Borel fixed-point theorem
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.
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 [math]\displaystyle{ \mathbb G_a }[/math] or the multiplicative group [math]\displaystyle{ \mathbb G_m }[/math]. 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.[1]
References
- ↑ Borel (1991), Proposition 15.2
- Borel, Armand (1956). "Groupes linéaires algébriques". Ann. Math.. 2 (Annals of Mathematics) 64 (1): 20–82. doi:10.2307/1969949.
- Borel, Armand (1991) [1969], Linear Algebraic Groups (2nd ed.), New York: Springer-Verlag, ISBN 0-387-97370-2
External links
- Hazewinkel, Michiel, ed. (2001), "Borel fixed-point theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=b/b017070
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