## Schubert varieties and Demazure’s character formula.(English)Zbl 0591.14036

Let G be a semi-simple algebraic group over a field k (of any characteristic), B its Borel subgroup. The author studies Schubert varieties in the flag variety G/B and proves the following theorem: Let X be a Schubert variety in G/B and $${\mathcal L}^ a$$line bundle of G/B with $$H^ 0(G/B,{\mathcal L})\neq 0$$. Then $$H^ i(X,{\mathcal L})=0$$, for all $$i>0$$ and the restriction map $$H^ 0(G/B,{\mathcal L})\to H^ 0(X,{\mathcal L})$$ is surjective.
From this follows, that, if char k$$=0$$, all the Schubert varieties have rational singularities, are normal and Cohen-Macaulay. By using the theorem above, the author gives the proof of the Demazure character formula in arbitrary characteristic.
Reviewer: S.Priščepionok

### MSC:

 14M17 Homogeneous spaces and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds 20G10 Cohomology theory for linear algebraic groups
Full Text:

### References:

 [1] Andersen, H.H.: Vanishing theorems and induced representations. J. Algebra62, 86-100 (1980) · Zbl 0439.20025 [2] Andersen, H.H.: The Frobenius morphism on the cohomology of homogeneous vector bundles onG/B. Ann. of Math.112, 113-121 (1980) · Zbl 0437.20035 [3] Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup,7, 53-88 (1974) · Zbl 0312.14009 [4] Demazure, M.: Une nouvelle formule des caractères. Bull. Sci. Math.98, 163-172 (1974) · Zbl 0365.17005 [5] Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math.11, 263-292 (1970) · Zbl 0202.07602 [6] Hansen, H.C.: On cycles on flag manifolds. Math. Scand.33, 269-274 (1973) · Zbl 0301.14019 [7] Joseph, A.: On the Demazure character formula. Ann. Sci. École Norm. Sup. (in press) · Zbl 0589.22014 [8] Kempf, G.: Linear systems on homogeneous spaces. Ann. of Math.103, 557-591 (1976) · Zbl 0327.14016 [9] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lect. Notes Math., Vol. 339, Berlin-Heidelberg-New York: Springer 1973 · Zbl 0271.14017 [10] Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. (In press) · Zbl 0601.14043 [11] Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. Math.79, 217-224 (1985) · Zbl 0553.14023 [12] Seshadri, C.S.: Line bundles on Schubert varieties. (In press) · Zbl 0688.14047
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.