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

Published 7 Jan 2016 in math.AG, math.AT, and math.KT | (1601.01615v1)

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.}