
  21. 21.
    B.B. Gordon, J.D. Lewis, Indecomposable higher Chow cycles, in The Arithmetic and Geometry of Algebraic Cycles, Banff, AB, 1998. Nato Science Series C: Mathematical and Physical Sciences, vol. 548 (Kluwer, Dordrecht, 2000), pp. 193–224Google Scholar
  22. 22.
    M. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map. J. Differ. Geom. 29, 545–555 (1989)MATHGoogle Scholar
  23. 23.
    M. Green, P. Griffiths, The regulator map for a general curve, in Symposium in Honor of C.H. Clemens, Salt Lake City, UT, 2000. Contemporary Mathematics, vol. 312 (American Mathematical Society, Providence, 2002), pp. 117–127Google Scholar
  24. 24.
    M. Green, S. Müller-Stach, Algebraic cycles on a general complete intersection of high multi-degree of a smooth projective variety. Comp. Math. 100(3), 305–309 (1996)MATHGoogle Scholar
  25. 25.
    P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)MATHGoogle Scholar
  26. 26.
    P.A. Griffiths, On the periods of certain rational integrals: I and II. Ann. Math. 90, 460–541 (1969)MATHCrossRefGoogle Scholar
  27. 27.
    U. Jannsen, Deligne cohomology, Hodge\(-\mathcal{D}-\)conjecture, and motives, in Beilinson’s Conjectures on Special Values of L-Functions, ed. by Rapoport, Schappacher, Schneider. Perspectives in Mathematics, vol. 4 (Academic, New York, 1988), pp. 305–372Google Scholar
  28. 28.
    U. Jannsen, in Mixed Motives and Algebraic K-Theory. Lecture Notes in Mathematics, vol. 1000 (Springer, Berlin, 1990)Google Scholar
  29. 29.
    B. Kahn, Groupe de Brauer et (2, 1)-cycles indecomposables. Preprint (2011)Google Scholar
  30. 30.
    K. Kato, Milnor K-theory and the Chow group of zero cycles. Contemp. Math. Part I 55, 241–253 (1986)CrossRefGoogle Scholar
  31. 31.
    M. Kerr, J.D. Lewis, The Abel-Jacobi map for higher Chow groups, II. Invent. Math. 170(2), 355–420 (2007)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    M. Kerr, J.D. Lewis, S. Müller-Stach, The Abel-Jacobi map for higher Chow groups. Compos. Math. 142(2), 374–396 (2006)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    M. Kerz, The Gersten conjecture for Milnor K-theory. Invent. Math. 175(1), 1–33 (2009)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    J. King, Log complexes of currents and functorial properties of the Abel-Jacobi map. Duke Math. J. 50(1), 1–53 (1983)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    M. Levine, Localization on singular varieties. Invent. Math. 31, 423–464 (1988)CrossRefGoogle Scholar
  36. 36.
    J.D. Lewis, in A Survey of the Hodge Conjecture, 2nd edn. Appendix B by B. Brent Gordon. CRM Monograph Series, vol. 10 (American Mathematical Society, Providence, 1999), pp. xvi+368Google Scholar
  37. 37.
    J.D. Lewis, Lectures on algebraic cycles. Bol. Soc. Mat. Mexicana (3) 7(2), 137–192 (2001)Google Scholar
  38. 38.
    J.D. Lewis, Real regulators on Milnor complexes. K-Theory 25(3), 277–298 (2002)Google Scholar
  39. 39.
    J.D. Lewis, Regulators of Chow cycles on Calabi-Yau varieties, in Calabi-Yau Varieties and Mirror Symmetry, Toronto, ON, 2001. Fields Institute Communications, vol. 38 (American Mathematical Society, Providence, 2003), pp. 87–117Google Scholar
  40. 40.
    S. Müller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces. J. Algebr. Geom. 6, 513–543 (1997)MATHGoogle Scholar
  41. 41.
    S. Müller-Stach, Algebraic cycle complexes, in Proceedings of the NATO Advanced Study Institute on the Arithmetic and Geometry of Algebraic Cycles, vol. 548, ed. by J.D. Lewis, N. Yui, B. Gordon, S. Müller-Stach, S. Saito (Kluwer, Dordrecht, 2000), pp. 285–305Google Scholar
  42. 42.
    D. Ramakrishnan, Regulators, algebraic cycles, and values of L-functions, in Contemporary Mathematics, vol. 83 (American Mathematical Society, Providence, 1989), pp. 183–310Google Scholar
  43. 43.
    M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math. 71, 207–233 (1983)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    J.H.M. Steenbrink, in A Summary of Mixed Hodge Theory, Motives, Seattle, WA, 1991. Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1 (American Mathematical Society, Providence, 1994), pp. 31–41Google Scholar
  45. 45.
    C. Voisin, The Griffiths group of a general Calabi-Yau threefold is not finitely generated. Duke Math. J. 102(1), 151–186 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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