A generalised Monge-Ampère equation

We consider a generalised complex Monge-Ampère equation on a compact Kähler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related real PDE in a ball), as long as the Hessian is bounded below by a pre-determined constant (whilst moving along the method of continuity path), a smooth solution exists. Finally, we prove existence for another real PDE in a 3-ball, which is a local, real version of a conjecture of X.X. Chen.

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