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Linearizing certain reductive group actions. (English) Zbl 0602.14047
Let \(X=Spec(A)\) be an affine (finite type) scheme over an algebraically closed characteristic zero field, and assume that the reductive algebraic group G acts on X. The main result of this paper, the normal linearization theorem, gives criteria to guarantee that X is a G-vector bundle: the necessary hypotheses are that there is a closed G-stable subscheme \(X_ 0\) of X which contains all closed orbits, which is a local complete intersection in X, and into which there is a G-equivariant retraction \(p: X\to X_ 0,\) then X is a G-vector bundle over \(X_ 0.\)
This result is applied in the case of fix pointed actions (i.e. when the projection of X to the categorical quotient X/G induces a bijection from fixed points). If the fixed points are a local complete intersection, then X is a vector bundle over X/G, and if vector bundles are trivial on X/G then X is G-isomorphic to (X/G)\(\times W\), where W is a G-module whose only closed orbit is a unique fixed point, i.e. G is one fix pointed on W. In particular, if G is one fix pointed on X and the fixed point x is regular on X then (X,x) is G-isomorphic to \((T_ x(X),0).\)
These results in turn generalize previous results of Białynicki-Birula and Panyushev which identify certain actions as conjugate to linear actions on vector spaces. The authors also indicate how their results could have applications to the Jacobian conjecture.
Reviewer: A.R.Magid

MSC:
14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
20G05 Representation theory for linear algebraic groups
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