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 .
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 formula_1 or the multiplicative group formula_2. 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".