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

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

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