Diagonalisable p-groups cannot fix exactly one point on projective varieties

TL;DR

This paper proves that diagonalisable p-groups cannot fix exactly one point on smooth projective varieties and shows that for groups of order two, the number of fixed points cannot be odd, extending classical topological results.

Contribution

It provides an algebraic proof of fixed point constraints for diagonalisable p-groups on projective varieties, including cases over fields of characteristic p.

Findings

Diagonalisable p-groups cannot fix exactly one point.

For groups of order two, the number of fixed points is never odd.

The approach extends classical topology results to algebraic geometry.

Abstract

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove that the number of fixed points cannot be odd. The main tool is a construction originally used by Rost in the context of the degree formula. The framework of diagonalisable groups allows us to include the case of base fields of characteristic p.