Geometric criteria for ${\mathbb{A}}^1$-connectedness and applications to norm varieties

Abstract

We show that ${\mathbb{A}}^1$-connectedness of a large class of varieties over a field $k$ can be characterized as the condition that their generic point can be connected to a $k$-rational point using (not necessarily naive) ${\mathbb{A}}^1$-homotopies. We also show that symmetric powers of ${\mathbb{A}}^1$-connected varieties (over an arbitrary field), as well as smooth proper models of them (over an algebraically closed field of characteristic $0$), are ${\mathbb{A}}^1$-connected. As an application of these results, we show that the standard norm varieties over a field $k$ of characteristic 0 become ${\mathbb{A}}^1$-connected (and consequently, universally $R$-trivial) after base change to an algebraic closure of $k$.

v4: 17 pages, final version before page proofs, accepted for publication in Journal of Algebraic Geometry