  Given a vector bundle of arbitrary rank with ample determinant line bundle on
a projective manifold, we propose a new elliptic system of differential
equations of Hermitian-Yang-Mills type for the curvature tensor. The system is
designed so that solutions provide Hermitian metrics with positive curvature in
the sense of Griffiths-and even in the dual Nakano sense. As a consequence, if
an existence result could be obtained for every ample vector bundle, the
Griffiths conjecture on the equivalence between ampleness and positivity of
vector bundles would be settled.
