Birational invariants and A^1-connectedness

Details Birational invariants and A^1-connectedness Birational invariants and A^1-connectedness

2010-01-26

arxiv.org/abs/1001.4574v5

Mathematics

Algebraic Geometry

24 pages; To appear Crelle's journal. Part II of ArXiV v2 (regarding the Luroth problem) is being reworked and will be uploade

Scientific paper

We study some aspects of the relationship between A^1-homotopy theory and birational geometry. We study the so-called A^1-singular chain complex and zeroth 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 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 \'etale cohomology. In particular, we deduce a number of vanishing results for cohomology of A^1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of A^1-connectedness by means of vanishing of unramified invariants.

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