Geometric criteria for 𝐴�1-connectedness and applications to norm varieties

Abstract

We show that 𝐴1-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) 𝐴1-homotopies. We also show that symmetric powers of 𝐴1-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 𝐴1-connected. As an application of these results, we show that the standard norm varieties over a field 𝑘 of characteristic 0 become 𝐴1-connected (and consequently, universally 𝑅-trivial) after base change to an algebraic closure of 𝑘.