Borel fixed-point theorem

Version for algebraically closed fields
Suppose $$G$$ is a solvable connected linear algebraic group over an algebraically closed field $$K$$. Consider a regular action $$\tau$$ of $$G$$ on a non-empty complete algebraic variety $$V$$ (here, regular simply means that the map $$G \times V \to V$$ 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 $$G$$ is a solvable split connected linear algebraic group over a field $$K$$. Consider a regular action $$\tau$$ of $$G$$ on a non-empty complete algebraic variety $$V$$ (here, regular simply means that the map $$G \times V \to V$$ 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 $$G$$ (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 $$K$$ in constructing $$G$$. 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.

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 $$G$$ 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.