A variational approach to complex Monge-Ampère equations

Abstract

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kahler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kahler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kahler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.