  In this paper we look at two naturally occurring situations where the
following question arises. When one can find a metric so that a Chern-Weil form
can be represented by a given form ? The first setting is semi-stable
Hartshorne-ample vector bundles on complex surfaces where we provide evidence
for a conjecture of Griffiths by producing metrics whose Chern forms are
positive. The second scenario deals with a particular rank-2 bundle (related to
the vortex equations) over a product of a Riemann surface and the sphere.
