The Frobenius morphism of Schubert schemes
\textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math., II. Ser. 122, 27-40 (1985; Zbl 0601.14043)] proved that the flag variety is Frobenius split with all its Schubert subvarieties compatibly split. As a consequence they obtained simple proofs of Kempf's and Demazure's vanishing theorems. \textit{S. Ramanan} and \textit{A. Ramanathan} [Invent. Math. 79, 217-224 (1985; Zbl 0553.14023)] refined the theory by introducing \({\mathcal L}\)-split varieties, where \({\mathcal L}\) is an effective invertible sheaf. The purpose of the present article is to give a new proof, using representation theory of algebraic groups, of Ramanan and Ramanathan's result that the Schubert subvarieties of a flag variety are all Frobenius \({\mathcal L}\)-split. A special feature of the proof is that it avoids the Demazure-Hansen desingularization.
scientific article; zbMATH DE number 783872