Borel fixed-point theorem
Encyclopedia
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 .
, solvable
algebraic group
acting regularly on a non-empty, complete
algebraic variety
V over an algebraically closed field
k. Then G has a fixed point in V.
Fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...
in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
generalizing the Lie-Kolchin theorem. The result was proved by .
Statement of the theorem
Let G be a connectedConnected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
acting regularly on a non-empty, complete
Complete algebraic variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphismis a closed map, i.e. maps closed sets onto closed sets....
algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
k. Then G has a fixed point in V.