Monge–Ampère equations in big cohomology classes

Sébastien Boucksom Institut de Mathématiques de Jussieu, CNRS–Université Pierre et Marie Curie Philippe Eyssidieux Institut Fourier, Université Grenoble 1 Vincent Guedj Institut de Mathématiques de Toulouse, Université Paul Sabatier Ahmed Zeriahi Institut de Mathématiques de Toulouse, Université Paul Sabatier

Analysis of PDEs Differential Geometry mathscidoc:1701.03002

Acta Mathematica, 205, (2), 199-262, 2009.1
We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.
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@inproceedings{sébastien2009monge–ampère,
  title={Monge–Ampère equations in big cohomology classes},
  author={Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355567965730},
  booktitle={Acta Mathematica},
  volume={205},
  number={2},
  pages={199-262},
  year={2009},
}
Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. Monge–Ampère equations in big cohomology classes. 2009. Vol. 205. In Acta Mathematica. pp.199-262. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355567965730.
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