Borel fixed-point theorem

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A connected solvable algebraic group acting regularly (cf. Algebraic group of transformations) on a non-empty complete algebraic variety over an algebraically-closed field has a fixed point in . It follows from this theorem that Borel subgroups (cf. Borel subgroup) of algebraic groups are conjugate (The Borel–Morozov theorem). The theorem was demonstrated by A. Borel [1]. Borel's theorem can be generalized to an arbitrary (not necessarily algebraically-closed) field : Let be a complete variety defined over a field on which a connected solvable -split group acts regularly, then the set of rational -points is either empty or it contains a point which is fixed with respect to . Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field is perfect, the maximal connected solvable -split subgroups of a connected -defined algebraic group are mutually conjugate by elements of the group of -points of [2].


[1] A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2) , 64 : 1 (1956) pp. 20–82
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969)
[3] V.V. Morozov, Dokl. Akad. Nauk SSSR , 36 : 3 (1942) pp. 91–94
How to Cite This Entry:
Borel fixed-point theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article