zbMATH — the first resource for mathematics

Flag varieties as equivariant compactifications of \(\mathbb{G}_{a}^{n}\). (English) Zbl 1217.14032
The author classifies all (generalized) flag varieties \(G/P\) which can be realized as an equivariant compactification of the commutative unipotent affine algebraic group \(\mathbb{G}_a^n\), i.e. \(\mathbb{G}_a^n\) acts on \(G/P\) with an open orbit. Here, \(G\) is a connected semisimple linear group of adjoint type over an algebraically closed field of characteristic zero and \(P\) is a parabolic subgroup of \(G\).
Note that, given any \(G/P\), the unipotent radical \(P_u^-\) of the opposite of \(P\) acts with an open orbit on \(G/P\). Moreover, \(P_u^-\) and such an orbit are affine spaces; in particular they are isomorphic to \(\mathbb{G}_a^n\) as varieties (with \(n=\dim \,G/P\)). So when \(P_u^-\) is commutative, it is isomorphic to \(\mathbb{G}_a^n\) and the desired action exists.
Suppose now that \(G\) is simple. There is another natural class of flag varieties with an action of \(\mathbb{G}_a^n\) as desired. Indeed, there are some \(G/P\) such that \(G\) is strictly contained in the connected component \(\widetilde{G}\) of the automorphism group of \(G/P\). If we write \(G/P\cong \widetilde{G}/\widetilde{P}\), then \(\widetilde{P}^-_u\) is commutative while \(P^-_u\) is not. So, \(G/P\) has an action of \(\mathbb{G}_a^n\) with an open orbit but \(\mathbb{G}_a^n\) is not isomorphic to \(P^-_u\).
The author proves that the only \(G/P\) with an action of \(\mathbb{G}_a^n\) as requested are products of flag varieties in the previous two classes. Finally, for fixed \(G/P\), there can exist infinitely many equivalence classes of actions of \(\mathbb{G}_a^n\) on \(G/P\) with an open orbit.

14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
Full Text: DOI
[1] I. V. Arzhantsev and E. V. Sharoyko, Hassett-Tschinkel correspondence: modality and projective hypersurfaces, arXiv:0912.1474 [math.AG]. · Zbl 1248.14053
[2] 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, No. 1337, Hermann, Paris, 1968 (French). · Zbl 0186.33001
[3] M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179 – 186. · Zbl 0406.14030 · doi:10.1007/BF01390108 · doi.org
[4] Brendan Hassett and Yuri Tschinkel, Geometry of equivariant compactifications of \?\?\(^{n}\), Internat. Math. Res. Notices 22 (1999), 1211 – 1230. · Zbl 0966.14033 · doi:10.1155/S1073792899000665 · doi.org
[5] 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. · Zbl 1137.14036
[6] A. L. Oniščik, On compact Lie groups transitive on certain manifolds, Soviet Math. Dokl. 1 (1960), 1288 – 1291. · Zbl 0098.36602
[7] A. L. Oniščik, Inclusion relations between transitive compact transformation groups, Trudy Moskov. Mat. Obšč. 11 (1962), 199 – 242 (Russian). · Zbl 0192.12601
[8] Arkadi L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. · Zbl 0796.57001
[9] Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649 – 671. · Zbl 0786.20029 · doi:10.1007/BF01231348 · doi.org
[10] 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. · Zbl 1205.13030 · doi:10.1070/SM2009v200n11ABEH004056 · doi.org
[11] Перестановочные матрицы., Наука и Техника, Минск, 1966 (Руссиан). · Zbl 1103.16302
[12] J. Tits, Espaces homogènes complexes compacts, Comment. Math. Helv. 37 (1962/1963), 111 – 120 (French). · Zbl 0108.36302 · doi:10.1007/BF02566965 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.