Abstract
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 P ⊂ G 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.
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References
Atiyah, M.F.: On the Krull-Schmidt theorem with application to sheaves. Bull. Soc. Math. France 84, 307–317 (1956)
Azad, H., Biswas, I.: A note on the tangent bundle of G/P. Proc. Indian Acad. Sci. Math. Sci. 120, 69–71 (2010)
Besse, A.L.: Einstein Manifolds. Reprint of the 1987 Edition. Classics in Mathematics. Springer, Berlin (2008)
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)
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 Pressley
Brion, M.: Spherical varieties, notes of a course available at https://www-fourier.ujf-grenoble.fr/~mbrion/notes_bremen.pdf
Carter, R.W.: Simple Groups of Lie Type, vol. 28. Wiley, London (1972). Pure and applied mathematics
Chaput, P.-E., Romagny, M.: On the adjoint quotient of Chevalley groups over arbitrary base schemes. J. Inst. Math. Jussieu 9, 673–704 (2010)
Demazure, M.: automorphismes et déformations des variétés de Borel. Invent. Math. 39, 179–186 (1977)
Jantzen, J.C.: Representations of algebraic groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence (2003)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves Aspects of Mathematics, vol. E31. Friedr. Vieweg & Sohn, Braunschweig (1997)
Katz, N.: Nilpotent Connections and the Monodromy Theorem Applications of a Result of Turrittin. Inst. Hautes Études Sci. Publ. Math. 39, 175–232 (1970)
Koszul, J.-L.: Sur la forme hermitienne canonique des espaces homogènes complexes. Canad. J. Math. 7, 562–576 (1955)
Lange, H., Pauly, C.: On Frobenius-destabilized rank-2 vector bundles over curves. Comment. Math. Helv. 83, 179–209 (2008)
Ramanan, S.: Holomorphic vector bundles on homogeneous spaces. Topology 5, 159–177 (1966)
Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tohoku Math. Jour. 36, 269–291 (1984)
Acknowledgements
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.
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Presented by Michel Brion.
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Biswas, I., Chaput, PE. & Mourougane, C. Stability of the Tangent Bundle of G/P in Positive Characteristics. Algebr Represent Theor 22, 239–247 (2019). https://doi.org/10.1007/s10468-018-9764-x
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DOI: https://doi.org/10.1007/s10468-018-9764-x