Contractions of the actions of reductive algebraic groups

V. L. Popov

Sbornik. Mathematics, Tome 58 (1987) no. 2, pp. 311-335

Résumé

It is shown that each algebraic action of a simply connected reductive algebraic group $G$ on an affine algebraic variety $X$ can be contracted (in a flat one-dimensional family of actions) to a canonical action of $G$ on a certain affine variety $\operatorname{gr}X$ having some very special properties. It is shown that $X$ and $\operatorname{gr}X$ have many algebro-geometric properties in common. As an application, we prove the Procesi–Kraft conjecture to the effect that the singularities of the closures of orbits in the case of spherical stabilizer are rational. It is assumed that the ground field has characteristic zero. Bibliography: 37 titles.

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