Abstract
Proper homogeneous G-spaces (where G is semisimple algebraic group) over positive characteristic fields k can be divided into two classes, the first one being the flag varieties G/P and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the D -affinity of Beilinson and Bernstein fails for varieties of unseparated flags.
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References
Andersen, H. H.: Review of[13], Math. Rev. # 22094 58 (1979).
Beilinson, B. and Bernstein, J.: Localisation de g-modules, C. R. Acad. Sci. Paris292 (1981) 15–18.
Deligne, P. and Illusie, L.: Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math.89 (1987) 247–270.
Demazure, M.: Invariants symmetriques entiers des groupes de Weyl et torsion, Invent. Math.21 (1973) 287–301.
Ibid.: Désingularisation des variétés de Schubert généralisées, Ann. scient. Éc. Norm. Sup.7 (1974) 52–88.
Demazure, M.: and Gabriel, P: Groupes Algébriques, North-Holland, Amsterdam, 1970.
Fulton, W.: Intersection Theory, Springer-Verlag, 1984.
Haastert, B.: Über Differentialoperatoren und D-Moduln in positiver Characteristik, Manus. math.58 (1987) 385–415.
Haboush, W. J. and Lauritzen, N.: Varieties of unseparated flags, Linear Algebraic Groups and Their Representations(Richard S. Elman, ed.), Contemp. Math., vol. 153, Amer. Math. Soc., 1993 pp. 35–57.
Jantzen, J. C.: Representations of Algebraic Groups, Academic Press, 1987.
Lauritzen, N.: Embeddings of homogeneous spaces in prime characteristics, Amer. J. Math. 118 (1996), no. 2, 377–387.
Ibid.: The Euler characteristic of a homogeneous line bundle, C. R. Acad. Sci. Paris315 (1992) 715–718.
Marlin, R.: Cohomologie de de Rham des variétés des drapeaux, Bull. Soc. math. France105 (1977) 89–96.
Srinivas, V. Gysin maps and cycle classes for Hodge cohomology, Proc. Ind. Acad. Sci.103 (1993) 209–247.
Wenzel, C.: Classification of all parabolic subgroup schemes of a reductive linear algebraic group over an algebraically closed field, Trans. Amer. Math. Soc.337 (1993) 211–218.
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LAURITZEN, N. Schubert cycles, differential forms and D-modules on varieties of unseparated flags. Compositio Mathematica 109, 1–11 (1997). https://doi.org/10.1023/A:1000117902922
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DOI: https://doi.org/10.1023/A:1000117902922