Algebras and Representation Theory

, Volume 22, Issue 1, pp 239–247 | Cite as

Stability of the Tangent Bundle of G/P in Positive Characteristics

  • Indranil BiswasEmail author
  • Pierre-Emmanuel Chaput
  • Christophe Mourougane


Let G be an almost simple simply-connected affine algebraic group over an algebraically closed field k of characteristic p >  0. If G has type Bn, Cn or F4, we assume that p >  2, and if G has type G2, we assume that p >  3. Let PG be a parabolic subgroup. We prove that the tangent bundle of G/P is Frobenius stable with respect to the anticanonical polarization on G/P.


Rational homogeneous space Tangent bundle Stability Frobenius 

Mathematics Subject Classification (2010)

14M17 14G17 14J60 


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We are very grateful to G. Ottaviani for pointing out an error in a previous version. He also brought [4] to our attention. The second and third authors thank the Tata Institute of Fundamental Research, while the first author thanks Institut de Mathématiques de Jussieu for hospitality during various stages of this work.


  1. 1.
    Atiyah, M.F.: On the Krull-Schmidt theorem with application to sheaves. Bull. Soc. Math. France 84, 307–317 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Azad, H., Biswas, I.: A note on the tangent bundle of G/P. Proc. Indian Acad. Sci. Math. Sci. 120, 69–71 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein Manifolds. Reprint of the 1987 Edition. Classics in Mathematics. Springer, Berlin (2008)Google Scholar
  4. 4.
    Boralevi, A.: On Simplicity and Stability of the Tangent Bundle of Rational Homogeneous Varieties. Geometric Methods in Representation Theory. II, 275–297, Sémin. Congr., 24-II, Soc. Math., France (2012)Google Scholar
  5. 5.
    Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 6–9. Elements of Mathematics (Berlin). Springer, Berlin (2005). Translated from the 1975 and 1982 French originals by Andrew PressleyzbMATHGoogle Scholar
  6. 6.
    Brion, M.: Spherical varieties, notes of a course available at
  7. 7.
    Carter, R.W.: Simple Groups of Lie Type, vol. 28. Wiley, London (1972). Pure and applied mathematicsGoogle Scholar
  8. 8.
    Chaput, P.-E., Romagny, M.: On the adjoint quotient of Chevalley groups over arbitrary base schemes. J. Inst. Math. Jussieu 9, 673–704 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Demazure, M.: automorphismes et déformations des variétés de Borel. Invent. Math. 39, 179–186 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jantzen, J.C.: Representations of algebraic groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence (2003)Google Scholar
  11. 11.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves Aspects of Mathematics, vol. E31. Friedr. Vieweg & Sohn, Braunschweig (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Katz, N.: Nilpotent Connections and the Monodromy Theorem Applications of a Result of Turrittin. Inst. Hautes Études Sci. Publ. Math. 39, 175–232 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Koszul, J.-L.: Sur la forme hermitienne canonique des espaces homogènes complexes. Canad. J. Math. 7, 562–576 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lange, H., Pauly, C.: On Frobenius-destabilized rank-2 vector bundles over curves. Comment. Math. Helv. 83, 179–209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ramanan, S.: Holomorphic vector bundles on homogeneous spaces. Topology 5, 159–177 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tohoku Math. Jour. 36, 269–291 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Indranil Biswas
    • 1
    Email author
  • Pierre-Emmanuel Chaput
    • 2
  • Christophe Mourougane
    • 3
  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia
  2. 2.Domaine Scientifique Victor Grignard, 239, Boulevard des AiguillettesUniversité Henri Poincaré Nancy 1Vandoeuvre-lès-Nancy CédexFrance
  3. 3.Département de Mathématiques, Campus de Beaulieu, bât. 22-23Université de Rennes 1Rennes CédexFrance

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