Relative versions of theorems of Bogomolov and Sukhanov over perfect fields. (English) Zbl 1167.14030

The authors state and prove relative versions of theorems of F. A. Bogomolov [Math. USSR, Izv. 13, 499–555 (1979); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227–1287 (1978; Zbl 0439.14002)] and A. A. Sukhanov [Math. USSR, Sb. 65, No. 1, 97–108 (1990); translation from Mat. Sb., Nov. Ser. 137(179), No. 1(9), 90–102 (1988; Zbl 0663.20043)] concerning observable subgroups of linear algebraic groups over non-algebraically closed perfect fields. A closed subgroup \(H\) of a linear algebraic group \(G\) is observable if the homogeneous space \(G/H\) is a quasi-affine variety. Sukhanov gave a criterion for a closed subgroup of an algebraic group over an algebraically closed field to be observable. An algebraic subgroup of \(G\) is observable if and only if it is subparabolic.
In this article, the authors give a relative version of the theorem of Sukhanov for a non-algebraically closed perfect field. In order to do this, they prove a relative version of the theorem of Bogomolov, stating that if \(G\) is a connected reductive group, \(V\) is a finite dimensional \(G\)-module, and \(0\in \overline{Gv}\), then the isotropy group \(G_v\) is contained in a proper quasiparabolic subgroup of \(G\).


14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
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