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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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