Birational invariants and 𝔸1-connectedness

Abstract

We study some aspects of the relationship between 𝔸1-homotopy theory and birational geometry. We study the so-called 𝔸1-singular chain complex and zeroth 𝔸1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite, the zeroth 𝔸1-homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified étale cohomology. In particular, we deduce a number of vanishing results for cohomology of 𝔸1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of 𝔸1-connectedness by means of vanishing of unramified invariants.