Fixed point theorems involving numerical invariants

TL;DR

This paper introduces invariants for smooth projective algebraic varieties that, when non-zero modulo p, prevent certain finite p-group actions from being fixed point free, including cases over fields of characteristic p.

Contribution

It presents new integer-valued invariants that detect fixed point properties of finite p-group actions on algebraic varieties, extending to characteristic p fields.

Findings

Invariants prevent fixed point free actions when non-zero mod p.

Counterexamples demonstrate the sharpness of the invariants.

Results apply to varieties over fields of characteristic p.

Abstract

We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo p prevents the existence of an action without fixed points of certain finite p-groups. The case of base fields of characteristic p is included. Counterexamples are systematically provided to test the sharpness of our results.