Geometric criteria for

Abstract

We show that 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) A ¹ -homotopies. We also show that symmetric powers of A ¹-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic 0) are A 1-connected. As an application of these results, we show that the standard norm varieties over a field k of characteristic 0 become A ¹-connected (and consequently, universally R-trivial) after base change to an algebraic closure of k.