  Demailly showed that the Hodge conjecture is equivalent to the statement that
any (p,p)-dimensional closed current with rational cohomology class can be
approximated by linear combinations of integration currents associated to
subvarieties, and asked whether any strongly positive (p,p)-dimensional closed
current with rational cohomology class can be approximated by positive linear
combinations of integration currents associated to subvarieties. Using tropical
geometry, we construct a (p,p)-dimensional current on a smooth projective
variety that does not satisfy the latter statement.
