// Workers AI · dad joke modeWhat did the Borel fixed-point theorem say? "I'm stuck on this point.

In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by Armand Borel (1956).

Statement

If G is a connected, solvable, linear algebraic group acting regularly on a non-empty, complete algebraic variety V over an algebraically closed field k, then there is a G fixed-point of V.

The Lie-Kolchin theorem proves this result under the stronger hypothesis that V is a projective variety.

A more general version of the theorem holds over a field k that is not necessarily algebraically closed. A solvable algebraic group G is split over k or k-split if G admits a composition series whose composition factors are isomorphic (over k) to the additive group $\mathbb {G} _{a}$ or the multiplicative group $\mathbb {G} _{m}$. If G is a connected, k-split solvable algebraic group acting regularly on a complete variety V having a k-rational point, then there is a G fixed-point of V.

References

• Borel, Armand (1956). "Groupes linéaires algébriques". Ann. Math. 2. 64 (1). Annals of Mathematics: 20–82. doi:10.2307/1969949. JSTOR 1969949. MR 0093006.

• Borel, Armand (1991) [1969], Linear Algebraic Groups (2nd ed.), New York: Springer-Verlag, ISBN 0-387-97370-2, MR 1102012

External links

• V.P. Platonov (2001) [1994], "Borel fixed-point theorem", Encyclopedia of Mathematics, EMS Press