Borel fixed-point theorem

A connected solvable algebraic group $G$ acting regularly (cf. Algebraic group of transformations) on a non-empty complete algebraic variety $V$ over an algebraically-closed field $k$ has a fixed point in $V$. It follows from this theorem that Borel subgroups 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 $k$: Let $V$ be a complete variety defined over a field $k$ on which a connected solvable $k$-split group $G$ acts regularly, then the set of rational $k$-points $V(k)$ is either empty or it contains a point which is fixed with respect to $G$. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field $k$ is perfect, the maximal connected solvable $k$-split subgroups of a connected $k$-defined algebraic group $H$ are mutually conjugate by elements of the group of $k$-points of $H$ [2].

References

[1]	A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2) , 64 : 1 (1956) pp. 20–82 MR0093006 Zbl 0070.26104
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3]	V.V. Morozov, Dokl. Akad. Nauk SSSR , 36 : 3 (1942) pp. 91–94