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On Frobenius splitting of orbit closures of spherical subgroups in flag varieties. (English) Zbl 1264.14066
Let \(G\) be a semisimple algebraic group over an algebraically closed field \(k\) and let \(H\) be a spherical subgroup of \(G\), namely a subgroup acting with finitely many orbits on the flag variety \(G/B\) associated with \(G\). In the present paper, under the assumption that \(H\) is reductive and connected, the authors establish a criterion for \(H\)-orbit closures in \(G/B\) to have nice geometric and cohomological properties. The main tools which are used are Frobenius splitting, introduced by V. B. Mehta and A. Ramanathan [Ann. Math. (2) 122, 27–40 (1985; Zbl 0601.14043)], and global F-regularity, introduced by K. E. Smith [Mich. Math. J. 48, Spec. Vol., 553–572 (2000; Zbl 0994.14012)]. In particular, any Frobenius split variety is weakly normal and, if projective, the higher cohomology groups of ample line bundles are zero, whereas any globally F-regular projective variety is normal, Cohen-Macaulay and the higher cohomology groups of nef line bundles are zero.
More precisely, assume that the characteristic of \(k\) is sufficiently large and let \(B\) be a Borel subgroup of \(G\) intersecting \(H\) in a Borel subgroup of \(H\). Let \(W\) be the Weyl group of \(G\) and, if \(w \in W\), denote by \(X(w) = \overline{BwB/B}\) the corresponding Schubert variety in \(G/B\). Given a parabolic subgroup \(P\) of \(G\) containing \(B\), denote \(W_P\) the corresponding parabolic subgroup of \(W\). The authors deal with the orbit closures of the shape \(\overline{HP/B}\): they give conditions for these varieties to admit a Frobenius splitting along a divisor whose restriction to \(P/B\) is ample that is compatible with all the subvarieties \(\overline{HX(w)}\) with \(w \in W_P\), and they give conditions for these subvarieties to be globally F-regular. Although the subgroup \(H\) occurring in the criterion is not assumed to be spherical, in many cases the relevant orbit closures \(\overline{HX(w)}\) coincide with closures of orbits under spherical subgroups.
As a special case, if \(H\) is trivial then the theorem reduces to the well known results that the flag variety admits a Frobenius splitting along an ample divisor which is compatible with all Schubert varieties, and that any Schubert variety is globally F-regular. Another special case is when \((G,H)\) is in the list of minimal rank pairs of N. Ressayre [Adv. Math. 224, No. 5, 1784–1800 (2010; Zbl 1203.14054)]. In this case, every \(H\)-orbit closure in the flag variety is of the shape \(\overline{HX(w)}\) for some \(w \in W\), and the theorem implies that the flag variety admits a Frobenius splitting along an ample divisor that is compatible with all the \(H\)-orbit closures.
MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
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