On the cohomology supported by line bundles over complete symmetric varieties
type: Item
instance of: scholarly article
title: On the cohomology supported by line bundles over complete symmetric varieties
zbMATH Open document ID: 1217.14035
author: Alexis Tchoudjem
published in: Transformation Groups
publication date: 2010-10-29T00:00:00Z
review text: Let \(G\) be a semisimple linear algebraic group over an algebraically closed field of characteristic zero. If \(X\) is a smooth projective \(G\)-variety and \(L\) is a \(G\)-linearized line bundle over \(X\), then all cohomology groups \(H^d(X,L)\) are finite-dimensional \(G\)-modules. In this article, the author studies the case where \(X\) is a complete symmetric variety. Given a Bialynicki-Birula cell \(C\) of codimension \(c\), the Lie algebra \({\mathbf g}\) of \(G\) acts naturally on the cohomology group \(H_C^c(L)\). In this article, the author gives a necessary condition on the cell for the existence of a finite dimensional simple subquotient of this \({\mathbf G}\)-module. As an application, one can calculate the Euler-Poincaré characteristic of \(L\) over \(X\) and give estimates of the higher cohomology groups \(H^d(X,L)\).
reviewed by: Lucy Moser-Jauslin
Mathematics Subject Classification ID: 14M27
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5808455
zbMATH Keywords: complete symmetric varieties
zbMATH Keywords: almost homogeneous varieties
zbMATH Keywords: wonderful compactification
zbMATH Keywords: Bialynicki-Birula cellular decomposition
MaRDI profile type: Publication
full work available at URL: s00031-010-9105-6
OpenAlex ID: W1748489451
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cites work: Q4126461
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DOI: 10.1007/S00031-010-9105-6
description: scientific article; zbMATH DE number 5808455