Rational smoothness and fixed points of torus actions
type: Item
instance of: scholarly article
title: Rational smoothness and fixed points of torus actions
zbMATH Open document ID: 0953.14004
DOI: 10.1007/BF01237356
author: Michel Brion
published in: Transformation Groups
publication date: 2001-01-23T00:00:00Z
review text: A rationally smooth point is a point, where the local cohomology with constant coefficients is the same as for a smooth point. In this paper the author studies rational smoothness of complex algebraic varieties \(X\) on which a torus \(T\) acts. His main result is, that for an attractive fixed point of \(T\) (i.e. the weights of \(T\) on the tangent space at \(x\) are contained in an open half space) rational smoothness of \(X\) at \(x\) is equivalent to the following three conditions: (1) a punctured neighborhood of \(x\) in \(X\) is rationally smooth, (2) for any subtorus \(T'\) of \(T\) of codimension one, the fixed point set \(X^{T'}\) of \(T'\) in \(X\) is rationally smooth at \(x\), (3) \(\dim_x(X) = \sum_{T'}\dim_x(X^{T'})\), the sum over all subtori of codimension one. Several applications of this result are given, for example, a generalization of a result of Springer concerning inner involutions. Another application is the existence of attractive slices to \(\overline{BgB}\), where \(B\) is a Borel subgroup of a connected reductive group \(G\), and \(\overline{BgB}\) is the closure in a smooth \(G \times G\) equivariant and regular completion of \(G\).
Mathematics Subject Classification ID: 14B15
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 1347438
zbMATH Keywords: algebraic groups
zbMATH Keywords: torus actions
zbMATH Keywords: fixed points
zbMATH Keywords: local cohomology
zbMATH Keywords: rational smoothness
zbMATH Keywords: slices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1347438