×

Geometry of \(B\times B\)-orbit closures in equivariant embeddings. (English) Zbl 1133.14053

Let \(G\) be a connected reductive group over an algebraically closed field. An equivariant embedding is a normal \(G\times G\) variety which contains an open subset which is \(G\times G\)-equvarariantly isomorphic to \(G\) (where the \(G\times G\)-action is given by left and right translation). Let \(X\) be an equivariant embedding. In this article, the authors study the geometry of \(B\times B\) orbit closures in \(X\), where \(B\) is a Borel subgroup of \(G\). These varieties generalize toric varieties and Schubert varieties. It is shown that \(B\times B\)-orbit closures are normal and Cohen-Macauley. Also when the underlying field is of characteristic zero, then these orbit closures have rational singularities. These results generalise previous results of the second author with M. Brion and J. F. Thomsen [Am. J. Math. 128, 949–962 (2006; Zbl 1105.14064)], where the case of large Schubert varieties was treated. As in the previous article, the main tool is to use the concept of global \(F\)-regularity introduced by K. E. Smith [Mich. Math. J. 48, 553–572 (2000; Zbl 0994.14012)] for varieties over a field of finite characteristic. Globally \(F\)-regular varieties are Cohen-Macauley and normal. In this paper, it is shown that every \(B\times B\)-orbit closure in a projective embedding \(X\) of a reductive group is globally \(F\)-regular.

MSC:

14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
14B05 Singularities in algebraic geometry
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Appel, K., Standard monomials for wonderful group compactifications · Zbl 1120.20051
[2] Brion, M.; Kumar, S., Frobenius splittings methods in geometry and representation theory, Progr. math., (2004), Birkhäuser Boston
[3] Brion, M., The behaviour at infinity of the Bruhat decomposition, Comment. math. helv., 73, 137-174, (1998) · Zbl 0935.14029
[4] Brion, M., Multiplicity-free subvarieties of flag varieties, (), 13-23 · Zbl 1052.14055
[5] Brion, M.; Polo, P., Large Schubert varieties, Represent. theory, 4, 97-126, (2000) · Zbl 0947.14026
[6] Brion, M.; Thomsen, J.F., F-regularity of large Schubert varieties, Amer. J. math., 128, 949-962, (2006) · Zbl 1105.14064
[7] Demazure, M.; Gabriel, P., Groupes algébriques. tome I: Géométrie algébrique, généralités, groupes commutatifs, (1970), Masson & Cie · Zbl 0203.23401
[8] Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture notes in math., vol. 156, (1970), Springer-Verlag · Zbl 0208.48901
[9] He, X., The G-stable pieces of the wonderful compactification, Trans. amer. math. soc., 359, 3005-3024, (2007) · Zbl 1124.20033
[10] Hochster, M.; Huneke, C., Tight closure and strong F-regularity, Mem. soc. math. fr., 38, 119-133, (1989) · Zbl 0699.13003
[11] Hochster, M.; Huneke, C., F-regularity, test elements, and smooth base change, Trans. amer. math. soc., 346, 1-62, (1994) · Zbl 0844.13002
[12] M. Hochster, C. Huneke, Tight closure in equal characteristic zero, preprint · Zbl 0741.13002
[13] Lyubeznik, G.; Smith, K.E., Weak and strong F-regularity are equivalent in graded rings, Amer. math. soc., 121, 1279-1290, (1999) · Zbl 0970.13003
[14] Lauritzen, N.; Thomsen, J.F., Line bundles on Bott-Samelson varieties, J. algebraic geom., 13, 461-473, (2004) · Zbl 1080.14056
[15] Lauritzen, N.; Pedersen, U.R.; Thomsen, J.F., Global F-regularity of Schubert varieties with applications to D-modules, J. amer. math. soc., 19, 345-355, (2006) · Zbl 1098.14038
[16] Ramanathan, A., Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. hautes études sci. publ. math., 65, 61-90, (1987) · Zbl 0634.14035
[17] Smith, K.E., F-rational rings have rational singularities, Amer. J. math., 119, 159-180, (1997) · Zbl 0910.13004
[18] Smith, K.E., Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan math. J., 48, 553-572, (2000) · Zbl 0994.14012
[19] Springer, T.A., Intersection cohomology of \(B \times B\)-orbits closures in group compactifications, J. algebra, 258, 71-111, (2002) · Zbl 1110.14047
[20] Thomsen, J.F., Frobenius splitting of equivariant closures of regular conjugacy classes, Proc. London math. soc. (3), 93, 570-592, (2006) · Zbl 1111.14051
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.