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 \'etale 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.