Borel fixed-point theorem
Contents
Statement
Version for algebraically closed fields
Suppose is a solvable connected linear algebraic group over an algebraically closed field
. Consider a regular action
of
on a non-empty complete algebraic variety
(here, regular simply means that the map
is a morphism of algebraic varieties, and has nothing to do with the regular group action as used in group theory). Then, there is a fixed point of the action.
Note that any projective variety is complete, so the Borel fixed-point theorem applies to all projective varieties.
Note the important hypotheses: the group is solvable, the field is algebraically closed, and the variety is a complete variety.
Version for arbitrary fields
Suppose is a solvable split connected linear algebraic group over a field
. Consider a regular action
of
on a non-empty complete algebraic variety
(here, regular simply means that the map
is a morphism of algebraic varieties, and has nothing to do with the regular group action as used in group theory). Then, there is a fixed point of the action.
The split condition basically says that the only composition factors allowed in (in a composition series as an algebraic group) are the additive and multiplicative groups. In particular, we are not allowed to use field extensions of
in constructing
. For an algebraically closed field, any solvable connected algebraic group is automatically split, because of the classification of connected one-dimensional algebraic groups over an algebraically closed field.
Note that any projective variety is complete, so the Borel fixed-point theorem applies to all projective varieties.
Related facts
Breakdown of variations
- Breakdown when the variety is not complete: The result does not apply if we are considering actions on non-complete varieties. For instance, the regular group action of
on itself does not have any fixed points. Loosely speaking, the variety being complete is what guarantees the existence of a point at infinity that cannot be moved by the group. Note again that any projective variety is complete.
- Breakdown when the field is not algebraically closed and the group is not split: See Borel fixed-point theorem fails for non-split linear algebraic groups
Applications
- Borel-Morozov theorem: This states that all Borel subgroups of a linear algebraic group over an algebraically closed field are conjugate.