Abstract: Motivated by mirror symmetry considerations (the deformed Hermitian-Yang-Mills equation due to Jacob-Yau) and a desire to study stability conditions involving higher Chern forms, a vector bundle version of the usual complex Monge-Ampere equation (studied by Calabi, Yau, Aubin, etc) will be discussed. I shall also discuss a Kobayashi-Hitchin type correspondence for a special case (a dimensional reduction to Riemann surfaces).