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

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 $C{H}_0$. 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.

12 pages, Comments welcome!

References in corpus (2)

• Unirational threefolds with no universal codimension 2 cycle
• The motive of a classifying space