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.
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