Vol. 86, No. 1, 1980

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On action of SL(2) on complete algebraic varieties

A. Białynicki-Birula

Vol. 86 (1980), No. 1, 53–58

The aim of this paper is to provide full proofs of results announced in [2]. Some theorems are proved under weaker assumptions. The results lead to a decomposition theorem for actions of SL(2) similar to that proved in [1] for torus actions. However even in this case of SL(2) and in a greater extent in the case of actions of arbitrary semisimple groups the results are not so full and many questions are left open. Some of them are mentioned in the paper.

All considered algebraic varieties and morphisms are assumed to be defined over an algebraically closed field k (of any characteristic). Let G denote a connected semisimple algebraic group. Let α : G × X X be an action of G on a complete algebraic variety X. For g G and x X we shall write g(x) or gx instead of α(g,x). The subvariety of fixed points of the action is denoted by XG. The orbit Gx of x X is said to be closed if it is closed in X. A closed orbit is said to be nontrivial if it is not composed of one point.

Mathematical Subject Classification 2000
Primary: 14L30
Secondary: 20G15, 32M05, 57S25
Received: 14 August 1978
Published: 1 January 1980
A. Białynicki-Birula