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Pré-Publication, Document De Travail Année : 2008

Monge-Ampère equations in big cohomology classes

Philippe Eyssidieux
Ahmed Zeriahi
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Résumé

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 $\a$ on $X$ (i.e.~a class that can be represented by a strictly positive current) and a positive measure $\mu$ on $X$ of total mass equal to the volume of $\a$ and putting no mass on pluripolar subsets, we show that $\mu$ can be written in a unique way as the top degree self-intersection in the non-pluripolar sense of a closed positive current in $\a$. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if $\mu$ has $L^{1+\e}$-density with respect to Lebesgue measure. If $\mu$ is smooth and positive everywhere, we prove that $T$ is smooth on the ample locus of $\a$ provided $\a$ 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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Dates et versions

hal-00360760 , version 1 (11-02-2009)

Identifiants

  • HAL Id : hal-00360760 , version 1

Citer

Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. Monge-Ampère equations in big cohomology classes. 2008. ⟨hal-00360760⟩
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