Geometric criteria for A1 -connectedness and applications to norm varieties

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.