Séminaire d'arithmétique à Lyon
# On the existence of non-vanishing sections of vector bundles

→
Europe/Paris

Description

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.