Duke Mathematical Journal

Standard conjectures for the arithmetic Grassmannian G(2,N) and Racah polynomials

Andrew Kresch and Harry Tamvakis

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G(2,N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials.

Article information

Source
Duke Math. J., Volume 110, Number 2 (2001), 359-376.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087574861

Digital Object Identifier
doi:10.1215/S0012-7094-01-11027-2

Mathematical Reviews number (MathSciNet)
MR1865245

Zentralblatt MATH identifier
1072.14514

Subjects
Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Citation

Kresch, Andrew; Tamvakis, Harry. Standard conjectures for the arithmetic Grassmannian G (2, N ) and Racah polynomials. Duke Math. J. 110 (2001), no. 2, 359--376. doi:10.1215/S0012-7094-01-11027-2. https://projecteuclid.org/euclid.dmj/1087574861


Export citation

References

  • R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6$-$j$ symbols, SIAM J. Math. Anal. 10 (1979), 1008--1016.
  • L. C. Biedenharn and J. D. Louck, The Racah-Wigner Algebra in Quantum Theory, Encyclopedia Math. Appl. 9, Addison-Wesley, Reading, Mass., 1981.
  • J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903--1027.
  • G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), 387--424.
  • W. Fulton, Intersection Theory, 2d ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • H. Gillet and C. Soulé, ``Arithmetic analogs of the standard conjectures'' in Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, 1994, 129--140.
  • A. Grothendieck, ``Standard conjectures on algebraic cycles'' in Algebraic Geometry (Bombay, 1968), Oxford Univ. Press, London, 1969, 193--199.
  • H. Hochstadt, The Functions of Mathematical Physics, Pure Appl. Math. 23, Wiley-Interscience, New York, 1971.
  • P. Hriljac, Heights and Arakelov's intersection theory, Amer. J. Math. 107 (1985), 23--38.
  • S. L. Kleiman, ``The standard conjectures'' in Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, 1994, 3--20.
  • K. Künnemann, Some remarks on the arithmetic Hodge index conjecture, Compositio Math. 99 (1995), 109--128.
  • K. Künnemann and V. Maillot, ``Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire'' in Regulators in Analysis, Geometry and Number Theory, Progr. Math. 171, Birkhäuser, Boston, 2000, 197--205.
  • K. Künnemann and H. Tamvakis, The Hodge star operator on Schubert forms, to appear in Topology.
  • A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (in Russian), ``Nauka,'' Moscow, 1985, ; English translation in Springer Ser. Comput. Phys., Springer, Berlin, 1991.
  • G. Racah, Theory of complex spectra, II, Physical Rev. 62 (1942), 438--462.
  • R. Roy, Binomial identities and hypergeometric series, Amer. Math. Monthly 94 (1987), 36--46.
  • C. Soulé, ``Hermitian vector bundles on arithmetic varieties'' in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, 1997, 383--419.
  • R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge Univ. Press, Cambridge, 1999.
  • G. Szegö, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, 1975.
  • Y. Takeda, A relation between standard conjectures and their arithmetic analogues, Kodai Math. J. 21 (1998), 249--258.
  • H. Tamvakis, Schubert calculus on the arithmetic Grassmannian, Duke Math. J. 98 (1999), 421--443.
  • N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and Special Functions, Vol. 1: Simplest Lie Groups, Special Functions and Integral Transforms, trans. V. A. Groza and A. A. Groza, Math. Appl. (Soviet Ser.) 72, Kluwer, Dordrecht, 1991.
  • F. J. W. Whipple, Well-poised series and other generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1926), 525--544.
  • J. A. Wilson, Hypergeometric series, recurrence relations and some new orthogonal functions, Ph.D. thesis, University of Wisconsin, Madison, 1978.
  • S. Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), 187--221.