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Complexity of action of reductive groups. (English. Russian original) Zbl 0601.14038
Funct. Anal. Appl. 20, 1-11 (1986); translation from Funkts. Anal. Prilozh. No. 1, 1-13 (1986).
Let $$T$$ be a reductive algebraic group acting on an algebraic variety $$X$$ and let $$B$$ be a Borel subgroup of $$G$$. The paper is devoted to the study of the connection between the complexity of actions of $$B$$ on $$X$$ and the modality of these actions (in the sense of singularity theory). If an algebraic group $$H$$ operates on an algebraic variety $$X$$, then the minimal codimension of $$H$$-orbits in $$X$$ is equal to the transcendence degree of the field $$k(X)\beta H$$ and is denoted by $$d(X,H)$$. If $$B$$ is a Borel subgroup of a reductive algebraic group $$G$$ acting on an algebraic variety $$X$$, then the complexity of the action of $$B$$ on $$X$$ equals $$d(X,B)$$.
In the paper the following definition of the modality of an action is suggested: The modality of the action of an algebraic group $$H$$ on an algebraic variety $$X$$ is the maximum of $$d(X',H)$$ on all subvarieties $$X'\subset X$$. This number is denoted by $$\mathrm{mod}(X,H)$$.
Theorem 1. If there is an open $$B$$-orbit in $$X$$, then there are only finitely many $$B$$-orbits in $$X$$. In particular, if $$d(X,B)=0$$, then $$\mathrm{mod}(X,B)=0$$.
Theorem 2. $$d(X,B)=\mathrm{mod}(X,B)$$. The same is valid for any maximal unipotent subgroup of $$G$$.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 20G15 Linear algebraic groups over arbitrary fields
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