A Priori Estimates of the Degenerate Monge-Ampere Equation on Kahler Manifolds of Nonnegative Bisectional Curvature

Abstract

The regularity theory of the degenerate complex Monge-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold \((M,g)\) with nonnegative orthogonal bisectional curvature of dimension \(m\). Given a solution \(\phi\) of the degenerate complex Monge-Amp\`{e}re equation \(\det(g_{i \bar{j}} + \phi_{i \bar{j}}) = f \det(g_{i \bar{j}})\), it is shown that the Laplacian of \(\phi\) can be controlled by a constant depending on \((M,g)\), \(\sup f\), and \(\inf_M \Delta f^{1/(m-1)}\).