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2010 Monge–Ampère equations in big cohomology classes
Sébastien Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi
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Acta Math. 205(2): 199-262 (2010). DOI: 10.1007/s11511-010-0054-7


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.


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Sébastien Boucksom. Philippe Eyssidieux. Vincent Guedj. Ahmed Zeriahi. "Monge–Ampère equations in big cohomology classes." Acta Math. 205 (2) 199 - 262, 2010.


Received: 23 January 2009; Published: 2010
First available in Project Euclid: 31 January 2017

zbMATH: 1213.32025
MathSciNet: MR2746347
Digital Object Identifier: 10.1007/s11511-010-0054-7

Rights: 2010 © Institut Mittag-Leffler

Vol.205 • No. 2 • 2010
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