# Involutions and Chern numbers of varieties

## TL;DR

This paper investigates the relationship between involutions on smooth projective varieties and their fixed loci using cobordism, revealing conditions under which the fixed locus's dimension exceeds its codimension based on Chern number divisibility.

## Contribution

It provides new results linking Chern numbers and the dimension of fixed loci under involutions, employing an elementary cobordism approach without resolution of singularities.

## Key findings

- Fixed locus dimension exceeds codimension when certain Chern numbers are not divisible by two or four.
- Results include analogues of classical algebraic topology theorems by Conner-Floyd and Boardman.
- Addresses vanishing loci of idempotent global derivations in characteristic two.

## Abstract

Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism. We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but not all, are analogues of theorems in algebraic topology obtained by Conner-Floyd and Boardman in the sixties. We include versions of our results concerning the vanishing loci of idempotent global derivations in characteristic two. Our approach to cobordism, following Merkurjev's, is elementary, in the sense that it does not involve resolution of singularities or homotopical methods.