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 {{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.

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 $\mathbb G_a$ or the multiplicative group $\mathbb G_m$ . 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.Borel (1991), Proposition 15.2

References

{{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