Diameter estimates in K?hler geometry

Diameter estimates for Kahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for L infinity$L<^>\infty$ estimates for the Monge-Ampere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long-standing problem of uniform diameter bounds and Gromov-Hausdorff convergence of the Kahler-Ricci flow, for both finite-time and long-time solutions.