Flag varieties as equivariant compactifications of $\mathbb {G}_{a}^{n}$
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- by Ivan V. Arzhantsev PDF
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Abstract:
Let $G$ be a semisimple affine algebraic group and $P$ a parabolic subgroup of $G$. We classify all flag varieties $G/P$ which admit an action of the commutative unipotent group $\mathbb {G}_{a}^{n}$ with an open orbit.References
- I. V. Arzhantsev and E. V. Sharoyko, Hassett-Tschinkel correspondence: modality and projective hypersurfaces, arXiv:0912.1474 [math.AG].
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179–186. MR 435092, DOI 10.1007/BF01390108
- Brendan Hassett and Yuri Tschinkel, Geometry of equivariant compactifications of $\textbf {G}_a^n$, Internat. Math. Res. Notices 22 (1999), 1211–1230. MR 1731473, DOI 10.1155/S1073792899000665
- Venkatramani Lakshmibai and Komaranapuram N. Raghavan, Standard monomial theory, Encyclopaedia of Mathematical Sciences, vol. 137, Springer-Verlag, Berlin, 2008. Invariant theoretic approach; Invariant Theory and Algebraic Transformation Groups, 8. MR 2388163
- A. L. Oniščik, On compact Lie groups transitive on certain manifolds, Soviet Math. Dokl. 1 (1960), 1288–1291. MR 0150238
- A. L. Oniščik, Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč. 11 (1962), 199–242 (Russian). MR 0153779
- Arkadi L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. MR 1266842
- Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649–671. MR 1189494, DOI 10.1007/BF01231348
- E. V. Sharoĭko, The Hassett-Tschinkel correspondence and automorphisms of a quadric, Mat. Sb. 200 (2009), no. 11, 145–160 (Russian, with Russian summary); English transl., Sb. Math. 200 (2009), no. 11-12, 1715–1729. MR 2590000, DOI 10.1070/SM2009v200n11ABEH004056
- D. A. Suprunenko and R. I. Tyškevič, Perestanovochnye matritsy, “Nauka i Tekhnika”, Minsk, 1966 (Russian). MR 0201472
- J. Tits, Espaces homogènes complexes compacts, Comment. Math. Helv. 37 (1962/63), 111–120 (French). MR 154299, DOI 10.1007/BF02566965
Additional Information
- Ivan V. Arzhantsev
- Affiliation: Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia
- MR Author ID: 359575
- Email: arjantse@mccme.ru
- Received by editor(s): March 14, 2010
- Published electronically: October 22, 2010
- Additional Notes: The author was supported by RFBR Grants 09-01-00648-a, 09-01-90416-Ukr-f-a, and the Deligne 2004 Balzan Prize in Mathematics.
- Communicated by: Harm Derksen
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 783-786
- MSC (2010): Primary 14M15; Secondary 14L30
- DOI: https://doi.org/10.1090/S0002-9939-2010-10723-2
- MathSciNet review: 2745631