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Birational invariants and $$\mathbb{A}^1$$-connectedness. (English) Zbl 1328.14036
Summary: We study some aspects of the relationship between $$\mathbb{A}^1$$-homotopy theory and birational geometry. We study the so-called $$\mathbb{A}^1$$-singular chain complex and zeroth $$\mathbb{A}^1$$-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 $$\mathbb{A}^1$$-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 $$\mathbb{A}^1$$-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of $$\mathbb{A}^1$$-connectedness by means of vanishing of unramified invariants.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14E08 Rationality questions in algebraic geometry 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 14E05 Rational and birational maps 18E30 Derived categories, triangulated categories (MSC2010)
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