The cone theorem and the vanishing of Chow cohomology

TL;DR

This paper establishes a cone theorem linking ${ m A}^1$-homotopy invariance to the vanishing of positive degree Chow cohomology in affine varieties, impacting the understanding of GIT quotient classes.

Contribution

It introduces a cone theorem for ${ m A}^1$-homotopy invariant functors that implies vanishing results for Chow cohomology, connecting homotopy theory and algebraic geometry.

Findings

Positive degree Chow cohomology vanishes for large classes of affine varieties.

The cone theorem relates ${ m A}^1$-homotopy invariance to Chow cohomology vanishing.

Implications for representing Chow classes of GIT quotients.

Abstract

We show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.