  We relate the existence of Noether global conserved currents associated with
locally variational field equations to existence of global solutions for a
local variational problem generating global equations. Both can be
characterized as the vanishing of certain cohomology classes.
  In the case of a 3-dimensional Chern-Simons gauge theory, the variationally
featured cohomological obstruction to the existence of global solutions is
sharp and equivalent to the usual obstruction in terms of the Chern
characteristic class for the flatness of a principal connection.
  We suggest a parallelism between the geometric interpretation of
characteristic classes as obstruction to the existence of flat principal
connections and the interpretation of certain de Rham cohomology classes to be
the obstruction to the existence of global extremals for a local variational
principle.
