In
mathematics, the Borel fixed-point theorem is a
fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...
in
algebraic geometry generalizing the
Lie–Kolchin theorem. The result was proved by .
Statement
If ''G'' is a
connected,
solvable, linear
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
acting regularly on a
non-empty,
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''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 In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ...
whose composition factors are isomorphic (over ''k'') to the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structure ...
or the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
. 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
*
*
External links
*
Fixed-point theorems
Group actions (mathematics)
Theorems in algebraic geometry
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