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18.28 Partial Bergman kernels065B04510A8A

065B04510A8A

The classical Bergman kernel of a Hermitian line bundle $L$ on a compact Kähler manifold $X$ encodes, through its asymptotic expansion as the tensor power $L^ k$ goes to infinity, deep geometric information about the underlying metric. This chapter develops the corresponding partial Bergman kernel. The word partial refers to the fact that we only use sections cut out by an ideal, typically the multiplier ideal of an auxiliary psh weight on a closed subset $K\subseteq X$.

The main result is the convergence theorem theorem 18.28.49. Given a $\theta $-psh potential $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, a closed non-pluripolar weighted subset $(K,v)$, and a Bernstein–Markov measure $\mu $, the normalized partial Bergman measures on $X$ converge weakly to the partial equilibrium measure of $(K,v,\varphi )$ as $k\to \infty $. This is the relative version of the theorem of Berman–Boucksom–Witt Nyström, corresponding to the case $\varphi =V_\theta $. The assumptions are essentially the mildest ones under which the statement makes sense.

The first part of the chapter introduces the relative envelope operators. In subsection 18.28.1 we define the relative $P$-envelope $P_{\theta ,K}[\varphi ](v)$ and the relative $\mathcal{I}$-envelope $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$ by replacing the constraint $\eta \leq 0$ in the usual $P$- and $\mathcal{I}$-envelopes with the constraint $\eta |_ K\leq v$. We establish their basic properties, including monotonicity, mass-preservation, and the relation with the envelope operators of section 16.1.

The second part develops the equilibrium theory associated with a weighted subset $(K,v)$. In subsection 18.28.19 we show that the Monge–Ampère measure $\theta _{P_{\theta ,K}[\varphi ](v)}^ n$ is carried by the contact locus $\{ P_{\theta ,K}[\varphi ](v)|_ K=v\} \subseteq K$, which justifies the name partial equilibrium measure. The chapter concludes in subsection 18.28.37 with the proof of the weak convergence of partial Bergman measures towards this partial equilibrium.

18.28.1 Partial envelopes9E595DC6361B

Let $X$ be a connected compact Kähler manifold of dimension $n$ and $K \subseteq X$ be a closed non-pluripolar set. Let $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a pseudo-effective cohomology class.

Definition 18.28.2.28F090ED5EFA Given a function $f\colon K\rightarrow [-\infty ,\infty ]$ and $\varphi \in \mathrm{PSH}(X,\theta )$, we introduce the relative $P$-envelope of $\varphi $ (with respect to $K,f,\theta $) as

19BBFEFEB969 18.28.3
\begin{equation} P_{\theta ,K}[\varphi ](f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq \varphi \Bigr \} . \end{equation}

Similarly, we define the relative $\mathcal{I}$-envelope of $\varphi $ (with respect to $K,f,\theta $) as

1F7034C27B5A 18.28.4
\begin{equation} P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _{\mathcal{I}} \varphi \Bigr \} . \end{equation}
By $\eta |_ K\leq f$ quasi-everywhere, we mean that there is a pluripolar set $S\subseteq X$ so that $\eta |_{K\setminus S}\leq f|_{K\setminus S}$. In general the upper envelope in (18.28.3) or (18.28.4) may be identically $\infty $.

Observe that when $f$ is bounded from below, neither envelope is identically $-\infty $.

The envelopes contain essentially all envelope operators that we have studied here:

Example 18.28.5.D0D0EA7C293D When $K=X$ and $f=0$, we have

\[ P_{\theta ,X}[\varphi ](0)=P_{\theta }[\varphi ],\quad P_{\theta ,X}[\varphi ]_{\mathcal{I}}(0)=P_{\theta }[\varphi ]_{\mathcal{I}}. \]

When $K=X$, we have

\[ P_{\theta ,X}[\varphi ](f)=P_{\theta }[\varphi ](f). \]

These definitions reduce to the envelopes studied in section 16.1. On the other hand, when $\varphi =V_{\theta }$, we have

\[ \begin{aligned} P_{\theta ,K}[V_{\theta }](f) & =P_{\theta ,K}[V_{\theta }]_{\mathcal{I}}(f)\\ & =\operatorname *{sup}_{}{}\! {}^{*}\left\{ \eta \in \mathrm{PSH}(X,\theta ): \begin{array}{l} \eta |_ K \leq f \textup{ quasi-everywhere} \end{array} \right\} . \end{aligned} \]

In an even more special case where $\varphi =V_{\theta }$ and $K=X$, we find that

\[ P_{\theta ,X}[V_{\theta }](f)=\operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta \leq f \textup{ quasi-everywhere}\Bigr \} . \]

This is nothing but $P_{\theta }(f)$ as we studied in definition 13.13.18.

Proposition 18.28.6.C9561913A3E8 Let $f,g\colon K\rightarrow [-\infty ,\infty ]$ be functions and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then we have the following:

  1. 33B49F7D21A7

    We have

    \[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f); \]
  2. 6B1193ABBD59

    assume that $f$ is bounded from above, then

    \[ \begin{aligned} \left.P_{\theta ,K}[\varphi ](f)\right|_ K\leq & f\textup{ quasi-everywhere},\\ \left. P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\right|_ K\leq & f\textup{ quasi-everywhere}; \end{aligned} \]
  3. 27E6FD2D7332

    suppose that $\varphi \preceq _{\mathcal{I}}\psi $, then

    \[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\psi ]_{\mathcal{I}}(f); \]
  4. B3B86BE1530D

    if $f\leq g$, then

    \[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ](g),\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(g); \]
  5. 13DB8823929A

    for any $t \in (0,1)$, we have

    \[ \begin{aligned} P_{\theta ,K}\left[t\varphi +(1-t)\psi \right](f)\geq & tP_{\theta ,K}[\varphi ](f)+(1-t)P_{\theta ,K}[\psi ](f),\\ P_{\theta ,K}\left[t\varphi +(1-t)\psi \right]_{\mathcal{I}}(f)\geq & tP_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)+(1-t)P_{\theta ,K}[\psi ]_{\mathcal{I}}(f). \end{aligned} \]

Proof. 1 This follows from $\eta \preceq \varphi \Rightarrow \eta \preceq _{\mathcal I}\varphi $.

2 This follows from proposition 5.14.5 and 5.17.4.

3 This follows by inclusion of the defining candidate sets.

4 This follows by inclusion of the defining candidate sets.

5 If $\eta ,\eta '\in \mathrm{PSH}(X,\theta )$ are such that

\[ \begin{split} & \eta |_ K\leq f\textup{ quasi-everywhere},\quad \eta '|_ K\leq f\textup{ quasi-everywhere},\\ & \eta \preceq \varphi ,\quad \eta '\preceq \psi , \end{split} \]

then for $t\in (0,1)$,

\[ \left.\left(t\eta +(1-t)\eta '\right)\right|_ K\leq f\textup{ quasi-everywhere},\quad t\eta +(1-t)\eta ' \preceq t\varphi +(1-t)\psi . \]

Therefore,

\[ t\eta +(1-t)\eta '\leq P_{\theta ,K}\left[t\varphi +(1-t)\psi \right](f). \]

Since $\eta $ and $\eta '$ are arbitrary, the first inequality follows. The second follows from the same argument. $\square$

Similar to the usual envelopes studied in section 16.1, the $P$-partial envelopes are well-behaved only when $\varphi $ has positive mass.

Lemma 18.28.7.0F9B18404274 Suppose that $f\colon K\rightarrow (-\infty ,\infty ]$ is bounded from below, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

96B9EFFFBDA7 18.28.8
\begin{equation} P_{\theta ,K}[\varphi ](f)= \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq \varphi \Bigr \} . \end{equation}
In other words, in this situation, in the definition (18.28.3), we may restrict to only those $\eta $ with positive mass.

Proof. By our assumption, we can find $C>0$ so that $\varphi |_ K-C\leq f$. Then if $\eta $ is a candidate in (18.28.3), $\eta \lor (\varphi -C)$ is also a candidate, and $\eta \lor (\varphi -C)\in \mathrm{PSH}(X,\theta )_{>0}$. Therefore, $\eta \lor (\varphi -C)$ is also a candidate for the right-hand side of (18.28.8). The equality (18.28.8) follows. $\square$

Note that in the most important cases, we can actually replace the condition $\eta \leq f$ quasi-everywhere in (18.28.3) by everywhere.

Lemma 18.28.9.A86A369D4CBF Let $f\colon K\rightarrow (-\infty ,\infty ]$ be bounded from below, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

226F49B219B2 18.28.10
\begin{equation} \begin{split} P_{\theta ,K}[\varphi ](f)=& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ and } \eta \preceq \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ and } \eta \preceq \varphi \Bigr \} . \end{split} \end{equation}

Proof. Both $\geq $ directions in (18.28.10) are trivial.

Conversely, let $\eta $ be a candidate in (18.28.8). Let $S\subseteq X$ be a pluripolar set so that $\eta |_{K\setminus S}\leq f|_{K\setminus S}$.

We now claim that we can choose $\chi \in \mathrm{PSH}(X,\theta )_{>0}$ with $\chi |_ S\equiv -\infty $ and $\chi \leq \eta $.

Assume this for the moment. For any $\delta \in (0,1)$, we have

\[ (1-\delta )\eta |_ K+\delta \chi |_ K\leq f,\quad (1-\delta )\eta +\delta \chi \preceq \varphi . \]

Hence, $(1-\delta )\eta +\delta \chi $ is dominated by the right-hand side of (18.28.10). Letting $\delta \to 0+$, we conclude that $\eta $ is dominated by the right-hand side of (18.28.10) and hence, (18.28.10) follows.

