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

Abstract

We establish the monotonicity property for the mass of non-pluripolar 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-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.Without the non-zero mass condition X θ n φ > 0 this characterization cannot hold (see Remark 3.3). The equivalence between (i) and (iii) in the above theorem shows that P θ [u] is the same potential for any u ∈ E(X, θ, φ), and equals to P θ [φ]. Given this and the inclusion E(X, θ, φ) ⊂ E(X, θ, P θ [φ]), one is tempted to consider only potentials φ in the image of the operator ψ → P θ [ψ], when studying the classes of relative full mass E(X, θ, φ). These potentials seemingly play the same role as V θ , the potential with minimal V * K,θ is supported on K, we thus haveand the desired inequality holds in this case. If M := M φ (K) ≥ 1, then by (12) we have φ ≤ M −1 V * K,φ + (1 − M −1 )φ ≤ φ + 1, and by definition of the relative capacity we can write:

Pluripotential theory and convex bodies: large deviation principle

“…It was shown in [11,Theorem 3.8] that the nonpluripolar Monge-Ampère measure of P ω [φ] is dominated by Lebesgue measure:…”

“…This fact plays a crucial role in solving the complex Monge-Ampère equation. For the reader's convenience, we note that in the notation of [11] (on the left)…”

“…ϕ ∈ E(X, ω)) then P ω [ϕ] = 0. We will use the following property of model potentials proved in [11,Theorem 3.12]: if φ is a model potential then…”

“…It was shown in [11,Theorem 4.10] (by adapting the arguments of [7]) that E φ is non-decreasing and concave along affine curves, giving rise to its trivial extension to P SH(X, ω, φ). We define…”

“…Remark 2.9. It was shown in [11,Theorem 4.28] that a unique (normalized) solution u in E(X, ω, φ) always exists (without the finite energy assumption on µ). But that proof does not give a solution in E 1 (X, ω, φ).…”