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Semistable sheaves on homogeneous spaces and abelian varieties

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Abstract

In this paper we prove that semistable sheaves with zero Chern classes on homogeneous spaces are trivial and semistable sheaves on abelian varieties with zero Chern classes are filtered by line bundles numerically equivalent to zero. The method consists in reducing modp and then showing that the Frobenius morphism preserves semistability on the above class of varieties. For technical reasons, we have to assume boundedness of semistable sheaves in charp.

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References

  1. Atiyah M 1957Proc. London. Math. Soc. 7 414

    Article  MATH  MathSciNet  Google Scholar 

  2. Barth W 1977Math. Ann. 226 125

    Article  MATH  MathSciNet  Google Scholar 

  3. Demazure M 1974Ann. Sci. Ec. Norm. Super t.7 53

    MATH  MathSciNet  Google Scholar 

  4. Ellencwajg G and Forster OBounding cohomology groups of vector bundles on P n, preprint

  5. Forster O, Hirschowitz A and Schneider M 1979Type de Scindage Generalise pour les fibres stables, preprint

  6. Gieseker D 1977Ann. Math. 106 45

    Article  MathSciNet  Google Scholar 

  7. Gieseker D 1973Ann. Sci. Ec. Norm. Super t.6 95

    MATH  MathSciNet  Google Scholar 

  8. Grothendieck A 1960/61 Techniques de construction et theorems d'existence en geometrie algebrique IV, les schemes de Hilbert, Seminaire Bourbake, No. 221

  9. Grothendieck A 1959/60 Technique de descente et theorems d'existence en geometrie algebrique I, Seminaire Bourbake No. 190

  10. Kleiman S 1968Algebraic cycles and the Weil conjectures in Dix exposes sur la cohmologie des schemas (Amsterdam: North Holland)

    Google Scholar 

  11. Lange H and Stuhler U 1917Math. Z. 156 73

    Article  MathSciNet  Google Scholar 

  12. Langton S 1975Ann. Math. 101 88

    Article  MathSciNet  Google Scholar 

  13. Maruyama M 1978J. Math. Kyoto Univ. 17 91

    MathSciNet  Google Scholar 

  14. Maruyama M 1980Nagoya Math. J. 78 65

    MATH  MathSciNet  Google Scholar 

  15. Maruyama M 1980On boundedness of families of torsion free sheaves, preprint

  16. Maruyama M 1978J. Math. Kyoto Univ. 18 557

    MATH  MathSciNet  Google Scholar 

  17. Mehta V B and Ramanathan A 1980An extension of Langton's theorem on valuative criteria for vector bundles, preprint

  18. Mumford D 1970Abelian varieties (Bombay: Oxford University Press)

    MATH  Google Scholar 

  19. Nori M V 1980The fundamental group-scheme, Ph.D. Thesis Bombay University, Bombay

    Google Scholar 

  20. Peskine C and Szpiro L 1973Publ. Math. I.H.E.X. 42 Paris

  21. Spindler H 1979Math. Ann. 243 131

    Article  MATH  MathSciNet  Google Scholar 

  22. Takemoto F 1972Nagoya Math. J. 47 29

    MATH  MathSciNet  Google Scholar 

  23. Miyanishi M 1973Some remarks on algebraic homogeneous vector bundles, number theory, algebraic geometry and commutative algebra in honour of Y Akizusi, Tokyo

  24. Mukai S 1978J. Math. Kyoto Univ. 18 239

    MATH  MathSciNet  Google Scholar 

  25. Mukai S 1981Nagoya Math. J. 81 153

    MATH  MathSciNet  Google Scholar 

  26. Ramanan S and Ramanathan A 1980Some remarks on the instability flag, preprint

  27. Ein L 1980Math. Ann. 254 53

    Article  MATH  MathSciNet  Google Scholar 

  28. Schwarzenberger R L E 1961Proc. London Math. Soc. 11 623

    Article  MATH  MathSciNet  Google Scholar 

  29. Umemura H 1975Nagoya Math. J. 59 107

    MATH  MathSciNet  Google Scholar 

  30. Oda T 1971Invent. Math. 13 247

    Article  MATH  MathSciNet  Google Scholar 

  31. Mehta V B and Nori M V 1980Translation invariant and semistable sheaves on abelian varieties, preprint

  32. Narasimhan M S and Ramanan S 1975J. Indian Math. Soc. 39 1

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005, Bombay, India

    V B Mehta & M V Nori

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  1. V B Mehta
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  2. M V Nori
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Mehta, V.B., Nori, M.V. Semistable sheaves on homogeneous spaces and abelian varieties. Proc. Indian Acad. Sci. (Math. Sci.) 93, 1–12 (1984). https://doi.org/10.1007/BF02861830

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  • DOI: https://doi.org/10.1007/BF02861830

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