2010/01/26 by Aravind Asok, Asok, Aravind · 1 citation
Mathematics · #14F35 #14F43 #14J10 #57R80 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.AG #math.AT #math.KT #msc:14F35 #msc:14F43 #msc:14J10 #msc:57R80
paper · pdf · doi:10.48550/arxiv.1001.4574
24 pages; To appear Crelle's journal. Part II of ArXiV v2 (regarding the Luroth problem) is being reworked and will be uploaded separately; the old version is still available at http://www-bcf.usc.edu/~asok
openalex publication_date 2010/01/26 · arxiv created 2011/11/19 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.