A variational approach to complex Monge-Ampere equations
Journal article, 2013

We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler 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 Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-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.

Author

Robert Berman

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Sebastien Boucksom

Pierre and Marie Curie University (UPMC)

Viincent Guedj

Universite Paul Sabatier Toulouse III

Ahmed Zeriahi

Universite Paul Sabatier Toulouse III

Publications Mathématiques

0073-8301 (ISSN)

Vol. 117 1 179-245

Subject Categories

Mathematics

Roots

Basic sciences

DOI

10.1007/s10240-012-0046-6

More information

Latest update

5/29/2018