×

zbMATH — the first resource for mathematics

Quotients of group completions by spherical subgroups. (English) Zbl 1052.14061
The author is concerned with describing projective embeddings of \(G/H\), where \(G\) is a semi-simple algebraic group and \(H\) is a spherical subgroup. The author exploits knowledge of \(G\times G\)-equivariant embeddings \(X\) of \(G\), and in fact a main thrust of the article is to determine a condition under which a projective embedding \(Y\) of \(G/H\) is a quotient of \(X\) by \(H\) (or rather a quotient by \(H\) of an open subset of \(X\), denoted \(X^{ss}({\mathcal L})\), which is determined by a \(G\times H\)-linearized line bundle over \(X\)).
The condition for having \(Y=X^{ss}({\mathcal L})//H\) requires the existence of a \(G\times G\)-equivariant embedding \(X_{{\mathcal O}}\) together with an equivariant surjective rational mapping \(\phi: X_{{\mathcal O}}\rightarrow G/H \cup {\mathcal O}\) for each \(G\) orbit \({\mathcal O}\) of codimension one in \(Y\). The author points out that the quotient \(X^{ss}({\mathcal L})//H\) is not a space of orbits in general. However, when the index of \(H\) in its normalizer is finite, together with other conditions, it is shown that there is a surjective, \(G\)-equivariant mapping \(\pi:X^{ss}({\mathcal L})\rightarrow X^{ss}({\mathcal L})//H\) whose fibers are the orbits of \(\{1\}\times H\).

MSC:
14M17 Homogeneous spaces and generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brion, M., The behaviour at infinity of the Bruhat decomposition, Comment. math. helv., 73, 1, 137-174, (1998) · Zbl 0935.14029
[2] de Concini, C.; Procesi, C., Complete symmetric varieties, (), 1-44 · Zbl 0581.14041
[3] Littelmann, P.; Procesi, C., Equivariant cohomology of wonderful compactifications, (), 219-262 · Zbl 0741.14029
[4] Bifet, E.; de Concini, C.; Procesi, C., Cohomology of regular embeddings, Adv. math., 82, 1, 1-34, (1990) · Zbl 0743.14018
[5] Ressayre, N., Variations de quotients Géométriques et applications, Thèse, Université Grenoble I, 2000
[6] Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory, (1994), Springer-Verlag New York · Zbl 0797.14004
[7] Luna, D.; Vust, T., Plongements d’espaces homogènes, Comment. math. helv., 58, 2, 186-245, (1983) · Zbl 0545.14010
[8] Knop, F., The luna – vust theory of spherical embeddings, (), 225-249 · Zbl 0812.20023
[9] Brion, M., Variétés sphériques, (), 1-60
[10] Brion, M.; Procesi, C., Action d’un tore dans une variété projective, (), 509-539 · Zbl 0741.14028
[11] Brion, M., Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke math. J., 58, 2, 397-424, (1989) · Zbl 0701.14052
[12] Knop, F.; Kraft, H.; Vust, T., The Picard group of a G-variety, (), 77-87
[13] Grosshans, J.D., Algebraic homogeneous spaces and invariant theory, Lect. notes in math., 1673, (1997), Springer-Verlag New York
[14] Bourbaki, N.; Bourbaki, N., Valuations, (), Chapitre 6 · Zbl 0205.34302
[15] Popov, V.L.; Vinberg, È.B., Invariant theory, (), 137-314, 315 (in Russian) · Zbl 0789.14008
[16] Bourbaki, N.; Bourbaki, N., Idéaux premiers associés et décomposition primaire, (), Chapitre 4 · Zbl 0547.13001
[17] Hartshorne, R., Algebraic geometry, (1977), Springer-Verlag New York · Zbl 0367.14001
[18] Renner, L.E., Reductive embeddings, (), 175-192
[19] Kannan, S., Remarks on the wonderful compactification of semisimple algebraic groups, Proc. Indian acad. sci. math. sci., 109, 3, 241-256, (1999) · Zbl 0946.14024
[20] Fulton, W., Introduction to toric varieties, (1993), Princeton University Press NJ
[21] Oda, T., Convex bodies and algebraic geometry. an introduction to the theory of toric varieties, (1988), Springer-Verlag Berlin, translated from the Japanese · Zbl 0628.52002
[22] Bourbaki, N., Systemes de racines, (), Chapitre VI · Zbl 0186.33001
[23] Brion, M., Vers une généralisation des espaces symétriques, J. algebra, 134, 1, 115-143, (1990) · Zbl 0729.14038
[24] Luna, D., Toute variété magnifique est sphérique, Transform. groups, 1, 3, 249-258, (1996) · Zbl 0912.14017
[25] Wasserman, B., Wonderful varieties of rank two, Transform. groups, 1, 4, 375-403, (1996) · Zbl 0921.14031
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.