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Projective normality of flag varieties and Schubert varieties. (English) Zbl 0553.14023
We prove in this paper the following results. Let G be a semisimple algebraic group over an algebraically closed field k and Q a parabolic subgroup containing a Borel subgroup B. Let X be a Schubert variety (i.e. the closure of a B orbit) in G/Q. Then (a) If L is a line bundle on G/Q such that $$H^ 0(G/Q,L)\neq 0$$ then $$H^ i(X,L)=0$$ for $$i>0$$ and the restriction map $$H^ 0(G/Q,L)\to H^ 0(X,L)$$ is surjective; (b) X is normal; (c) X is projectively normal in any embedding given by an ample line bundle on G/Q. - If we prove the results for fields of positive characteristic they follow for fields of characteristic zero by semicontinuity. When char k$$=0$$ we have the absolute Frobenius morphism $$F:X\to X$$ defined by raising functions on X to the p-th power. In the preprint ”Frobenius splitting and cohomology vanishing for Schubert varieties” by V. B. Mehta and A. Ramanathan it was shown using duality for the Frobenius morphism of the Bott-Samelson-Demazure variety (constructed in the paper of Demazure cited below) that the p-th power map $$0_ X\to F_*0_ X$$ admits a section. This quickly gives (a) for ample line bundles L. In this paper we extend this method, by a closer examination of the splitting, to the general case of $$H^ 0(G/Q,L)\neq 0$$. We then deduce (b) from (a) by an inductive argument involving the $${\mathbb{P}}^ 1$$-fibrations $$G/B\to G/P$$ for suitable minimal parabolic subgroups P containing B.
These results prove the conjectures of M. Demazure in his paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009). In particular his character formula for $$H^ 0(X,L)$$ for fields of arbitrary characteristic also follows. Incidentally our results uphold the main claims in Demazure’s paper in spite of the falsity of proposition 11, §2 of that paper.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14L30 Group actions on varieties or schemes (quotients)
Zbl 0312.14009
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##### References:
 [1] Andersen, H.H.: The Frobenius morphism on the cohomology of homogeneous vector bundles onG/B. Ann. Math.112, 113-121 (1980) · Zbl 0437.20035 [2] Demazure, M.: Désingularisations des variétés de Schubert généralisés, Ann. E.N.S.7, 53-88 (1974) · Zbl 0312.14009 [3] Kempf, G.: Linear systems on homogeneous spaces. Ann. Math.103, 557-591 (1976) · Zbl 0327.14016 [4] Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. · Zbl 0601.14043 [5] Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. (To appear) · Zbl 0541.14039 [6] Seshadri, C.S.: Line bundles on Schubert varieties. (To appear) · Zbl 0688.14047
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