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

Citations:

Zbl 1105.14064; Zbl 0994.14012
Full Text:

References:

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