Anabelian aspects in the étale homotopy theory of Brauer-Severi varieties

Abstract

We study the etale 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.