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In a closed background, the information lost by the scalar functional can be described exactly: for the model references below, two functionals agree precisely when the masses agree. This also gives examples in which the supporting probabilities stay fixed while the potentials have different limits.
In this section $X$ is a compact Kähler manifold and $\theta $ is any smooth closed big real $(1,1)$-form. Let $\varphi \in \mathrm{PSH}(X,\theta )$ satisfy $P_\theta [\varphi ]=\varphi $, have small unbounded locus in the sense of section 5.33, and have positive mass
\[ m_\varphi =\int _ X\mathrm{MA}_\theta (\varphi )>0. \]
Write $B_{\theta ,\varphi }$ and $\mathcal A_{\theta ,\varphi }$ for the normalization scalar and the supporting sets of the equation (11.33.1) in the ordinary type of $\varphi $, and put $Z_ F=\int _ X\mathrm{e}^ F\, \mathrm{d}V$.
Theorem 11.42.1.1DCF668EBCCF For every such pair $(\theta ,\varphi )$ and every bounded measurable $F$, the equation in the ordinary type of $\varphi $ has a unique constant and a solution unique modulo constants. Its normalization scalar is
\[ B_{\theta ,\varphi }(F)=\log m_\varphi -\log Z_ F. \]
For continuous real $F$, its supporting set is
\[ \mathcal A_{\theta ,\varphi }(F)=\{ \mathrm{e}^ F\mathrm{d}V/Z_ F\} . \]
For two such pairs on the same $X$, with the same $\mathrm{d}V$ and possibly in different classes,
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11.42.2
\begin{equation} \sup _{F\in L^\infty }|B_{\theta ,\varphi }(F)-B_{\theta ',\psi }(F)| =|\log m_\varphi -\log m_\psi |. \end{equation}
Thus an invariant of these pairs is determined by the whole functional if and only if it is constant on every fiber of the map $(\theta ,\varphi )\mapsto m_\varphi $.
Proof.
Existence follows from corollary 10.31.56. Indeed, a Kähler manifold has the bounded mass property by lemma 19.9.12, and since in a closed background all potentials of the ordinary type of $\varphi $ have the same mass $m_\varphi >0$, the reference satisfies the positive-volume model condition (13.20.5). Uniqueness modulo constants is the classical theorem on equations with prescribed singularities in closed big classes, which applies under the model and small-unbounded-locus hypotheses; see the notes in section 25.13.
Every potential of the ordinary type of $\varphi $ has mass $m_\varphi $, so integrating the equation forces $cZ_ F=m_\varphi $, which is the scalar formula. For continuous $F$, Jensen's inequality shows, as for (11.37.2), that the normalized exponential measure is a supporting probability, and differentiating at zero in both signs of every continuous direction shows that it is the only one. Subtracting the scalar formulas gives (11.42.2). Finally, $m_\varphi =\mathrm{d}V(X)\exp B_{\theta ,\varphi }(0)$, so the functional and the mass determine each other, and an invariant factors through either one exactly when it is constant on the fibers of the mass map.
$\square$
For an ordinary divisorial type whose residual form $\beta $ is positive and closed, theorem 11.41.1 gives the same formula with mass $\int _ X\beta ^ n$, since Stokes' theorem for bounded products in a closed background fixes the mass of every bounded $\beta $-plurisubharmonic function. This computation does not require the real weighted sum of divisors to have a model representative. If $\beta _ j\to \beta $ uniformly, the masses converge by continuity of $\beta \mapsto \int _ X\beta ^ n$; under the two-sided bounds $(1-\varepsilon _ j)\beta \leq \beta _ j\leq (1+\varepsilon _ j)\beta $ one has the sharper inequalities $(1-\varepsilon _ j)^ n\int \beta ^ n\leq \int \beta _ j^ n\leq (1+\varepsilon _ j)^ n\int \beta ^ n$.
Example 11.42.3.C6588E30CFF6 Let $X=\mathbb P^ n$ with $n\geq 1$, let $\theta $ be the Fubini–Study form, normalized by $\int _ X\theta ^ n=1$, and let $D$ be a hyperplane. Let $s_ D$ be a section of $\mathcal O(1)$ defining $D$, measured with the Fubini–Study metric and rescaled so that $a_ D=\log |s_ D|^2\leq 0$, and choose $t\in (0,1)$. Then
\[ \mathrm{dd}^{\mathrm{c}}a_ D=[D]-\theta ,\qquad \theta +\mathrm{dd}^{\mathrm{c}}(ta_ D)=(1-t)\theta +t[D]. \]
Put $\varphi _{t,D}=P_\theta [ta_ D]$. By proposition 18.4.8, this envelope fixed point has the ordinary type of $ta_ D$. Moreover $ta_ D\in \mathrm{PSH}(X,t\theta )$ and $ta_ D\leq \varphi _{t,D}$, so $\varphi _{t,D}$ satisfies (16.28.67) with $\varepsilon =t$. Off $D$, the potential $ta_ D$ has Monge–Ampère measure $(1-t)^ n\theta ^ n$, and its non-pluripolar measure puts no mass on $D$.
