Secondary characteristic classes and the Euler class

  • Aravind Asok

  • Jean Fasel

Abstract

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if XX is a smooth affine scheme of dimension dd over a field kk of finite 2-cohomological dimension (with char(k)2)\mathrm{char}(k)\neq 2) and EE is a rank dd vector bundle over XX, vanishing of the Chow-Witt theoretic Euler class of EE is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when kk is of small 2-cohomological dimension.