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.
