Representations of reductive groups in cohomology spaces
type: Item
instance of: scholarly article
title: Representations of reductive groups in cohomology spaces
zbMATH Open document ID: 0818.14018
DOI: 10.1007/BF01450503
author: Michel Brion
published in: Mathematische Annalen
publication date: 1995-08-08T00:00:00Z
full work available at URL: https://eudml.org/doc/165269
review text: Consider a connected complex reductive group \(G\) acting on a complex algebraic variety \(X\). Recall that for any \(G\)-linearized coherent sheaf \({\mathcal F}\) on \(X\), the cohomology spaces \(H^ i (X, {\mathcal F})\) are rational \(G\)-modules: they decompose into a direct sum of simple \(G\)- modules, with (finite or infinite) multiplicities. We prove the following alternative: (1) If a Borel subgroup of \(G\) has a dense orbit in \(X\), then there is a (finite) upper bound \(C(X, {\mathcal F})\) for all multiplicities of the \(G\)-modules \(H^ i (X, {\mathcal F})\). In fact, there exists a constant \(C(X)\) such that for any \(G\)-linearized vector bundle \(E\) over \(X\), and for any integer \(i \geq 0\), the multiplicities of \(H^ i (X,E)\) are at most \(C(X) \cdot\text{rk} (E)\). (2) If no Borel subgroup has a dense orbit on \(X\), then there exists a \(G\)-variety \(Y\) which is \(G\)-birational to \(X\), and a \(G\)-linearized invertible sheaf \(L\) over \(X\), such that the subspace of \(G\)-invariants in \(H^ 0 (X,L)\) or \(H^ 1 (X,L)\) is infinite-dimensional. The first statement is a qualitative analog of the classical Borel-Weil-Bott theorem, which describes the cohomology groups of line bundles on flag varieties.
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14F25
zbMATH DE Number: 734588
zbMATH Keywords: subspace of invariants
zbMATH Keywords: Borel-Weil-Bott theorem
zbMATH Keywords: cohomology groups of line bundles on flag varieties
MaRDI profile type: Publication
cites work: Actions with a finite number of orbits
cites work: Q4697141
cites work: Cohomology of regular embeddings
cites work: On affine algebraic homogeneous spaces
cites work: Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I
cites work: Homogeneous vector bundles
cites work: Q4308255
cites work: Valuations des espaces homogènes sphériques. (Valuations of spherical homogeneous spaces)
cites work: Induced modules and affine quotients
cites work: A very simple proof of Bott's theorem
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili
cites work: The Grothendieck-Cousin complex of an induced representation
cites work: Q3993222
cites work: The \(K\)-theory of a toric variety
cites work: Q4039830
OpenAlex ID: W4251772148
description: scientific article; zbMATH DE number 734588