Rational smoothness and fixed points of torus actions
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 \(G\).
scientific article; zbMATH DE number 1347438
scientific article