It remains to construct $\chi $. Since $\varphi $ has positive mass, the class $\{ \theta \} $ is big. Fix a Kähler form $\omega _0$ on $X$ and choose $\rho \in \mathrm{PSH}(X,\theta )$ with $\theta _{\rho }\geq 2\epsilon _0\omega _0$ for some $\epsilon _0>0$. By theorem 5.13.20, there is a quasi-psh function $q$ such that $q|_ S\equiv -\infty $. Choose $A>0$ so that $A\omega _0+\mathrm{dd}^{\mathrm{c}}q\geq 0$. For $a>0$ small enough, $\rho +aq\in \mathrm{PSH}(X,\theta )$ and still has pole $-\infty $ on $S$. Replacing $\rho $ by $\rho +aq$, we may therefore assume that $\rho |_ S\equiv -\infty $. Since $\int _ X \theta _{\eta }^ n>0$, fixing such a $\rho $ and taking $\epsilon \in (0,1)$ small enough, we have

\[ \int _ X \theta ^ n_{\epsilon \rho +(1-\epsilon )V_{\theta }}+\int _ X \theta _{\eta }^ n>\int _ X \theta _{V_{\theta }}^ n. \]

Thus, by proposition 16.3.25 and theorem 16.3.35, we have

\[ \Bigl (\epsilon \rho +(1-\epsilon )V_{\theta }\Bigr ) \land \eta \in \mathrm{PSH}(X,\theta )_{>0}. \]

It suffices to define this function as $\chi $. $\square$

Corollary 18.28.11.712499DFC37F Suppose that $f\colon K\rightarrow \mathbb {R}$ is bounded, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then there is a constant $C=C(\sup _ K f,K,X)>0$ such that

\[ P_{\theta ,K}[\varphi ](f)\leq C,\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq C. \]

Proof. Since $K$ is non-pluripolar, 5.17.4 then gives a constant $C>0$, depending only on $\sup _ K f$, $K$ and $X$, such that every $\eta \in \mathrm{PSH}(X,\theta )$ satisfying $\eta |_ K\leq f$ quasi-everywhere satisfies $\eta \leq C$ on $X$.

This applies to the candidates defining both envelopes. Hence both candidate families are locally uniformly bounded above, so their upper semicontinuous suprema are genuine $\theta $-psh functions. By proposition 18.28.6 2, both envelopes are dominated by $f$ quasi-everywhere on $K$. Applying 5.17.4 once more gives

\[ P_{\theta ,K}[\varphi ](f)\leq C,\quad P_{\theta ,K}[\varphi ]_{\mathcal I}(f)\leq C. \]
$\square$

Proposition 18.28.12.A1C36A6249C7 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Suppose that $\varphi \preceq _{P} \psi $. Then

\[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\psi ](f). \]

Proof. Let $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ be such that $\eta |_ K\leq f$ quasi-everywhere and $\eta \preceq \varphi $. Then $\eta \preceq _ P \psi $, and thanks to corollary 16.8.30 for any $C>0$, we have $\eta \land (\psi +C)\in \mathrm{PSH}(X,\theta )_{>0}$. But then $\eta \land (\psi +C)$ is a candidate in the definition of $P_{\theta ,K}[\psi ](f)$, and we find

\[ \eta \land (\psi +C)\leq P_{\theta ,K}[\psi ](f). \]

By corollary 16.8.30 again, we get

\[ \eta =P_{\theta }[\psi ](\eta )\leq P_{\theta ,K}[\psi ](f). \]

Our assertion follows now from 18.28.7. $\square$

Corollary 18.28.13.72097BEFEC59 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

\[ \begin{aligned} P_{\theta ,K}[\varphi ](f)= & \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} \\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ and } \eta \preceq _ P \varphi \Bigr \} . \end{aligned} \]

Proof. We first apply proposition 18.28.12,

\[ \begin{split} P_{\theta ,K}[\varphi ](f)= & P_{\theta ,K}\left[P_{\theta }[\varphi ]\right](f)\\ =& \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \end{split} \]

Next observe that by 18.28.7, the latter can also be written as

\[ \operatorname *{sup}_{}{}\! {}^{*}\Bigl \{ \eta \in \mathrm{PSH}(X,\theta )_{>0}: \eta |_ K \leq f \textup{ quasi-everywhere and } \eta \preceq _ P \varphi \Bigr \} . \]

Next we apply 18.28.9; the remaining statements follow. $\square$

Corollary 18.28.14.96B842B9BF11 Let $f\colon K\rightarrow (-\infty ,\infty ]$ be a function bounded from below and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ be $\mathcal{I}$-good. Then

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)=P_{\theta ,K}[\varphi ](f). \]

Proof. In fact, since $\varphi $ is $\mathcal{I}$-good, a potential $\eta \in \mathrm{PSH}(X,\theta )$ satisfies $\eta \preceq _ P \varphi $ if and only if $\eta \preceq _{\mathcal{I}} \varphi $. Therefore, our assertion follows from corollary 18.28.13 and (18.28.4). $\square$

Next, we make the following observations about the singularity types of our envelopes:

Lemma 18.28.15.18E5130ACB80 Let $f \colon K\rightarrow \mathbb {R}$ be a bounded function, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. We have

EC4A564BADAB 18.28.16
\begin{equation} P_{\theta ,K}[\varphi ](f)\sim P_{\theta }[\varphi ],\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\sim P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}

Proof. We first prove the $P$-part of (18.28.16).

If $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta \leq 0$, then $\eta +\inf _ K f$ is a candidate in the definition (18.28.3), and hence,

\[ \eta +\inf _ K f \leq P_{\theta ,K}[\varphi ](f). \]

It follows that

\[ P_{\theta }[\varphi ]+\inf _ K f \leq P_{\theta ,K}[\varphi ](f). \]

Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be such that

\[ \eta |_ K\leq f,\quad \eta \preceq \varphi . \]

Since $K$ is non-pluripolar, by 5.17.4 we can find $C=C(\sup _ K f,K, X)>0$ so that $\eta \leq C$. Hence $\eta -C$ is a candidate in the definition of $P_{\theta }[\varphi ]$, and hence

\[ \eta -C\leq P_{\theta }[\varphi ]. \]

By 18.28.9, this implies that

\[ P_{\theta ,K}[\varphi ](f) \leq P_{\theta }[\varphi ] + C. \]

The proof of the $\mathcal{I}$-part is similar, but we spell it out since this comparison will be used below. If $\eta \in \mathrm{PSH}(X,\theta )$ satisfies $\eta \leq 0$ and $\eta \preceq _{\mathcal{I}}\varphi $, then $\eta +\inf _ K f$ is a candidate in (18.28.4). Hence

\[ P_{\theta }[\varphi ]_{\mathcal{I}}+\inf _ K f\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f). \]

Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be a candidate in (18.28.4). Since $K$ is non-pluripolar and $f$ is bounded, the same compactness estimate gives a constant $C>0$, independent of $\eta $, such that $\eta \leq C$. Then $\eta -C\leq 0$ and $\eta -C\preceq _{\mathcal{I}}\varphi $. By corollary 16.8.16,

\[ \eta -C\leq P_{\theta }[\varphi ]_{\mathcal{I}}. \]

Taking the upper envelope over all such $\eta $ gives

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta }[\varphi ]_{\mathcal{I}}+C. \]

This completes the proof. $\square$

Corollary 18.28.17.CE1417F951E3 Let $f\colon K\rightarrow \mathbb {R}$ be a bounded function, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

A99869D43853 18.28.18
\begin{equation} P_{\theta ,K}[\varphi ](f)=P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \end{equation}

Proof. Take any $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta |_ K\leq f$ quasi-everywhere, then

\[ \eta \leq P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[V_{\theta }](f). \]

Therefore,

\[ \eta \leq P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \]

So we conclude the $\leq $ direction in (18.28.18).

For the reverse direction, take $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ such that

\[ \eta \preceq \varphi ,\quad \eta \leq P_{\theta ,K}[V_{\theta }](f) \quad \textup{quasi-everywhere}. \]

Then $\eta |_ K\leq f$ quasi-everywhere by proposition 18.28.6. Hence

\[ \eta \leq P_{\theta ,K}[\varphi ](f). \]

By corollary 18.28.13, the $\geq $ direction in (18.28.18) follows. $\square$

18.28.19 Partial equilibrium measuresB13AC61C497D

Let $X$ be a connected compact Kähler manifold of dimension $n$, and $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a pseudo-effective cohomology class.

Definition 18.28.20.E4ED4F8B05A8 A weighted subset of $X$ is a pair $(K,v)$ consisting of a closed non-pluripolar subset $K\subseteq X$ and a function $v\in C^0(K)$.

The next result motivates our terminology to call the measures $\theta _{P_{\theta ,K}[\varphi ](f)}^ n$ the partial equilibrium measures of our context:

Lemma 18.28.21.D7B3984EBF7A Let $(K,v)$ be a weighted subset of $X$, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

2F195F41BD70 18.28.22
\begin{equation} \int _{X\setminus K}\theta _{P_{\theta ,K}[\varphi ](v)}^ n=0,\quad \int _ K \mathds {1}_{\left\{ P_{\theta ,K}[\varphi ](v)|_ K\neq v\right\} }\theta _{P_{\theta ,K}[\varphi ](v)}^ n=0. \end{equation}

More precisely, we have

D22CB4C7F0CC 18.28.23
\begin{equation} \theta _{P_{\theta ,K}[\varphi ](v)}^ n\leq \mathds {1}_{\left\{ P_{\theta ,K}[\varphi ](v)|_ K=P_{\theta ,K}[V_{\theta }](v)|_ K=v\right\} }\, \theta _{P_{\theta ,K}[V_{\theta }](v)}^ n. \end{equation}
Here $\{ P_{\theta ,K}[\varphi ](v)|_ K\neq v\} $ and $\{ P_{\theta ,K}[\varphi ](v)|_ K=P_{\theta ,K}[V_{\theta }](v)|_ K=v\} $ are understood as consisting of points of $K$ satisfying these constraints.

Proof. Step 1. We first prove (18.28.22) when $\varphi = V_{\theta }$.

