Chern classes of complex Galois representations (Q1199152)

Let \(F\) be a field with absolute Galois group \(G_ F\). \(F\) is called exceptional if \(\text{Gal}(F(\mu_{2^ \infty})/F)\) contains torsion. If \(\rho:G_ F\to GL_ r(\mathbb{C})\) is a finite dimensional, continuous representation and if \(c_ i(\rho)\in H^{2i}(G_ F;\mathbb{Z})\) is the \(i\)th Chern class of \(\rho\) then the author proves that \(c_ i(\rho)=0\) if \(i\geq 2\) and \(F\) is nonexceptional. In any case, \(2c_ i(\rho)=0\). The author relates this to generalizations of a conjecture of K. Kato and suggests a generalization of his result to smooth varieties and their arithmetic filtration.