Borel fixed-point theorem

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A fixed-point theorem in algebraic geometry

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).

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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 G a {\displaystyle \mathbb {G} _{a}} or the multiplicative group G m {\displaystyle \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.¹

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

References

Borel (1991), Proposition 15.2 ↩