Let $S\subseteq X$ be a proper analytic subset, such that $V_{\theta }$ is locally bounded on $X\setminus S$. Such $S$ exists by 13.13.8. Fix holomorphic balls $B\Subset B'\Subset X\setminus S$.

By proposition 5.14.3 and 18.28.9, we can take an increasing sequence $(\eta _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ such that

\[ \eta _ j\nearrow P_{\theta ,K}[V_{\theta }](v)\textup{ a.e.},\quad \eta _ j|_ K\leq v\textup{ for all }j\geq 1. \]

Note that $P_{\theta ,K}[V_{\theta }](v)\sim V_{\theta }$ by 18.28.15. Take $C>0$ so that

\[ \left.\left( P_{\theta ,K}[V_{\theta }](v) \right)\right|_ K-C\leq v. \]

Then we may replace $\eta _ j$ by $\eta _ j\lor (P_{\theta ,K}[V_{\theta }](v)-C)$ and assume that $\eta _ j\sim V_{\theta }$ for each $j$. In particular, $\eta _ j$ is locally bounded on $X\setminus S$.

For each $j\geq 1$, let $\gamma _ j\in \mathrm{PSH}(X,\theta )$ be the balayage of $\eta _ j$ on $B$, defined in proposition 5.17.9. Note that $(\gamma _ j)_ j$ is increasing due to proposition 5.14.22.

For the first equality in (18.28.22), we may assume that $B\Subset X\setminus (S \cup K)$. It suffices to show that $\theta _{P_{\theta ,K}[V_{\theta }](v)}^ n$ does not charge $B$.

Since $B$ lies outside $K$ by assumption, we also have $\gamma _ j|_ K=\eta _ j|_ K \leq v$ for all $j\geq 1$. In particular, $\gamma _ j\leq P_{\theta ,K}[V_{\theta }](v)$.

It follows that $\gamma _ j \nearrow P_{\theta ,K}[V_{\theta }](v)$ almost everywhere as well. Testing against non-negative continuous functions supported in $B$ and using theorem 13.13.9, we find that $\theta ^ n_{P_{\theta ,K}[V_{\theta }](v)}$ does not charge $B$, as desired.

We next prove the second equality in (18.28.22). The proof is essentially the same as that of 13.13.21.

It suffices to show that $\theta ^ n_{P_{\theta ,K}[V_{\theta }](v)}$ does not charge the set $K\cap \{ P_{\theta ,K}[V_{\theta }](v)|_ K<v\} $.

Let $x\in (X\setminus S) \cap K$ be a point such that $P_{\theta ,K}[V_{\theta }](v)(x)<v(x)-\epsilon $ for some $\epsilon >0$. We may assume that $x\in B$ and $B'$ is small enough, so that

  1. B2B18BBFA88D

    $P_{\theta ,K}[V_{\theta }](v)|_{\overline{B'}}<v(x)-\epsilon $.

  2. D0D5ED334E0E

    $v|_{\overline{B'}\cap K}>v(x)-\epsilon $.

By 1, we have $\gamma _ j|_{\partial B}\leq v(x)-\epsilon $, hence by proposition 5.14.22, $\gamma _ j|_ B\leq v(x)-\epsilon $. By 2, we have $\gamma _ j|_{B\cap K}\leq v|_{B\cap K}$. On the other hand, $\gamma _ j|_{K\setminus B}=\eta _{j}|_{K\setminus B}\leq v|_{K\setminus B}$. Therefore, $\gamma _ j|_ K\leq v$. It follows again that $\gamma _ j\leq P_{\theta ,K}[V_{\theta }](v)$. Therefore, we conclude as in the previous paragraph that $\gamma _ j\nearrow P_{\theta ,K}[V_{\theta }](v)$. Thus, testing against non-negative continuous functions supported in $B$, we conclude that $\theta ^ n_{P_{\theta ,K}[V_{\theta }](v)}$ does not charge $B$ again by theorem 13.13.9.

Step 2. We handle the general case.

By corollary 18.28.17,

\[ P_{\theta ,K}[\varphi ](v) = P_{\theta ,X}[\varphi ]\left(P_{\theta ,K}[V_{\theta }](v)\right). \]

Now corollary 16.3.14 gives

\[ \begin{aligned} \theta _{P_{\theta ,K}[\varphi ](v)}^ n \leq & \mathds {1}_{\left\{ P_{\theta ,K}[\varphi ](v) = P_{\theta ,K}[V_{\theta }](v)\right\} } \theta _{P_{\theta ,K}[V_{\theta }](v)}^ n \\ \leq & \mathds {1}_{\left\{ P_{\theta ,K}[\varphi ](v)|_ K=P_{\theta ,K}[V_{\theta }](v)|_ K=v\right\} }\, \theta _{P_{\theta ,K}[V_{\theta }](v)}^ n, \end{aligned} \]

where in the second inequality we have used Step 1. Hence (18.28.23) follows, and a fortiori (18.28.22) follows as well. $\square$

Corollary 18.28.24.9790F808BF37 Let $v\in C^0(K)$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

FFCF7AC9E703 18.28.25
\begin{equation} \begin{aligned} \int _{(X\setminus K)\cup \left\{ P_{\theta ,K}[\varphi ](v)<v\right\} }\theta ^ n_{P_{\theta ,K}[\varphi ](v)}=& 0,\\ \int _{(X\setminus K)\cup \left\{ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)<v\right\} }\theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}=& 0. \end{aligned} \end{equation}

Proof. The first equation in (18.28.25) follows from 18.28.21. For the second, it suffices to observe that

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)=P_{\theta ,K}\left[P_{\theta }[\varphi ]_{\mathcal{I}}\right]_{\mathcal{I}}(v)=P_{\theta ,K}\left[P_{\theta }[\varphi ]_{\mathcal{I}}\right](v) \]

by corollary 18.28.14. $\square$

Definition 18.28.26.12F5599D8900 Given $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, we define the partial equilibrium energy functional $\mathcal{E}_{\theta ,K}^{\varphi }\colon C^0(K)\rightarrow \mathbb {R}$ as follows: given $v \in C^0(K)$, we set

14.30
\begin{equation} \mathcal{E}_{\theta ,K}^{\varphi }(v)\coloneqq E_{\theta }^{P_{\theta }[\varphi ]_{\mathcal{I}}}\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right). \end{equation}
Recall that the energy $E_{\theta }^{P_{\theta }[\varphi ]_{\mathcal{I}}}$ functional is defined in definition 16.5.10.

Note that by 18.28.15, we have

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\in \mathcal{E}^{\infty }\left(X,\theta ;P_{\theta }[\varphi ]_{\mathcal{I}}\right), \]

so $\mathcal{E}_{\theta ,K}^{\varphi }(v)\in \mathbb {R}$.

Proposition 18.28.27.765C3766A729 Let $v,f \in C^0(K)$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then $\mathbb {R}\ni t \mapsto \mathcal{E}_{\theta ,K}^{\varphi }(v+tf)$ is differentiable and

0B2DE1BB9170 18.28.28
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}_{\theta ,K}^{\varphi }(v+tf) = \int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)} \end{equation}

for all $t\in \mathbb {R}$.

Proof. We may assume that $\varphi $ is $\mathcal{I}$-model by replacing $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$.

Note that it suffices to prove (18.28.28) at $t=0$, which is equivalent to

94F4F91642C8 18.28.29
\begin{equation} \lim _{t\to 0}\frac{E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)}{t}=\int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}. \end{equation}

It suffices to prove the right derivative at $0$ for an arbitrary $f$; applying the same argument to $-f$ then gives the left derivative. In other words, we need to prove that

8353FF09478E 18.28.30
\begin{equation} \lim _{t\to 0+}\frac{E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)}{t}=\int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}. \end{equation}

We first reduce the proof of (18.28.30) to the case $f\leq 0$. Assume that this case is known. Choose $A\in \mathbb R$ such that $f-A\leq 0$ on $K$. Since for any $t>0$,

\[ P_{\theta ,K}[\varphi ]_{\mathcal I}\Bigl ( v+t(f-A) \Bigr )= P_{\theta ,K}[\varphi ]_{\mathcal I}(v+tf)-tA, \]

then we have

\[ \int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}=\int _ K (f-A)\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}+A\int _ X \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}. \]

By (16.5.13), for any $t>0$,

\[ E_{\theta }^{\varphi }\Bigl (P_{\theta ,K}[\varphi ]_{\mathcal{I}}\left(v+t(f-A)\right)\Bigr )=E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-tA\int _ X\theta _{\varphi }^ n. \]

So the limit in (18.28.30) exists and is equal to the left-hand side of (18.28.30) with $f-A$ in place of $f$ plus $A\int _ X\theta _{\varphi }^ n$. So (18.28.30) for $f-A$ implies the desired formula for $f$ in view of 18.28.15 and the monotonicity theorem theorem 13.13.11. Hence we may assume that $f\leq 0$.

