The Frobenius morphism of Schubert schemes
type: Item
instance of: scholarly article
title: The Frobenius morphism of Schubert schemes
zbMATH Open document ID: 0867.14023
author: Masaharu Kaneda
published in: Journal of Algebra
publication date: 1997-08-10T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 783872
zbMATH Keywords: Schubert varieties
zbMATH Keywords: Frobenius splitting
zbMATH Keywords: vanishing theorem
zbMATH Keywords: split varieties
zbMATH Keywords: flag variety
reviewed by: Dan Laksov
MaRDI profile type: Publication
full work available at URL: jabr.1995.1135
OpenAlex ID: W1979768829
DOI: 10.1006/JABR.1995.1135
description: scientific article; zbMATH DE number 783872