Motivated by mirror symmetry considerations (the deformed Hermitian-Yang-Mills equation due to Jacobs-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, Aubin, Yau, etc) will be discussed. I shall also discuss a Kobayashi-Hitchin type correspondence for a special case (a dimensional reduction to Riemann surfaces).

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Last updated: 14 Feb 2019