Fix $t>0$. Set

\[ \eta _ t\coloneqq P_{\theta ,K}[\varphi ]_{\mathcal I}(v+tf),\quad \eta _0\coloneqq P_{\theta ,K}[\varphi ]_{\mathcal I}(v). \]

By the comparison principle (16.5.6) and proposition 16.5.30, we find

\[ \begin{aligned} E_{\theta }^{\varphi }(\eta _ t)-E_{\theta }^{\varphi }(\eta _0) ={}& \frac{1}{n+1}\sum _{i=0}^ n\int _ X(\eta _ t-\eta _0)\, \theta _{\eta _ t}^ i\wedge \theta _{\eta _0}^{n-i}\\ \leq {}& \int _ X(\eta _ t-\eta _0)\, \theta _{\eta _0}^ n. \end{aligned} \]

By 18.28.21 and corollary 18.28.14, the measure $\theta _{\eta _0}^ n$ is carried by $K\cap \{ \eta _0=v\} $, and the measure $\theta _{\eta _ t}^ n$ is carried by $K\cap \{ \eta _ t=v+tf\} $. On $K\cap \{ \eta _0=v\} $, we have $\eta _ t\leq v+tf=\eta _0+tf$, hence

\[ \eta _ t-\eta _0\leq tf. \]

On $K\cap \{ \eta _ t=v+tf\} $, we have $\eta _0\leq v$, hence

\[ \eta _ t-\eta _0\geq tf. \]

It follows that

\[ \int _ X(\eta _ t-\eta _0)\, \theta _{\eta _0}^ n \leq t\int _ K f\, \theta _{\eta _0}^ n \]

and

\[ \int _ X(\eta _ t-\eta _0)\, \theta _{\eta _ t}^ n \geq t\int _ K f\, \theta _{\eta _ t}^ n. \]

Thus, letting $t\to 0+$, we get the inequality

\[ \varlimsup _{t\to 0+}\frac{E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)}{t}\leq \int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}. \]

Similarly, for $t>0$, we have

\[ \begin{aligned} & E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)\\ \geq & \int _ X \left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)-P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)\, \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)}^ n\\ \geq & t\int _ K f\, \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)}^ n. \end{aligned} \]

Letting $t\to 0+$, we find

\[ \varliminf _{t\to 0+}\frac{E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)\right)-E_{\theta }^{\varphi }\left(P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)}{t}\geq \int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}. \]

Here we used the following elementary continuity consequence:

\[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)-t\| f\| _{C^0(K)} \leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf) \leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v) \]

for $t>0$, and hence $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf) \nearrow P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$ almost everywhere when $t\to 0+$. Next we claim that

\[ \int _ K f\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}\leq \lim _{t\to 0+} \int _ K f\, \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v+tf)}^ n. \]

Choose a continuous extension $\tilde f\in C^0(X)$ of $f$. Since 18.28.21, corollary 18.28.14 show that all the measures appearing here are carried by $K$, the integrals over $K$ against $f$ are the same as the integrals over $X$ against $\tilde f$. Thus corollary 13.13.12 applies.

We conclude (18.28.29). $\square$

Lemma 18.28.31.D21332713DB8 Assume that $\{ \theta \} $ represents a big cohomology class. Fix a Kähler form $\omega $ on $X$ and a weighted subset $(K,v)$ of $X$. There exists a uniformly bounded increasing sequence $(v^-_ j)_ j$ in $C^\infty (X)$ and a uniformly bounded decreasing sequence $(v^+_ j)_ j$ in $C^\infty (X)$, such that

32CDA8E3EB57 18.28.32
\begin{equation} \sup _ X |v_ j^-|\leq C,\quad \sup _ X |v_ j^+|\leq C \end{equation}

for some constant $C$ depending only on $\| v\| _{C^0(K)}$, $X$, $K$ and $\theta +\omega $. Furthermore, for any $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ and any $\delta \in [0,1]$ we have the following:

  1. 9C89F797B0BF

    We have

    \[ P_{\theta +\delta \omega ,X}[\varphi ](v_ j^+) \searrow P_{\theta +\delta \omega ,K}[\varphi ](v), \]

    and

    \[ P_{\theta +\delta \omega ,X}[\varphi ](v^-_ j) \nearrow P_{\theta +\delta \omega ,K}[\varphi ](v) \]

    almost everywhere.

  2. 4E87F51F9E7D

    Suppose in addition that $\varphi $ is $\mathcal{I}$-good, then

    \[ \lim _{j\to \infty }\mathcal{E}_{\theta ,X}^{\varphi }(v_ j^-)=\mathcal{E}_{\theta ,K}^{\varphi }(v),\quad \lim _{j\to \infty }\mathcal{E}_{\theta ,X}^{\varphi }(v_ j^+)=\mathcal{E}_{\theta ,K}^{\varphi }(v). \]

Proof. Step 1. First we construct $(v^-_ j)_ j$.

Let

\[ C_{K,v} \coloneqq \sup \left\{ \sup _ X \eta : \eta \in \mathrm{PSH}(X,\theta +\omega ), \eta |_ K \leq v\right\} . \]

Since $K$ is non-pluripolar, we have $C_{K,v} \in \mathbb {R}$ by 5.17.4. Set

\[ M_{K,v}\coloneqq \max \{ C_{K,v}+1,\sup _ K v\} . \]

Now define $\widetilde v \colon X \to \mathbb {R}$ as

\[ \widetilde{v}(x)=\left\{ \begin{aligned} v(x),& \quad x\in K;\\ M_{K,v},& \quad x\in X\setminus K. \end{aligned} \right. \]

Since $M_{K,v}\geq \sup _ K v$, the function $\widetilde v$ is lower semicontinuous. Therefore there exists an increasing and uniformly bounded sequence $(v^-_ j)_ j$ in $C^\infty (X)$ such that $v^-_ j \nearrow \widetilde v$ and

\[ \| v_ j^-\| _{C^0(X)}\leq \| \tilde{v}\| _{L^{\infty }(X)}+1. \]

The right-hand side can clearly be bounded by a constant as in (18.28.32).

Step 2. We construct $(v_ j^+)_ j$.

Since

\[ h \coloneqq P_{\theta + \omega ,K}[V_{\theta +\omega }](v) \lor \left(\inf _ K v -1\right) \]

is usc, there exists a decreasing and uniformly bounded sequence $(v^+_ j)_ j$ in $C^\infty (X)$ such that $v^+_ j \searrow h$. We may guarantee that

\[ \| v^+_ j\| _{C^0(X)}\leq \| h\| _{L^{\infty }(X)}+1. \]

Note that $\inf _ K v-1\leq h\leq C_{K,v} \lor (\inf _ K v-1)$. So (18.28.32) is satisfied.

Step 3. We finish the proof. Fix $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ and $\delta \in [0,1]$.

Note that 2 follows from corollary 18.28.14, 18.28.15, proposition 16.5.37 and 1. Let us prove 1.

Step 3.1. We prove 1 for $(v^-_ j)_ j$.

Observe that $P_{\theta + \delta \omega ,X}[\varphi ](v^-_ j)$ is increasing in $j\geq 1$, and

\[ P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right)\leq P_{\theta + \delta \omega ,K}[\varphi ](v). \]

We claim that

A58ACF533BAE 18.28.33
\begin{equation} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right) \nearrow P_{\theta + \delta \omega ,K}[\varphi ](v)\quad \textup{a.e.}. \end{equation}

Let $\eta \in \mathrm{PSH}(X,\theta +\delta \omega )$ be a potential such that $\eta \preceq \varphi $ and $\eta |_ K\leq v$. Take some $\epsilon >0$. Then, since $\eta $ is upper semicontinuous, $\widetilde v$ is lower semicontinuous and $\eta < \epsilon +\widetilde v$, compactness gives $j_0>0$ such that $\eta < \epsilon + v^-_ j$ for $j \geq j_0$. Hence,

\[ \eta \leq \epsilon +P_{\theta + \delta \omega ,X}[\varphi ](v^-_ j) \]

for $j\geq j_0$. Therefore,

\[ \eta \leq \operatorname *{sup}_{j\geq 1}{}\! {}^{*} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right)+\epsilon . \]

Letting $\epsilon \to 0+$, we find

\[ \eta \leq \operatorname *{sup}_{j\geq 1}{}\! {}^{*} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right). \]

Since $\eta $ is arbitrary, (18.28.33) follows.

Step 3.2. We prove 1 for $(v^+_ j)_ j$.

Note that $P_{\theta + \delta \omega ,X}[\varphi ](v^+_ j)$ is decreasing in $j$, and

11BA40C76F9B 18.28.34
\begin{equation} \chi \coloneqq \lim _{j\to \infty } P_{\theta + \delta \omega ,X}[\varphi ]\left(v^+_ j\right) \geq P_{\theta + \delta \omega ,K}[\varphi ](v). \end{equation}

In particular, $\chi \in \mathrm{PSH}(X,\theta +\delta \omega )_{>0}$, since $P_{\theta + \delta \omega ,K}[\varphi ](v)$ has the same singularity types as $P_{\theta + \delta \omega }[\varphi ]$ by 18.28.15. Observe that

\[ P_{\theta + \delta \omega ,X}[\varphi ]\left(v_ j^+\right)\leq v_ j^+ \]

for all $j\geq 1$, since $v_ j^+$ is continuous. Thus, $\chi \leq h$. On the other hand, $h|_ K\leq v$ quasi-everywhere. So in particular, $\chi |_{K}\leq v$ quasi-everywhere. On the other hand,

\[ \chi \leq P_{\theta + \delta \omega ,X}[\varphi ]\left(v^+_1\right)\sim _ P \varphi \]

by 18.28.15. Hence, by corollary 18.28.13,

\[ \chi \leq P_{\theta + \delta \omega ,K}[\varphi ](v). \]

Therefore, (18.28.34) is an equality. $\square$

Proposition 18.28.35.9724C0B32032 Let $(K,v)$ be a weighted subset of $X$. Let $(\varphi _ j)_ j$ be a sequence in $\mathrm{PSH}(X,\theta )_{>0}$ and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then the following hold:

  1. BE5DB43FE077

    If $(\varphi _ j)_ j$ is decreasing, and $\varphi _ j\xrightarrow {d_ S}\varphi $, then

    1BDE5E6D4627 18.28.36
    \begin{equation} P_{\theta ,K}[\varphi _ j]_{\mathcal{I}}(v) \searrow P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v),\quad P_{\theta ,K}[\varphi _ j](v) \searrow P_{\theta ,K}[\varphi ](v). \end{equation}
  2. E712A3A2D87D

    If $\varphi _ j \nearrow \varphi $ almost everywhere then

    \[ P_{\theta ,K}[\varphi _ j]_{\mathcal{I}}(v) \nearrow P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\textup{ a.e.},\quad P_{\theta ,K}[\varphi _ j](v) \nearrow P_{\theta ,K}[\varphi ](v) \textup{ a.e.}. \]

Proof. We only give the proof for the relative $\mathcal{I}$-envelopes; the proof for the relative $P$-envelopes is identical, replacing $P_{\theta }[\bullet ]_{\mathcal{I}}$ by $P_{\theta }[\bullet ]$ and using corollary 18.28.13 for the final candidate condition.

