# 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.