{{short description|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 {{harvs |txt |authorlink=Armand Borel |first=Armand |last= Borel |year=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 <math>\mathbb G_a</math> or the multiplicative group <math>\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''.<ref>Borel (1991), Proposition 15.2</ref>
==References== {{reflist}} * {{cite journal | last = Borel | first = Armand | title = Groupes linéaires algébriques | journal = Ann. Math. |series=2 | year = 1956 | pages = 20–82 | volume = 64 | doi = 10.2307/1969949 | issue = 1 | publisher = Annals of Mathematics | jstor = 1969949 | mr =0093006 }}
*{{citation |last=Borel |first=Armand |author-link=Armand Borel |title=Linear Algebraic Groups |edition=2nd |location=New York |publisher=Springer-Verlag |isbn=0-387-97370-2 | year=1991 | orig-date=1969 | mr=1102012}}
==External links== * {{springer|id=b/b017070|title=Borel fixed-point theorem|author=V.P. Platonov}}
Category:Fixed-point theorems Category:Group actions Category:Theorems in algebraic geometry
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