Set

\[ \eta _ j\coloneqq P_{\theta ,K}[\varphi _ j]_{\mathcal{I}}(v),\quad \eta \coloneqq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v). \]

By 18.28.15, $d_ S(\eta _ j,P_{\theta }[\varphi _ j]_{\mathcal{I}})=0$ and $d_ S(\eta ,P_{\theta }[\varphi ]_{\mathcal{I}})=0$ for all $j$. Moreover, theorem 16.9.49 gives

\[ P_{\theta }[\varphi _ j]_{\mathcal{I}}\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}} \]

in case 1, and the same convergence holds in case 2 by corollary 16.9.12 and theorem 16.9.49. Consequently, $\eta _ j\xrightarrow {d_ S}\eta $ in both cases.

Assume first that $(\varphi _ j)_ j$ is decreasing. Then $(\eta _ j)_ j$ is decreasing and $\eta _ j\geq \eta $. Let $\eta _\infty \coloneqq \inf _ j \eta _ j$. The Monge–Ampère masses of $\eta _ j$ converge to the mass of $\eta $, by the preceding $d_ S$-convergence and theorem 16.9.30. Since $\eta _ j\geq \eta _\infty \geq \eta $, the monotonicity theorem theorem 13.13.11 gives

\[ \int _ X\theta _{\eta _\infty }^ n=\int _ X\theta _\eta ^ n. \]

Hence $\eta _ j\xrightarrow {d_ S}\eta _\infty $ by corollary 16.9.14. The uniqueness of the $d_ S$-limit gives $d_ S(\eta _\infty ,\eta )=0$. Since $\eta _\infty \geq \eta $, proposition 16.9.6 implies $\eta _\infty \sim _ P \eta $. In particular, $\eta _\infty \preceq _{\mathcal{I}}\varphi $ by corollary 16.8.14. The condition $\eta _\infty \leq v$ on $K$ quasi-everywhere is inherited from the $\eta _ j$'s, so $\eta _\infty $ is a candidate for $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$. Thus $\eta _\infty \leq \eta $, and the reverse inequality was already observed. This proves the $\mathcal{I}$-part of 1.

Assume next that $\varphi _ j\nearrow \varphi $ almost everywhere. Then $(\eta _ j)_ j$ is increasing and $\eta _ j\leq \eta $. Put $\tilde\eta \coloneqq \operatorname *{sup}_{}{}\! {}^{*}_ j \eta _ j$. By corollary 18.28.11 and corollary 16.9.12, we have $\eta _ j\xrightarrow {d_ S}\tilde\eta $. The $d_ S$-convergence above therefore gives $d_ S(\tilde\eta ,\eta )=0$, hence $\tilde\eta \sim _ P \eta $ by proposition 16.9.6. In particular, $\tilde\eta \preceq _{\mathcal{I}}\varphi $ by corollary 16.8.14. Also $\tilde\eta \leq v$ on $K$ quasi-everywhere by proposition 5.14.5. Hence $\tilde\eta $ is a candidate for $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$, so $\tilde\eta \leq \eta $. Since $\eta _ j\leq \eta $ for all $j$, this proves $\eta _ j\nearrow \eta $ almost everywhere. $\square$

18.28.37 Quantization of partial equilibrium measures0211344889F8

Let $X$ be a connected compact Kähler manifold of dimension $n$, let $L$ be a pseudo-effective line bundle on $X$, and let $h$ be a Hermitian metric on $L$. Set $\theta =c_1(L,h)$. Let $(T,h_ T)$ be a Hermitian line bundle on $X$.

18.28.38 Bernstein–Markov measuresB2D64A3DB08E

Let $K\subseteq X$ be a closed non-pluripolar subset, and $v\colon K\rightarrow [-\infty ,\infty )$ be a measurable function.

Definition 18.28.39.E365D575C322 Let $\mu $ be a Borel probability measure on $K$. We introduce the following functions on $\mathrm{H}^0(X,L^ k \otimes T)$ ($k\geq 1$), with values possibly equal to $\infty $:

46ED22785DA5 18.28.40
\begin{equation} \begin{aligned} N^ k_{v,\mu }(s) \coloneqq & \left(\int _ K h^ k \otimes h_ T(s,s) \mathrm{e}^{-kv} \, \mathrm{d}\mu \right)^{1/2},\\ N^ k_{v,K}(s) \coloneqq & \sup _{K\setminus \{ v=-\infty \} } \left( h^ k \otimes h_ T(s,s)\mathrm{e}^{-kv}\right)^{1/2}. \end{aligned} \end{equation}
Observe that if $\mu $ puts no mass on $\{ v=-\infty \} $ then we always have
F1111E54880E 18.28.41
\begin{equation} N^ k_{v,\mu }(s)\leq N^ k_{v,K}(s). \end{equation}
We start with the following elementary observation:

Lemma 18.28.42.E97173FEFAFD Let $v_1,v_2\colon K\rightarrow [-\infty ,\infty )$ be two measurable functions such that

  1. 7AB5A208122C

    $v_1\leq v_2$;

  2. 60A51A0A77E1

    $\{ v_1=-\infty \} =\{ v_2=-\infty \} $.

Then for any $s\in \mathrm{H}^0(X,L^ k\otimes T)$ ($k\geq 1$), we have

\[ N_{v_1,K}^ k(s)\geq N_{v_2,K}^ k(s). \]

Proof. This follows immediately from the definition (18.28.40). $\square$

Definition 18.28.43.48A0DBADBA86 Let $(K,v)$ be a weighted subset of $X$. A Borel probability measure $\mu $ on $K$ is Bernstein–Markov with respect to $(K,v)$ if for each $\epsilon >0$, there is a constant $C_\epsilon >0$ such that

FE92486453EB 18.28.44
\begin{equation} N^ k_{v,K}(s) \leq C_\epsilon \mathrm{e}^{\epsilon k} N^ k_{v,\mu }(s) \end{equation}

for any $s \in \mathrm{H}^0(X,L^ k \otimes T)$ and any $k\in \mathbb {Z}_{>0}$. We write $\mathrm{BM}(K,v)$ for the set of Bernstein–Markov measures with respect to $(K,v)$.

As pointed out in [BBWN11], any volume form on $X$ is Bernstein–Markov with respect to $(X,v)$, with $v \in C^\infty (X)$.

Proposition 18.28.45.A86573651304 Assume that $(K,v)$ is a weighted subset of $X$. Then

  1. A757D6ACA763

    $N^ k_{v,K}$ is a norm on $\mathrm{H}^0(X,L^ k\otimes T)$.

  2. 6EBD970292B6

    For any $\mu \in \mathrm{BM}(K,v)$, $N^ k_{v,\mu }$ is a norm on $\mathrm{H}^0(X,L^ k\otimes T)$.

Proof. 1 As $v$ is bounded, $N^ k_{v,K}$ is clearly finite on $\mathrm{H}^0(X,L^ k\otimes T)$. In order to show that it is a norm, it suffices to show that for any $s\in \mathrm{H}^0(X,L^ k\otimes T)$, $N^ k_{v,K}(s)=0$ implies that $s=0$. In fact, we have $s|_ K=0$, hence $s=0$ by the connectedness of $X$ and the assumption that $K$ is non-pluripolar.

2 As $v$ is bounded, clearly $N^ k_{v,\mu }$ is finite and satisfies the triangle inequality. Non-degeneracy follows from the fact that $N^ k_{v,K}$ is a norm and (18.28.44). $\square$

774E9C5AFE10

In this section, we fix a weighted subset $(K,v)$ of $X$ and $\mu \in \mathrm{BM}(K,v)$.

Definition 18.28.46.C3C647253911 For any $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, we introduce the partial Bergman kernels of $\varphi $ (with respect to $(K,v)$) as follows: For any $k\geq 1$, we introduce

14.50
\begin{equation} \begin{aligned} B^ k_{v,\varphi , \mu }(x) \coloneqq \sup \left\{ h^ k \otimes h_ T(s,s)\mathrm{e}^{-kv}(x):s \in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right),\right.\\ \left. N^ k_{v,\mu }(s) \leq 1 \right\} ,\quad x\in K. \end{aligned} \end{equation}

We extend $B^ k_{v,\varphi , \mu }$ to the whole $X$ by setting it to be $0$ outside $K$.

The partial Bergman measures of $\varphi $ (with respect to $(K,v)$) are defined as

14.53
\begin{equation} \beta ^ k_{v,\varphi ,\mu } \coloneqq \frac{n!}{k^ n} B^ k_{v,\varphi ,\mu } \, \mathrm{d}\mu \end{equation}

for each $k\geq 1$.

Remark 18.28.47.129D3B333F18 Recall that in the definition of $\beta ^ k_{v,\varphi ,\mu }$, the singular metric $\varphi $ only appears through the conditions $s \in \mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$. The pointwise norm used to define $\beta ^ k_{v,\varphi ,\mu }$ and $N^ k_{v,\mu }$ is defined via $v$ not $\varphi $.

This leads to the following seemingly paradoxical consequence, if $\psi \in \mathrm{PSH}(X,\theta )$ and $\varphi \preceq _{\mathcal{I}}\psi $, then

\[ \beta ^ k_{v,\varphi ,\mu }\leq \beta ^ k_{v,\psi ,\mu }\quad \forall \, k\geq 1. \]

We shall apply this consequence without further mention in the sequel.

Observe that

\[ \int _ X \beta ^ k_{v,\varphi ,\mu }=\frac{n!}{k^ n}h^0\left(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi )\right). \]

In particular,

889A68CCEC83 18.28.48
\begin{equation} \lim _{k\to \infty }\int _ X \beta ^ k_{v,\varphi ,\mu }=\operatorname{vol}\left(\theta _{\varphi }\right)=\int _ X \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n \end{equation}
by theorem 18.1.46 and 18.28.15. The goal of this section is to improve this result to the convergence of the Bergman kernels themselves:

Theorem 18.28.49.EA3E101DCA5E Suppose that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

A6F9F2DF4A0D 18.28.50
\begin{equation} \beta ^ k_{v,\varphi ,\mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n,\quad \textup{as }k\to \infty . \end{equation}

The proof will be given at the end of this subsection. We shall prove a few preliminaries at first.

