Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we can define the notions of Nakano positivity,
Griffiths positivity and ampleness. All these notions of positivity are equivalent for line bundles. In general, Nakano positivity
implies Griffiths positivity and Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies
Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, Demailly designed several systems of equations
of Hermitian-Yang-Mills type for the curvature tensor. In this talk, I will briefly discuss about the solution of these systems on the
vortex bundle using method of continuity.