%0 Generic
%A Schmidt, Johannes
%D 2013
%F heidok:14452
%K homotopie rationaler Punkt, homotopie Fixpunkt
%R 10.11588/heidok.00014452
%T Anabelian aspects in the étale homotopy theory of Brauer-Severi varieties
%U https://archiv.ub.uni-heidelberg.de/volltextserver/14452/
%X We study the étale homotopy theory of Brauer-Severi varieties over fields of characteristic 0.  We prove that the induced Galois representations on geometric homotopy invariants (e.g., l-adic cohomology or higher homotopy groups) are all isomorphic for Brauer-Severi varieties of the same dimension.  If the base field has cohomological dimension smaller or equal 2 then we can show more in the case of Brauer-Severi curves:  There is even an isomorphism between the Hochschild-Serre spectral sequences computing cohomology with local coefficients.  Further, we study homotopy rational and homotopy fixed points on Brauer-Severi varieties and their connections to genuine rational points.  In particular, we show that under a suitable assumption on the first profinite Chern class map an analogue of the weak section conjecture for Brauer-Severi varieties turns out to be true.  We can give a counter example to this analogue without the extra assumption over p-adic local fields.