11.32 Normalization and weak adjoint measures7091FEDB43F0

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Let $X$ be a connected compact complex manifold of dimension $n\geq 1$ with a smooth positive volume form $\mathrm{d}V$. Function spaces are real, and Lebesgue norms are taken with respect to $\mathrm{d}V$. Until section 11.40, $\theta $ is a smooth semipositive big real $(1,1)$-form and $\varphi \in \mathrm{PSH}(X,\theta )$ is a model reference in the sense of (16.28.67): it is a fixed point of the ordinary singularity envelope $P_\theta [\cdot ]$ of definition 5.40.4, and its envelope $P_{\varepsilon \theta }(\varphi )$ in a smaller background is not identically $-\infty $ for some $0<\varepsilon <1$. We write $u\sim \varphi $ when $u$ has the ordinary singularity type of $\varphi $ (definition 16.7.1), and $\mathrm{MA}^{\mathrm{ext}}_\theta $ for the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3; when its total mass is finite, it is the usual non-pluripolar measure, which we also denote by $\mathrm{MA}_\theta $.

In the ordinary type of $\varphi $, the equation $\mathrm{MA}^{\mathrm{ext}}_\theta (u)=c\, \mathrm{e}^ F\mathrm{d}V$ determines the constant $c>0$ uniquely, even when it has several normalized solutions. As a function of the logarithmic density $F$, the logarithm of this constant is concave. Its supporting functionals are probability measures that describe the first-order response of the constant to a change of density; for smooth Hermitian equations they are the Gauduchon adjoint probabilities of the linearized operator. The convex dual of the functional controls the concentration of these measures through entropy, and this gives compactness when the density, the reference or the positive background varies. The same measures provide the integral normalizations that single out the limit of the exponential equations as the exponent tends to zero. The scalar functional does not, however, determine the singularities of the solutions: examples with moving divisors have identical functionals but different limiting potentials. On closed projective backgrounds, convex geometry gives quantitative criteria for varying references, including references with nonanalytic poles or degenerate residuals.