Take a Kähler form $\omega $ on $X$ so that

\[ \int _ X \omega ^ n = 1. \]

Proposition 18.28.51.4C368CFA09EB Assume here that $K=X$. Let $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. If $v \in C^\infty (X)$, then

4DF720497AC6 18.28.52
\begin{equation} \beta ^ k_{v,\varphi ,\omega ^ n} \rightharpoonup \theta _{P_{\theta ,X}[\varphi ]_{\mathcal{I}}(v)}^ n=\theta _{P_{\theta ,X}[\varphi ](v)}^ n\quad \textup{as }k\to \infty . \end{equation}

Proof. The equality part in (18.28.52) follows from corollary 18.28.14. It remains to prove the weak convergence.

Let $\tau $ be the weak limit of a subsequence of $(\beta ^ k_{v,\varphi ,\omega ^ n})_ k$. In view of (18.28.48), it suffices to show that

14347C41585E 18.28.53
\begin{equation} \tau \leq \theta _{P_{\theta ,X}[\varphi ](v)}^ n. \end{equation}

We can obtain an upper bound of $\tau $ via the observation:

\[ \beta ^ k_{v,\varphi ,\omega ^ n} \leq \beta ^ k_{v,V_{\theta },\omega ^ n}. \]

After replacing $\varphi $ by $V_{\theta }$, the corresponding problem is handled by [Theorem 1.2, Ber11], giving us

\[ \beta ^ k_{v,V_{\theta },\omega ^ n} \rightharpoonup \theta _{P_{\theta ,X}[V_{\theta }](v)}^ n=\mathds {1}_{\left\{ v = P_{\theta ,X}[V_{\theta }](v)\right\} }\, \theta _ v^ n \quad \textup{as }k\to \infty , \]

where the last identity is due to proposition 16.3.41.

We obtain that

1B7624ED3973 18.28.54
\begin{equation} \tau \leq \mathds {1}_{\left\{ v = P_{\theta ,X}[V_{\theta }](v)\right\} }\, \theta _ v^ n. \end{equation}

Now observe that it suffices to show that

C700200CB6BE 18.28.55
\begin{equation} \tau \left\{ P_{\theta ,X}[\varphi ](v) < v\right\} =0. \end{equation}

Assuming this, then (18.28.54) further gives

\[ \tau \leq \mathds {1}_{\left\{ P_{\theta ,X}[\varphi ](v)= v\right\} }\, \theta _ v^ n= \theta _{P_{\theta ,X}[\varphi ](v)}^ n, \]

where the equality follows from proposition 16.3.41, and (18.28.53) follows.

It now remains to argue (18.28.55). Fix a Kähler form $\omega $ on $X$. Observe that

\[ P_{\theta +\epsilon \omega ,X}[\varphi ](v) \sim \varphi \]

for all $\epsilon \geq 0$ by 18.28.15 and proposition 16.6.22. The family $P_{\theta +\epsilon \omega ,X}[\varphi ](v)$ is decreasing as $\epsilon \to 0+$. Let

\[ \chi \coloneqq \inf _{\epsilon >0}P_{\theta +\epsilon \omega ,X}[\varphi ](v). \]

Then $\chi \in \mathrm{PSH}(X,\theta )$ since $\chi \geq P_{\theta ,X}[\varphi ](v)$. Moreover $\chi \leq v$ and $\chi \preceq \varphi $. Hence

\[ \chi \leq P_{\theta ,X}[\varphi ](v). \]

Thus, equality holds, and

\[ P_{\theta +\epsilon \omega ,X}[\varphi ](v)\searrow P_{\theta ,X}[\varphi ](v). \]

In other words,

\[ \left\{ P_{\theta ,X}[\varphi ](v) < v\right\} =\bigcup _{\epsilon \in \mathbb {Q}_{>0}}\left\{ P_{\theta +\epsilon \omega ,X}[\varphi ](v) < v\right\} . \]

So in order to show (18.28.55), we may fix $\epsilon >0$ and argue that

CE5263973CB7 18.28.56
\begin{equation} \tau \left\{ P_{\theta +\epsilon \omega ,X}[\varphi ](v) < v\right\} =0. \end{equation}

Let $k\geq 1$, and $s \in \mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal I(k\varphi ))$ be a section such that $N^ k_{v,\omega ^ n}(s) \leq 1$. Then by [Lemma 4.1, Ber11], there exists $C>0$ such that

\[ h^ k\otimes h_ T(s,s)\mathrm{e}^{-kv} \leq B^ k_{v,\varphi ,\omega ^ n} \leq B^ k_{v,V_{\theta },\omega ^ n} \leq k^ n C. \]

This implies that

\[ \frac{1}{k}\log h^ k \otimes h_ T(s,s) \leq v + \frac{\log C }{k} + n \frac{\log k}{ k }. \]

We define $\varphi _ k$ as in proposition 5.20.17. Take $\alpha _ k\nearrow 1$ as in proposition 5.20.17. After adding a constant to $\varphi $, we may assume that $\varphi \leq 0$ on $X$. After replacing $\alpha _ k$ by a subsequence, we also have $\alpha _ k\varphi \in \mathrm{PSH}(X,\theta +\epsilon \omega )$ for all $k>0$, and $\alpha _ k\varphi \searrow \varphi $. Then

\[ \frac{1}{k}\log h^ k \otimes h_ T(s,s)\preceq \varphi _ k\preceq \alpha _ k\varphi . \]

We notice that $\frac{1}{k}\log h^ k \otimes h_ T(s,s) \in \mathrm{PSH}(X, \theta + \epsilon \omega )$ for all $k \geq k_0(\epsilon )$. Indeed,

\[ \frac{1}{k}\log h^ k\otimes h_ T(s,s) \in \mathrm{PSH}\left(X,\theta + k^{-1} c_1(T,h_ T)\right) \]

by proposition 5.20.4.

In particular,

\[ \frac{1}{k}\log h^ k \otimes h_ T(s,s)-\frac{\log C}{k}-n\frac{\log k}{k}\leq P_{\theta +\epsilon \omega ,X}[\alpha _ k \varphi ](v). \]

Now taking supremum over all candidates $s$, we obtain that

725395815923 18.28.57
\begin{equation} B^ k_{v,\varphi ,\omega ^ n} \leq C k^ n \mathrm{e}^{k \left(P_{\theta +\epsilon \omega ,X}[\alpha _ k \varphi ](v)-v\right)},\quad k \geq k_0. \end{equation}

Now we prove (18.28.56). First observe that $\alpha _ k\varphi \xrightarrow {d_ S}\varphi $ by 16.9.25. Thus, by proposition 18.28.35,

\[ P_{\theta +\epsilon \omega ,X}[\alpha _ k \varphi ](v) \searrow P_{\theta +\epsilon \omega ,X}[\varphi ](v) \]

and we get that

\[ \bigcup _{k=1}^{\infty }\left\{ P_{\theta +\epsilon \omega ,X}[\alpha _ k \varphi ](v) < v\right\} =\left\{ P_{\theta +\epsilon \omega ,X}[\varphi ](v) < v\right\} . \]

It suffices to show that $\tau $ does not put mass on the set $\{ P_{\theta +\epsilon \omega ,X}[\alpha _ m \varphi ](v) < v\} $ for each fixed $m$. Fix a positive rational $\delta >0$, it suffices to show that $\tau $ puts no mass on $F\coloneqq \{ P_{\theta +\epsilon \omega ,X}[\alpha _ m \varphi ](v) < v-\delta \} $. Note that $F$ is open, and (18.28.57) gives $\beta ^\ell _{v,\varphi ,\omega ^ n}(F)\to 0$. Therefore, $\tau $ puts no mass on $F$ as well, proving (18.28.56). $\square$

Definition 18.28.58.A0F71B40AAE1 Take $k\geq 1$ and $\varphi \in \mathrm{PSH}(X,\theta )$, let $\operatorname{Norm}\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$ be the space of norms on the vector space $\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$.

Let $\mathcal{L}_{k,\varphi } \colon \operatorname{Norm}\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )) \to \mathbb R$ be the partial Donaldson functional:

14.62
\begin{equation} \mathcal{L}_{k,\varphi }(H) = \frac{n!}{k^{n+1}} \log \frac{\operatorname{vol}\{ s \in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) : H(s) \leq 1\} }{\operatorname{vol}\left\{ s\in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) : N^ k_{0,\omega ^ n}(s) \leq 1\right\} }, \end{equation}

where $\operatorname{vol}$ is a Lebesgue measure on $\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$. The normalization does not matter.

Proposition 18.28.59.F547344ACE10 Let $w,w' \in C^0(X)$ and $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Then

D29819040D75 18.28.60
\begin{equation} \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n}) - \mathcal{L}_{k,\varphi }(N^ k_{w',\omega ^ n}) \right)= \mathcal{E}_{\theta ,X}^{\varphi }(w)-\mathcal{E}_{\theta ,X}^{\varphi }(w'). \end{equation}

In particular,

04790CD84F49 18.28.61
\begin{equation} \lim _{k\to \infty }\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n})=\mathcal{E}_{\theta ,X}^{\varphi }(w)\, . \end{equation}

Proof. First observe that by proposition 18.28.45, for any $k\geq 1$, $N^ k_{w,\omega ^ n}$ and $N^ k_{w',\omega ^ n}$ are both norms, hence the expressions inside the limit in (18.28.60) make sense.

