A vector bundle version of the Monge-Ampère equation

Abstract

We introduce a vector bundle version of the complex Monge-Ampère equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA-positivity) which is necessary for the infinite-dimensional symplectic form to be Kähler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a “positively curved” solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lübke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.