Universally defined cycles I

TL;DR

This paper introduces the concept of universally defined cycles on smooth varieties, proving they are polynomial in Chern classes, and explores their properties on products and powers of such varieties.

Contribution

It establishes that universally defined cycles are given by polynomials in Chern classes and proposes a conjectural explicit form for these cycles on powers of smooth varieties.

Findings

Universally defined cycles are polynomials in Chern classes.

Results extend to products of smooth varieties.

A conjectural explicit form is proposed for cycles on powers of smooth varieties.

Abstract

We introduce and study the notion of universally defined cycles of smooth varieties of dimension $d$, and prove that they are given by polynomials in the Chern classes. A similar result is proved for universally defined cycles on products of smooth varieties. We also state a conjectural explicit form for universally defined cycles on powers of smooth varieties, and provide some steps towards establishing it.