Seminar

Speaker: Piotr Achinger

A smooth variety in characteristic zero is Zariski-locally a K(π; 1) space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin’s proof that l-adic cohomology agrees with singular cohomology over C. The characteristic p variant of this is not known—we do not even know whether the affine plane is a K(π; 1) in positive characteristic! I will show how to reduce this question to a “Bertini-type” statement regarding wild ramification of l-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank 1 and speculate on how one might treat the general case.