## Acta Mathematica

### Monge–Ampère equations in big cohomology classes

#### Abstract

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 L1+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.

#### Article information

Source
Acta Math., Volume 205, Number 2 (2010), 199-262.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892500

Digital Object Identifier
doi:10.1007/s11511-010-0054-7

Mathematical Reviews number (MathSciNet)
MR2746347

Zentralblatt MATH identifier
1213.32025

Rights

#### Citation

Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed. Monge–Ampère equations in big cohomology classes. Acta Math. 205 (2010), no. 2, 199--262. doi:10.1007/s11511-010-0054-7. https://projecteuclid.org/euclid.acta/1485892500

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