We first assume that $w,w'\in C^\infty (X)$. To start, we make the following observation:

\[ \begin{aligned} \mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n}) - \mathcal{L}_{k,\varphi }(N^ k_{w',\omega ^ n})& = \int _0^1 \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{L}_{k,\varphi }(N^ k_{w' + t(w-w'),\omega ^ n})\, \mathrm{d}t\\ & = \int _0^1 \int _ X (w-w') \, \beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}\, \mathrm{d}t. \end{aligned} \]

The second equality follows from differentiating the Gram matrix of the Hilbert norm $N^ k_{w'+t(w-w'),\omega ^ n}$:

\[ \frac{\mathrm d}{\mathrm dt}\mathcal L_{k,\varphi } \left(N^ k_{w'+t(w-w'),\omega ^ n}\right)=\int _ X (w-w')\, \beta ^ k_{w'+t(w-w'),\varphi ,\omega ^ n}. \]

By proposition 18.28.51, we have

\[ \lim _{k\to \infty }\int _ X (w-w') \, \beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}=\int _ X (w-w')\, \theta _{P_{\theta ,X}[\varphi ](w' + t(w-w'))}^ n. \]

By theorem 18.1.46, we have

\[ \bigl |\int _ X (w-w')\beta ^ k_{w' + t(w-w'),\varphi ,\omega ^ n}\bigr | \leq C \sup _ X |w - w'|. \]

Hence, by the dominated convergence theorem we obtain that

\[ \begin{aligned} \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n}) - \mathcal{L}_{k,\varphi }(N^ k_{w',\omega ^ n})\right) & = \int _0^1 \int _ X (w-w') \, \theta _{P_{\theta ,X}[\varphi ](w' + t(w-w'))}^ n \, \mathrm{d}t\\ & = \mathcal{E}_{\theta ,X}^{\varphi }(w)-\mathcal{E}_{\theta ,X}^{\varphi }(w'), \end{aligned} \]

where in the last line we have used proposition 18.28.27.

This proves (18.28.60) for smooth weights.

For general $w,w'\in C^0(X)$, choose smooth functions $w_ j,w'_ j$ converging uniformly to $w,w'$, respectively. If $\| w_ j-w\| _{C^0}\leq \delta _ j$, then

\[ \mathrm{e}^{-k\delta _ j/2}N^ k_{w,\omega ^ n}\leq N^ k_{w_ j,\omega ^ n}\leq \mathrm{e}^{k\delta _ j/2}N^ k_{w,\omega ^ n}, \]

and hence

\[ \left|\mathcal{L}_{k,\varphi }(N^ k_{w_ j,\omega ^ n})-\mathcal{L}_{k,\varphi }(N^ k_{w,\omega ^ n})\right| \leq \frac{n!}{k^ n}h^0\left(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi )\right)\delta _ j. \]

The same estimate holds for $w'_ j$ and $w'$. Moreover, proposition 18.28.27 shows that the energy side is Lipschitz with respect to the uniform norm of the weight. Letting first $k\to \infty $ in the smooth case, and then $j\to \infty $, proves (18.28.60) for continuous weights. Equation (18.28.61) again follows by taking $w'=0$. $\square$

Lemma 18.28.62.922E6C36BCC8 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Let $(K,v)$ be a weighted subset of $X$. Let $\mu \in \mathrm{BM}(K,v)$. Then

C3210B1F6DE6 18.28.63
\begin{equation} \lim _{k\to \infty } \big ( \mathcal{L}_{k,\varphi }(N^ k_{v,K}) - \mathcal{L}_{k,\varphi }(N^ k_{v,\mu })\big ) = 0\, . \end{equation}

Proof. By (18.28.41) and (18.28.44), for every $\epsilon >0$ and all $k\geq 1$,

\[ N^ k_{v,\mu }\leq N^ k_{v,K}\leq C_\epsilon \mathrm{e}^{\epsilon k}N^ k_{v,\mu }. \]

The corresponding unit balls therefore satisfy inclusions with radii differing by at most the factor $C_\epsilon \mathrm{e}^{\epsilon k}$. Taking logarithmic volumes and multiplying by $n!k^{-n-1}$ gives

\[ \left|\mathcal{L}_{k,\varphi }(N^ k_{v,K})-\mathcal{L}_{k,\varphi }(N^ k_{v,\mu })\right| \leq \frac{2n!}{k^{n+1}} h^0\left(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi )\right)(\log C_\epsilon +\epsilon k). \]

The right-hand side has limsup at most $2\epsilon \operatorname{vol}(\theta _\varphi )$ by theorem 18.1.46. Letting $\epsilon \to 0+$ proves (18.28.63). $\square$

Corollary 18.28.64.4CE59F3C4805 Let $w \in C^0(X)$, $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _\varphi $ is a Kähler current. Then

\[ \lim _{k\to \infty }\mathcal{L}_{k,\varphi }(N^ k_{w,X})=\mathcal{E}_{\theta ,X}^{\varphi }(w). \]

Proof. This follows from 18.28.62 and proposition 18.28.59: indeed $\omega ^ n \in \mathrm{BM}(X,0)$, hence $\omega ^ n\in \mathrm{BM}(X,w)$ as $w$ is bounded. $\square$

It would be helpful to consider the following auxiliary functions:

\[ \begin{aligned} P'_{\theta ,K}[\varphi ](v)\coloneqq & \sup \Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq v \textup{ and } \eta \preceq \varphi \Bigr \} ,\\ P'_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\coloneqq & \sup \Bigl \{ \eta \in \mathrm{PSH}(X,\theta ): \eta |_ K \leq v \textup{ and } \eta \preceq _{\mathcal{I}} \varphi \Bigr \} . \end{aligned} \]
We note the following maximum principles that follow from the above definitions:

Lemma 18.28.65.5C6401E18EB3 Let $v\in C^0(K)$, $\varphi \in \mathrm{PSH}(X,\theta )$, and $\eta \in \mathrm{PSH}(X,\theta )$. Assume that $\eta \preceq \varphi $. Then

2DBAE7A20C99 18.28.66
\begin{equation} \sup _ K (\eta -v) = \sup _{\{ \eta \neq -\infty \} } \left(\eta - P'_{\theta ,K}[\varphi ](v)\right)=\sup _{\left\{ P'_{\theta ,K}[\varphi ](v)\neq -\infty \right\} } \left(\eta - P'_{\theta ,K}[\varphi ](v)\right). \end{equation}

Proof. We prove the first equality first. We write $S=\{ \eta =-\infty \} $.

By definition, $P'_{\theta ,K}[\varphi ](v)|_ K\leq v$, so

\[ \left.\left(\eta -P'_{\theta ,K}[\varphi ](v)\right)\right|_{K\setminus S}\geq \eta |_{K\setminus S}-v|_{K\setminus S}\, . \]

This implies that

\[ \sup _ K (\eta -v) \leq \sup _{X\setminus S} \left(\eta - P'_{\theta ,K}[\varphi ](v)\right). \]

Conversely, observe that $\sup _ K(\eta -v)>-\infty $ as $K$ is non-pluripolar. It follows that $\eta -\sup _ K (\eta -v)$ is a candidate in the definition of $P'_{\theta ,K}[\varphi ](v)$. Hence,

3504DA07F50F 18.28.67
\begin{equation} \eta -\sup _ K (\eta -v)\leq P'_{\theta ,K}[\varphi ](v). \end{equation}

This implies that

\[ \sup _ K (\eta -v) \geq \sup _{X\setminus S} \left(\eta - P'_{\theta ,K}[\varphi ](v)\right), \]

finishing the proof of the first identity in (18.28.66).

In order to prove the latter identity, observe that

\[ \left\{ P'_{\theta ,K}[\varphi ](v) =-\infty \right\} \subseteq S \]

as a consequence of (18.28.67). On $\{ P'_{\theta ,K}[\varphi ](v)\neq -\infty \} \cap S$, we have $\eta -P'_{\theta ,K}[\varphi ](v)=-\infty $. Hence the last equality of (18.28.66) also follows. $\square$

Proposition 18.28.68.F431300C59C4 Let $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Let $(K,v)$, $(K',v')$ be two weighted subsets of $X$. Then

FFF9581DFDD2 18.28.69
\begin{equation} \lim _{k\to \infty } \left( \mathcal{L}_{k,\varphi }(N^ k_{v,K}) - \mathcal{L}_{k,\varphi }(N^ k_{v',K'}) \right) = \mathcal{E}_{\theta ,K}^{\varphi }(v)-\mathcal{E}_{\theta ,K'}^{\varphi }(v'). \end{equation}

In particular,

5AAAC502B9E5 18.28.70
\begin{equation} \lim _{k\to \infty }\mathcal{L}_{k,\varphi }(N^ k_{v,K})=\mathcal{E}_{\theta ,K}^{\varphi }(v). \end{equation}

Proof. First observe that by proposition 18.28.45, for any $k>0$, $N^ k_{v,K}$ and $N^ k_{v',K'}$ are both norms, hence the expressions inside the limit in (18.28.69) make sense. Moreover, (18.28.70) is just a special case of (18.28.69) for $K'=X$ and $v'= 0$.

To prove (18.28.69) it is enough to show that for any fixed $w \in C^\infty (X)$ we have

AC1B69834497 18.28.71
\begin{equation} \lim _{k\to \infty } \big ( \mathcal{L}_{k,\varphi }(N^ k_{v,K}) - \mathcal{L}_{k,\varphi }(N^ k_{w,X}) \big ) = \mathcal{E}_{\theta ,K}^{\varphi }(v) - \mathcal{E}_{\theta ,X}^{\varphi }(w)\, . \end{equation}

For $\epsilon \in (0,1)$ small enough, $\theta _{(1-\epsilon )\varphi }$ is still a Kähler current. Let us fix such $\epsilon $, along with an arbitrary $\epsilon ' \in (0,1)$.

Let $(v^-_{j})_ j$ and $(v^+_{j})_ j$ be the sequences of smooth functions constructed in 18.28.31 for the data $(K,v)$.

