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A variational approach to complex Monge-Ampère equations

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Abstract

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.

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Authors and Affiliations

  1. Chalmers Techniska Högskola, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden

    Robert J. Berman

  2. Institut de Mathématiques, CNRS-Université Pierre et Marie Curie, 75252, Paris Cedex, France

    Sébastien Boucksom

  3. I.M.T., Université Paul Sabatier and Institut Universitaire de France, 31062, Toulouse Cedex 09, France

    Vincent Guedj

  4. I.M.T., Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    Ahmed Zeriahi

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  1. Robert J. Berman
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  2. Sébastien Boucksom
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  3. Vincent Guedj
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  4. Ahmed Zeriahi
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Correspondence to Sébastien Boucksom.

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Berman, R.J., Boucksom, S., Guedj, V. et al. A variational approach to complex Monge-Ampère equations. Publ.math.IHES 117, 179–245 (2013). https://doi.org/10.1007/s10240-012-0046-6

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