Frobenius splitting methods in geometry and representation theory
type: Item
instance of: scholarly article
title: Frobenius splitting methods in geometry and representation theory
zbMATH Open document ID: 1072.14066
author: Michel Brion
author: Shrawan Kumar
published in: Progress in Mathematics
publication date: 2005-03-15T00:00:00Z
review text: The theory of Frobenius splitting, introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27--40 (1985; Zbl 0601.14043)] and refined by \textit{S. Ramanan} and \textit{A. Ramanathan} [Invent. Math. 79, 217--224 (1985; Zbl 0553.14023)], is a powerful method in the study of flag varieties and representation theory of linear algebraic groups in positive characteristic. One can also recover from it many of the analogous results in characteristic 0 by reduction to positive characteristic. The present monography is the first exposition in book form of this theory and of its major applications. It addresses to mathematicians and graduate students having a basic knowledge of algebraic geometry and representation theory of algebraic groups as provided by standard texts. The first chapter of the book is devoted to the general study of Frobenius split schemes. They are defined as follows : Let \(X\) be a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p > 0\). Consider the absolute Frobenius morphism \(F = (\text{id}_X, F^{\#}) : X\rightarrow X\), where \(F^{\#} : {\mathcal O}_X\rightarrow F_{\ast}{\mathcal O}_X (={\mathcal O}_X)\) maps \(f\) to \(f^p\). \(X\) is Frobenius split if there exists a morphism of \({\mathcal O}_X\)-modules \(\phi : F_{\ast}{\mathcal O}_X\rightarrow {\mathcal O}_X\) which is a left inverse of \(F^{\#}\). This has remarkable consequences: \(X\) is reduced (because \(F^{\#}\) is injective), weakly normal and, if \(X\) is projective, \(\text{H}^i(X,L) = 0\), \( \forall i > 0\), for every ample line bundle \(L\) on \(X\) (because \(F^{\ast}L\simeq L^{\otimes p}\), hence \(\text{H}^i(X,L)\) injects into \(\text{H}^i(X,L^{\otimes p})\)). Moreover, (variants of) the Kodaira, Grauert-Riemenschneider and Kawamata-Viehweg vanishing theorems are quick consequences of the existence of a Frobenius splitting. On the other hand, the existence of a Frobenius splitting is a quite restrictive co…
reviewed by: Iustin Coandă
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 2146008
zbMATH Keywords: flag varieties
zbMATH Keywords: Schubert varieties
zbMATH Keywords: representations of algebraic groups
zbMATH Keywords: Hilbert schemes of points
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2146008