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.
