Actions of diagonalizable $p$-groups and Chern numbers modulo $p$

TL;DR

This paper establishes lower bounds on the fixed point loci of diagonalizable p-groups acting on smooth projective varieties, linking these bounds to modulo p Chern numbers through equivariant localization and cobordism techniques.

Contribution

It introduces a novel approach connecting modulo p Chern numbers with fixed point bounds using a filtration on the cobordism ring and localization methods.

Findings

Lower bounds for fixed loci depending on modulo p Chern numbers

Introduction of a filtration on the modulo p cobordism ring

Application of localization and partition dividing techniques

Abstract

We obtain lower bounds for the dimension of fixed loci of diagonalizable $p$-groups acting on smooth projective varieties. Those bounds depend on the modulo $p$ Chern numbers of the ambient variety, and are expressed in a natural way by introducing an appropriate filtration on the "modulo $p$ cobordism ring" (for $p=2$ this is Thom's unoriented cobordism ring $MO^*$). They are obtained using equivariant localization methods, via the concentration theorem for the Chow ring, and by a technique of "partition dividing". As applications we derive statements in the spirit of Boardman's Five-Halves Theorem for involutions on manifolds.