Since mass is invariant under bounded changes of the potential in a closed background,
\[ m_{\varphi _{t,D}}=(1-t)^ n,\qquad B_{\varphi _{t,D}}(G)-B_0(G)=n\log (1-t)\quad (G\in C(X,\mathbb R)). \]
Thus the constant in proposition 11.39.1 is sharp.
Now let $n=1$, and take for $D$ two distinct points $p,q\in \mathbb P^1$. The corresponding references have distance zero, although $\varphi _{t,p}$ is unbounded near $p$ and bounded near $q$, while $\varphi _{t,q}$ behaves in the opposite way; in particular they are not ordinarily equivalent. For every positive integer $m$,
\[ \mathcal I(m\varphi _{t,p})=\mathcal O(-\lfloor mt\rfloor p),\qquad \mathcal I(m\varphi _{t,q})=\mathcal O(-\lfloor mt\rfloor q), \]
by lemma 5.37.1, and for $mt\geq 1$ these ideals differ. Thus the distance $\Delta $ of (11.39.3) does not even separate multiplier-ideal types, and it determines neither the ordinary nor the ideal singularity type.
Fix a smooth real function $F$ on $\mathbb P^1$, and let $w_ F$ be the solution of $\theta +\mathrm{dd}^{\mathrm{c}}w_ F=\mathrm{e}^ F\mathrm{d}V/Z_ F$ with $\int _ Xw_ F\, \mathrm{d}V=0$. The normalized solutions for $D=p$ and $D=q$ are
\[ u_{t,D}=ta_ D+(1-t)w_ F-\sup _ X\bigl (ta_ D+(1-t)w_ F\bigr ). \]
Their difference satisfies $\mathrm{dd}^{\mathrm{c}}(u_{t,p}-u_{t,q})=t([p]-[q])\ne 0$. Hence the sequence of solutions obtained by alternating between $D=p$ and $D=q$ has two different $L^1$ cluster points, although the scalar functional and all its supporting sets are the same at every step. Within this one-parameter family, the functional does recover $t$, since $t\mapsto (1-t)^ n$ is injective.
Example 11.42.4.0B7F3ACBCCC4 On $\mathbb P^ n$, $n\geq 2$, let $\omega $ be the Fubini–Study form, normalized as in example 11.42.3, and let $f$ be a nonconstant smooth real function. Fix $0<t<1$ and put
\[ \beta =\mathrm{e}^ f\omega ,\qquad \theta =\beta +t\omega ,\qquad \varphi _ D=t\log |s_ D|^2. \]
Here $s_ D$ defines a hyperplane $D$ as in example 11.42.3, and $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ D=\beta +t[D]$, so every hyperplane has the same positive residual form $\beta $. This form is not closed: $\mathrm{d}\beta =\mathrm{e}^ f\mathrm{d}f\wedge \omega \not\equiv 0$, since $\mathrm{d}f$ is nonzero somewhere and wedging a one-form with $\omega ^{n-1}$ is injective. By theorem 11.41.1, the ordinary types of all $\varphi _ D$ have the same functional $B_\beta $ on bounded tests, hence the same supporting sets. At a smooth test, the common supporting probability is the Gauduchon adjoint probability of theorem 11.34.9.
If $v_ F$ is the common smooth residual solution, the normalized solutions are the functions $\varphi _ D+v_ F-\sup _ X(\varphi _ D+v_ F)$. Two distinct hyperplanes $D_1,D_2$ give different poles, different divisorial coefficients and, for sufficiently large $k$, different multiplier ideals $\mathcal O(-\lfloor kt\rfloor D_ i)$, by the calculation in normal coordinates of lemma 5.37.1. The potentials differ by more than a constant, since the difference of their complex Hessians is $t([D_1]-[D_2])$, and alternating between the two hyperplanes again produces two distinct $L^1$ limits. These conclusions come from the common residual form; for general nonclosed backgrounds we do not claim that the functional is determined by a mass.
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