Stably A¹-connected varieties and universal triviality of CH₀

الأصل · EN

We study the relationship between several notions of connectedness arising in A¹-homotopy theory of smooth schemes over a field k: A¹-connectedness, stable A¹-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 CH₀. We show that stable A¹-connectedness coincides with motivic connectedness, under suitable hypotheses on k. Then, we observe that there exist stably A¹-connected smooth proper varieties over the field of complex numbers that are not A¹-connected.