Séminaire d'arithmétique à Lyon

On the existence of non-vanishing sections of vector bundles

by Alexey Ananyevskiy


It is well known that one may study the problem of existence of nowhere 
vanishing sections of vector bundles using methods of algebraic 
topology, in particular, characteristic classes. The simplest 
incarnation of this principle is that an oriented rank d real vector 
bundle over a dimension d compact manifold admits a nowhere vanishing 
continuous section if and only if the Euler class of this bundle 
vanishes. In the realm of algebraic geometry the picture is more 
intricate: a result of Murthy says that a rank d vector bundle over a 
smooth affine variety of dimension d over an algebraically closed field 
admits a nowhere vanishing section if and only if its top Chern class 
vanishes, while over a general field this does not hold with the 
counterexample given by the tangent bundle to the 2-sphere over the real 
numbers. Nevertheless, for the field of real numbers Bhatwadekar, Das, 
Mandal and Sridharan proved that triviality of the top Chern class is 
equivalent to the existence of a nowhere vanishing section provided that 
either d is odd, or when some additional assumption on the variety 
holds. In the talk I will outline how one can apply the methods of 
motivic homotopy theory, in particular, motivic Euler classes introduced 
by Fabien Morel, to this problem to give a new proof of these results 
and generalize them to the fields of virtual cohomological dimension at 
most 1.