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Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties

  • Torsten EkedahlEmail author
  • Gerard van der Geer
Chapter
Part of the Progress in Mathematics book series (PM, volume 269)

Summary

We introduce a stratification on the space of symplectic flags on the de Rham bundle of the universal principally polarized abelian variety in positive characteristic. We study its geometric properties, such as irreducibility of the strata, and we calculate the cycle classes. When the characteristic p is treated as a formal variable these classes can be seen as a deformation of the classes of the Schubert varieties for the corresponding classical flag variety (the classical case is recovered by putting p equal to 0). We relate our stratification with the E-O stratification on the moduli space of principally polarized abelian varieties of a fixed dimension and derive properties of the latter. Our results are strongly linked with the combinatorics of the Weyl group of the symplectic group.

Key words

moduli space abelian variety E-O-stratification cycle classes Weyl group 

References

  1. [BGG73]
    I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Schubert cells, and the cohomology of the spaces \(G/P\), Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26.zbMATHMathSciNetGoogle Scholar
  2. [BL00]
    S. Billey, V. Lakshmibai, Singular Loci of Schubert Varieties, Progs. in Math., vol. 182, Birkhäuser Boston Inc., Boston, MA, 2000.Google Scholar
  3. [Ch94]
    C. Chevalley, Sur les décompositions cellulaires des espaces \(G/B\), Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, With a foreword by Armand Borel, pp. 1–23.Google Scholar
  4. [Ek87]
    T. Ekedahl, On supersingular curves and abelian varieties, Math. Scand. 60 (1987), 151–178.zbMATHMathSciNetGoogle Scholar
  5. [FC90]
    G. Faltings, C.-L. Chai, Degeneration of abelian varieties., Ergebnisse der Math., no. 22, Springer-Verlag, 1990.Google Scholar
  6. [FP98]
    W. Fulton, P. Pragacz, Schubert varieties and degeneracy loci, SLN, vol. 1689, Springer-Verlag, Berlin, 1998, Appendix J by the authors in collaboration with I. Ciocan-Fontanine.Google Scholar
  7. [Fu96]
    W. Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom. 43 (1996), no. 2, 276–290.zbMATHMathSciNetGoogle Scholar
  8. [Ful]
    W. Fulton, Intersection Theory, vol. 3, Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 2, Springer-Verlag, 1984.Google Scholar
  9. [Ge99]
    G. van der Geer, Cycles on the moduli space of abelian varieties, Moduli of curves and abelian varieties (C. Faber and E. Looijenga, eds.), Aspects Math., E33, Vieweg, Braunschweig, 1999, pp. 65–89.Google Scholar
  10. [Gr65]
    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965), no. 24, 231.Google Scholar
  11. [GrLie4-6]
    N. Bourbaki, Groupes et algèbres de Lie. IV–VI, Hermann, Paris, 1968.Google Scholar
  12. [HW61]
    H. Hasse, E. Witt, Zyklische unverzweigte Erweiterungskörper von primzahlgrade p über einem algebraischen Funktionenkörper der Charakteristik p, Monatsh. Math. Phys. 43 (1963), 477–492.CrossRefMathSciNetGoogle Scholar
  13. [Ha07]
    S. Harashita, Ekedahl-Oort strata and the first Newton slope strata, J. of Algebraic Geom. 16 (2007), no. 1, 171–199.zbMATHMathSciNetGoogle Scholar
  14. [Jo93]
    A. J. de Jong, The moduli spaces of principally polarized abelian varieties with \(\Gamma\sb 0(p)\)-level structure, J. of Algebraic Geom. 2 (1993), no. 4, 667–688.zbMATHMathSciNetGoogle Scholar
  15. [KS03]
    S. Keel, L. Sadun, Oort's conjecture for \(A\sb g\otimes\mathbb C\), J. Amer. Math. Soc. 16 (2003), no. 4, 887–900.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [KT02]
    A. Kresch, H. Tamvakis, Double Schubert polynomials and degeneracy loci for the classical groups, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1681–1727.zbMATHMathSciNetGoogle Scholar
  17. [M2]
    D. R. Grayson, M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2.
  18. [MB81]
    L. Moret-Bailly, Familles de courbes et variétés abéliennes sur \(\mathbb P^1\), Séminaire sur les pinceaux de courbes de genre au moins deux. (L. Szpiro, ed.), Astérisque, vol. 86, Soc. math. de France, 1981, pp. 109–140.Google Scholar
  19. [MB85]
    L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque (1985), no. 129, 266.Google Scholar
  20. [MW04]
    B. Moonen, T. Wedhorn, Discrete invariants of varieties in positive characteristic, IMRN (2004), no. 72, 3855–3903, Eprint: math.AG/0306339.Google Scholar
  21. [Mo01]
    B. Moonen, Group schemes with additional structures and Weyl group cosets, Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 255–298.Google Scholar
  22. [Mu70]
    D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press., London, 1970.Google Scholar
  23. [Oo01]
    F. Oort, A stratification of a moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 345–416.Google Scholar
  24. [Oo95]
    F. Oort, Complete subvarieties of moduli spaces, Abelian varieties (Egloffstein, 1993), de Gruyter, Berlin, 1995, pp. 225–235.Google Scholar
  25. [PR97]
    P. Pragacz, J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; \(\tilde Q\)-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [PR99]
    H. Pittie, A. Ram, A Pieri-Chevalley formula in the K-theory of a G/B-bundle, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102–107.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [Pr91]
    P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, Topics in invariant theory (Paris, 1989/1990), Lecture Notes in Math., vol. 1478, Springer-Verlag, Berlin, 1991, pp. 130–191.Google Scholar
  28. [RR85]
    S. Ramanan, A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), no. 2, 217–224.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [S]
    N. Sloane, The on-line encyclopedia of integer sequences, http://www.research.att.com/njas/sequences.
  30. [Sj07]
    J. Sjöstrand, Bruhat intervals as rooks on skew Ferrers boards, J. Combin. Theory Ser. A, 114 (2007), 1182–1198.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Matematiska institutionenStockholms universitetStockholmSweden
  2. 2.Korteweg-de Vries InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands

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