By proposition 5.20.17 there exists $k_0(\epsilon ,\epsilon ') \in \mathbb N$ such that

\[ \frac{1}{k} \log h^ k \otimes h_ T (s,s) \preceq (1-\epsilon )\varphi , \]

and $\frac{1}{k} \log h^ k \otimes h_ T (s,s) \in \mathrm{PSH}(X,\theta + \epsilon '\omega )$ for any $s \in \mathrm{H}^0(X,T\otimes L^ k \otimes \mathcal{I}(k\varphi ))$, as long as $k \geq k_0(\epsilon ,\epsilon ')$.

In particular, 18.28.65 gives that

\[ \begin{aligned} N^ k_{P'_{\theta + \epsilon '\omega ,K}[(1-\epsilon )\varphi ](v),X}(s)=& N^ k_{v,K}(s),\\ N^ k_{P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^-),X}(s)=& N^ k_{v_{j}^-,X}(s),\\ N^ k_{P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^+),X}(s)=& N^ k_{v^+_{j},X}(s). \end{aligned} \]

As

\[ P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^-) \leq P'_{\theta + \epsilon '\omega ,K}[(1-\epsilon )\varphi ](v) \leq P'_{\theta + \epsilon '\omega ,X}[(1-\epsilon )\varphi ](v_{j}^+), \]

where the second inequality follows from 18.28.31 and the fact that $P'=P$ for the global continuous obstacles $v_ j^\pm $. By 18.28.42 we have

\[ N^ k_{v^+_{j},X}(s) \leq N^ k_{v,K}(s) \leq N^ k_{v_{j}^-,X}(s), \quad s \in \mathrm{H}^0(X,T \otimes L^ k \otimes \mathcal{I}(k\varphi )), k \geq k_0(\epsilon ,\epsilon '). \]

Composing with $\mathcal{L}_{k,\varphi }$ we arrive at

\[ \mathcal{L}_{k,\varphi }(N^ k_{v^-_{j},X})\leq \mathcal{L}_{k,\varphi }(N^ k_{v,K})\leq \mathcal{L}_{k,\varphi }(N^ k_{v^+_{j},X})\, , \quad k \geq k_0(\epsilon ,\epsilon '). \]

For any $j>0$, by corollary 18.28.64 we get

\[ \begin{aligned} \mathcal{E}_{\theta ,X}^{\varphi }(v^-_{j}) - \mathcal{E}_{\theta ,X}^{\varphi }(w)=& \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{v^-_{j},X})-\mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \varliminf _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \varlimsup _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ \leq & \lim _{k\to \infty } \left(\mathcal{L}_{k,\varphi }(N^ k_{v^+_{j},X})-\mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ =& \mathcal{E}_{\theta ,X}^{\varphi }(v^+_{j}) - \mathcal{E}_{\theta ,X}^{\varphi }(w). \end{aligned} \]

Using 18.28.31, we can let $j \to \infty $ to arrive at

\[ \begin{split} \mathcal{E}_{\theta ,K}^{\varphi }(v) -\mathcal{E}_{\theta ,X}^{\varphi }(w) & \leq \varliminf _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ & \leq \varlimsup _{k\to \infty }\left(\mathcal{L}_{k,\varphi }(N^ k_{v,K})- \mathcal{L}_{k,\varphi }(N^ k_{w,X})\right)\\ & \leq \mathcal{E}_{\theta ,K}^{\varphi }(v) -\mathcal{E}_{\theta ,X}^{\varphi }(w). \end{split} \]

Hence, (18.28.71) follows. $\square$

Corollary 18.28.72.B1F17F151050 Let $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Then

\[ \lim _{k\to \infty } \mathcal{L}_{k,\varphi }(N^ k_{v,\mu })= \mathcal{E}_{\theta ,K}^{\varphi }(v). \]

Proposition 18.28.73.AC691D964EDB Let $\varphi \in \mathrm{PSH}(X,\theta )$ be a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Then

\[ \beta ^ k_{v,\varphi , \mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n=\theta _{P_{\theta ,K}[\varphi ](v)}^ n\quad \textup{as }k\to \infty . \]

Proof. For $w\in C^0(X)$, let $f_ k,g\colon \mathbb {R}\rightarrow \mathbb {R}$ ($k\geq 1$) be defined as

\[ f_ k(t)\coloneqq \mathcal{L}_{k,\varphi }(N^ k_{v+tw,\mu }),\quad g(t)\coloneqq \mathcal{E}_{\theta ,K}^{\varphi }(v+tw). \]

By corollary 18.28.72, for all $t\in \mathbb {R}$, we have

\[ \lim _{k\to \infty } f_ k(t)= g(t). \]

Here we use that $\mu \in \mathrm{BM}(K,v+tw)$ for all fixed $t$, which follows from $\mu \in \mathrm{BM}(K,v)$ and the boundedness of $tw$ on $K$. Note that $f_ k$ is concave by Hölder's inequality. For each $k$, the function $f_ k$ is differentiable and

\[ f_ k'(0)=\int _ X w\, \beta ^ k_{v,\varphi ,\mu }. \]

This follows by differentiating the Gram matrix of the Hilbert norm $N^ k_{v+tw,\mu }$. The function $g$ is differentiable at $0$ by proposition 18.28.27, with

\[ g'(0)=\int _ K w\, \theta ^ n_{P_{\theta ,K}[\varphi ]_{\mathcal I}(v)}. \]

Applying 21.11.63 to the convex functions $-f_ k$ and $-g$ gives

\[ \lim _{k\to \infty } f_ k'(0)=g'(0), \]

which is equivalent to $\beta ^ k_{v, \varphi ,\mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ](v)}^ n$, by proposition 18.28.27. $\square$

Proposition 18.28.74.0DFF36A22512 Suppose that $\varphi \in \mathrm{PSH}(X,\theta )$ is such that $\theta _{\varphi }$ is a Kähler current. Let $(K,v)$ be a weighted subset of $X$ and $\mu \in \mathrm{BM}(K,v)$. Then

7EA15066EE63 18.28.75
\begin{equation} \beta ^ k_{v,\varphi ,\mu } \rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n\quad \textup{as } k\to \infty . \end{equation}

Proof. Let $\tau $ be the weak limit of a subsequence of $\beta ^ k_{v,\varphi ,\mu }$. We claim that

CBCF54FCD801 18.28.76
\begin{equation} \tau \leq \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n. \end{equation}

Observe that this claim implies the conclusion. In fact, by theorem 18.1.46, we have equality of the total masses, so equality holds in (18.28.76). As $\tau $ is an arbitrary cluster point of the sequence $(\beta ^ k_{v,\varphi ,\mu })_ k$, we get (18.28.75).

It remains to prove (18.28.76). Let $(\varphi _ j)$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We may assume that $\theta _{\varphi _ j}$ is a Kähler current for all $j\geq 1$. By corollary 18.1.14, we know that

\[ \varphi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}. \]

The quasi-equisingular approximation is decreasing, so applying the decreasing case of proposition 18.28.35 to the limit $P_{\theta }[\varphi ]_{\mathcal{I}}$, and using the definition of the relative $\mathcal{I}$-envelope, we get

\[ P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)\searrow P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v). \]

In particular,

FD0D14628CD8 18.28.77
\begin{equation} \lim _{j\to \infty } \int _ X \theta _{P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)}^ n= \int _ X \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n. \end{equation}

Observe that

\[ \beta ^ k_{v,\varphi ,\mu } \leq \beta ^ k_{v,\varphi _ j,\mu } \]

for any $k \geq 1$. As $\mu \in \mathrm{BM}(K,v)$, by proposition 18.28.73,

\[ \tau \leq \theta _{P_{\theta ,K}[\varphi _ j]_{\mathcal{I}}(v)}^ n, \]

for any $j \geq 1$ fixed. By (18.28.77) and the semicontinuity theorem theorem 13.13.9, we have

\[ \theta _{P_{\theta ,K}\left[\varphi _ j\right]_{\mathcal{I}}(v)}^ n\rightharpoonup \theta _{P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)}^ n \]

as $j\to \infty $. Therefore, (18.28.76) follows. $\square$

Proof of theorem 18.28.49. By 18.28.15 and proposition 16.6.11, the three potentials below have the same multiplier ideals. Hence

\[ \begin{aligned} \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) & = \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal I(kP_{\theta }[\varphi ]_{\mathcal{I}})\right)\\ & = \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal I\left(kP_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)\right)\right). \end{aligned} \]

This allows us to replace $\varphi $ with $P_{\theta ,K}[\varphi ]_{\mathcal{I}}(v)$.

By 13.13.47, there exists a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ j\nearrow \varphi $ a.e. and $\theta _{\varphi _ j}$ is a Kähler current for each $j\geq 1$. This gives

\[ \beta ^ k_{v,\varphi _ j,\mu } \leq \beta ^ k_{v,\varphi ,\mu }. \]

Let $\tau $ be the weak limit of a subsequence of $(\beta ^ k_{v,\varphi ,\mu })_ k$. Then by proposition 18.28.74,

\[ \theta _{P_{\theta ,K} [\varphi _ j]_{\mathcal{I}}(v)}^ n\leq \tau . \]

By proposition 18.28.35 and corollary 13.13.12, we have

\[ \theta _{P_{\theta ,K} [\varphi _ j]_{\mathcal{I}}(v)}^ n \rightharpoonup \theta _{P_{\theta ,K} [\varphi ]_{\mathcal{I}}(v)}^ n. \]

Hence,

8FD19D2B07C7 18.28.78
\begin{equation} \theta _{P_{\theta ,K} [\varphi ]_{\mathcal{I}}(v)}^ n\leq \tau . \end{equation}

A comparison of total masses using (18.28.48) gives that equality holds in (18.28.78). As $\tau $ is arbitrary, we obtain (18.28.50). $\square$

Remark 18.28.79.E9D132FDBF95 The results in this chapter can also be reformulated as the large deviation principle of a determinantal point process on $X$ using the Gärtner–Ellis theorem exactly as in [Ber14]. We do not pursue this reformulation here.

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