A variational approach to complex Monge-Ampère equations
Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 179-245.

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.

DOI: 10.1007/s10240-012-0046-6
Berman, Robert J. 1; Boucksom, Sébastien 2; Guedj, Vincent 3; Zeriahi, Ahmed 4

1 Chalmers Techniska Högskola, Chalmers University of Technology and University of Gothenburg Göteborg Sweden
2 Institut de Mathématiques, CNRS-Université Pierre et Marie Curie 75252, Paris Cedex France
3 I.M.T., Université Paul Sabatier and Institut Universitaire de France 31062, Toulouse Cedex 09 France
4 I.M.T., Université Paul Sabatier 31062, Toulouse Cedex 09 France
     author = {Berman, Robert J. and Boucksom, S\'ebastien and Guedj, Vincent and Zeriahi, Ahmed},
     title = {A variational approach to complex {Monge-Amp\`ere} equations},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {179--245},
     publisher = {Springer-Verlag},
     volume = {117},
     year = {2013},
     doi = {10.1007/s10240-012-0046-6},
     zbl = {1277.32049},
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     url = {http://www.numdam.org/articles/10.1007/s10240-012-0046-6/}
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Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent; Zeriahi, Ahmed. A variational approach to complex Monge-Ampère equations. Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 179-245. doi : 10.1007/s10240-012-0046-6. http://www.numdam.org/articles/10.1007/s10240-012-0046-6/

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