Borel fixed-point theorem

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Statement

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.

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 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