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Algebra, Arithmetic, and Geometry pp 567–636Cite as

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

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Part of the book series: Progress in Mathematics ((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.

2000 Mathematics Subject Classifications: 14K, 14G35, 14L

To Yuri Ivanovich Manin on the occasion of his 70th birthday

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References

  1. 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.

    MATH  MathSciNet  Google Scholar 

  2. S. Billey, V. Lakshmibai, Singular Loci of Schubert Varieties, Progs. in Math., vol. 182, Birkhäuser Boston Inc., Boston, MA, 2000.

    Google Scholar 

  3. 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. T. Ekedahl, On supersingular curves and abelian varieties, Math. Scand. 60 (1987), 151–178.

    MATH  MathSciNet  Google Scholar 

  5. G. Faltings, C.-L. Chai, Degeneration of abelian varieties., Ergebnisse der Math., no. 22, Springer-Verlag, 1990.

    Google Scholar 

  6. 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. W. Fulton, Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom. 43 (1996), no. 2, 276–290.

    MATH  MathSciNet  Google Scholar 

  8. W. Fulton, Intersection Theory, vol. 3, Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 2, Springer-Verlag, 1984.

    Google Scholar 

  9. 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. 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. N. Bourbaki, Groupes et algèbres de Lie. IV–VI, Hermann, Paris, 1968.

    Google Scholar 

  12. H. Hasse, E. Witt, Zyklische unverzweigte Erweiterungskörper von primzahlgrade p über einem algebraischen Funktionenkörper der Charakteristik p, Monatsh. Math. Phys. 43 (1963), 477–492.

    Article  MathSciNet  Google Scholar 

  13. S. Harashita, Ekedahl-Oort strata and the first Newton slope strata, J. of Algebraic Geom. 16 (2007), no. 1, 171–199.

    MATH  MathSciNet  Google Scholar 

  14. 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.

    MATH  MathSciNet  Google Scholar 

  15. S. Keel, L. Sadun, Oort's conjecture for \(A\sb g\otimes\mathbb C\), J. Amer. Math. Soc. 16 (2003), no. 4, 887–900.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Kresch, H. Tamvakis, Double Schubert polynomials and degeneracy loci for the classical groups, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1681–1727.

    MATH  MathSciNet  Google Scholar 

  17. 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. 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. L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque (1985), no. 129, 266.

    Google Scholar 

  20. B. Moonen, T. Wedhorn, Discrete invariants of varieties in positive characteristic, IMRN (2004), no. 72, 3855–3903, Eprint: math.AG/0306339.

    Google Scholar 

  21. 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. D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press., London, 1970.

    Google Scholar 

  23. 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. F. Oort, Complete subvarieties of moduli spaces, Abelian varieties (Egloffstein, 1993), de Gruyter, Berlin, 1995, pp. 225–235.

    Google Scholar 

  25. P. Pragacz, J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; \(\tilde Q\)-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.

    Article  MATH  MathSciNet  Google Scholar 

  26. 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.

    Article  MATH  MathSciNet  Google Scholar 

  27. 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. S. Ramanan, A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), no. 2, 217–224.

    Article  MATH  MathSciNet  Google Scholar 

  29. N. Sloane, The on-line encyclopedia of integer sequences, http://www.research.att.com/njas/sequences.

  30. J. Sjöstrand, Bruhat intervals as rooks on skew Ferrers boards, J. Combin. Theory Ser. A, 114 (2007), 1182–1198.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Matematiska institutionen, Stockholms universitet, SE-106 91, Stockholm, Sweden

    Torsten Ekedahl

  2. Korteweg-de Vries Instituut, Universiteit van Amsterdam, Postbus 94248, 1090 GE, Amsterdam, The Netherlands

    Gerard van der Geer

Authors
  1. Torsten Ekedahl
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  2. Gerard van der Geer
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Correspondence to Torsten Ekedahl .

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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

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Ekedahl, T., van der Geer, G. (2009). Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_13

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