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Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity

Tamás Darvas, Eleonora Di Nezza and Chinh H. Lu

Vol. 11 (2018), No. 8, 2049–2087
DOI: 10.2140/apde.2018.11.2049

We establish the monotonicity property for the mass of nonpluripolar products on compact Kähler manifolds, and we initiate the study of complex Monge–Ampère-type equations with prescribed singularity type. Using the variational method of Berman, Boucksom, Guedj and Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kähler–Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for nonpluripolar products with small unbounded locus.

Monge–Ampère equation, variational approach, pluripotential theory
Mathematical Subject Classification 2010
Primary: 32Q15, 32U05, 32W20
Secondary: 32Q20
Received: 27 June 2017
Revised: 4 February 2018
Accepted: 10 April 2018
Published: 6 June 2018
Tamás Darvas
Department of Mathematics
University of Maryland
College Park, MD
United States
Eleonora Di Nezza
Institut des Hautes Études Scientifiques
Université Paris-Saclay
Bures sur Yvette
Chinh H. Lu
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud
Université Paris-Saclay