Stably A^1-connected varieties and universal triviality of CH_0

TL;DR

This paper explores various notions of connectedness in ${\f A}^1$-homotopy theory, linking them to rationality and triviality of the Chow group, and establishes equivalences and distinctions among these concepts over different fields.

Contribution

It clarifies the relationships between ${\bf A}^1$-connectedness, stable ${\bf A}^1$-connectedness, and motivic connectedness, showing their equivalence under certain conditions and providing examples of differences.

Findings

Motivically connected smooth proper varieties have universally trivial $CH_0$.

Stable ${\bf A}^1$-connectedness coincides with motivic connectedness under certain hypotheses.

Existence of stably ${\bf A}^1$-connected varieties over complex numbers that are not ${\bf A}^1$-connected.

Abstract

We study the relationship between several notions of connectedness arising in ${\mathbb A}^1$-homotopy theory of smooth schemes over a field $k$: ${\mathbb A}^1$-connectedness, stable ${\mathbb A}^1$-connectedness and motivic connectedness, and we discuss the relationship between these notations and rationality properties of algebraic varieties. Motivically connected smooth proper $k$-varieties are precisely those with universally trivial $CH_0$. We show that stable ${\mathbb A}^1$-connectedness coincides with motivic connectedness, under suitable hypotheses on $k$. Then, we observe that there exist stably ${\mathbb A}^1$-connected smooth proper varieties over the field of complex numbers that are not ${\mathbb A}^1$-connected.