16.13 Concentration on the contact set954F538078ED

Proof. For $\varepsilon >0$ put $w_\varepsilon =\max \{ u,v-\varepsilon \} $. As $\varepsilon \downarrow 0$, $w_\varepsilon $ increases to $\max \{ u,v\} =v$, so $\mu _{w_\varepsilon }\to \mu _ v$ weakly along every sequence $\varepsilon \downarrow 0$, by corollary 13.8.9. On the plurifine open set $\{ u>v-\varepsilon \} $ we have $w_\varepsilon =u$, hence $\mathbf1_{\{ u>v-\varepsilon \} }\mu _{w_\varepsilon }=\mathbf1_{\{ u>v-\varepsilon \} }\mu _ u$. This set contains every point of $\{ u=v\} $ at which $u$ is finite, and $\mu _ u$ does not charge the pole set of $u$. Therefore

The right side is a fixed finite measure. Integrating a continuous function $\chi \geq 0$ against both sides and letting $\varepsilon \downarrow 0$, we obtain $\int _ X\chi \, \mu _ v\geq \int _ X\chi \mathbf1_{\{ u=v\} }\, \mu _ u$, so $\mu _ v\geq \mathbf1_{\{ u=v\} }\mu _ u$ by lemma 16.12.1. Restricting both sides to $\{ u=v\} $ proves the lemma. $\square$

Proof. By proposition 10.4.6, the big class contains a Kähler current $\theta +\mathrm{dd}^{\mathrm{c}}\psi $ whose potential $\psi \in \mathrm{PSH}(X,\theta )$ has analytic singularities, and we put $A=\{ \psi =-\infty \} $. Every point of $X$ has a connected coordinate neighborhood $V$ on which $\psi =c\log \sum _ i|g_ i|^2+b$, with $c>0$, $b$ bounded and holomorphic functions $g_ i$ that do not all vanish identically (definition 6.14.5). Thus $A\cap V$ is the common zero set of the $g_ i$, so $A$ is an analytic subset of $X$, different from $X$, and it is closed and pluripolar, being locally the pole set of the plurisubharmonic function $\log \sum _ i|g_ i|^2$. On $V\setminus A$ the continuous function $\sum _ i|g_ i|^2$ is positive, so $\psi $ is locally bounded on $X\setminus A$. If $\varphi $ has minimal singularities, then $\psi \leq \varphi +C'$ for a constant $C'$, so $\varphi $ is locally bounded below on $X\setminus A$; it is bounded above, being upper semicontinuous on the compact space $X$. The last assertion holds because non-pluripolar products give no mass to pluripolar sets. $\square$

Proof. Adding a constant to $f$, we may assume that $-C\leq f\leq 0$. Then $V_\theta -C\leq f$, so $V_\theta -C$ is a competitor, while every competitor is nonpositive and hence at most $V_\theta $. Thus $V_\theta -C\leq u\leq V_\theta $, and $u$ has minimal singularities.

Let $A$ be the proper analytic subset of lemma 16.13.2. Since $u$ has minimal singularities, it is locally bounded on $X\setminus A$, and $\mu _ u(A)=0$.

Let $x\in X\setminus A$ with $u(x)<f(x)$, and choose $\delta >0$ with $u(x)<f(x)-3\delta $. Since $u$ is upper semicontinuous and $f$ is lower semicontinuous, there is a coordinate ball $B\ni x$ with $\overline B\subset X\setminus A$ such that $u<u(x)+\delta $ and $f>f(x)-\delta $ on $\overline B$; thus $\sup _{\overline B}u\leq \inf _{\overline B}f-\delta $. Shrinking $B$, we may assume that $B\Subset B'$ for a coordinate ball $B'$ with $\overline{B'}\subset X\setminus A$ and that $\theta =\mathrm{dd}^{\mathrm{c}}h$ near $\overline{B'}$ for a smooth function $h$ (lemma 5.4.2), and then, shrinking $B$ further, that the oscillation $\operatorname {osc}_{\overline B}h=\sup _{\overline B}h-\inf _{\overline B}h$ is less than $\delta $.

We show that $\mu _ u$ vanishes on $B$. Since $u$ is bounded near $\overline{B'}$, corollary 10.3.10 gives $\widetilde u\in \mathrm{PSH}(X,\theta )$ with $\widetilde u\geq u$, $\widetilde u=u$ on $X\setminus B$ and $(\theta +\mathrm{dd}^{\mathrm{c}}\widetilde u)^ n=0$ on $B$. The function $\widetilde u+h$ is plurisubharmonic, hence subharmonic, on $B$, and at every point $\zeta \in \partial B$ its upper limit from $B$ is at most $(\widetilde u+h)(\zeta )=(u+h)(\zeta )\leq \sup _{\overline B}(u+h)$, by upper semicontinuity. The maximum principle (proposition 3.4.7) therefore gives

Hence $\widetilde u\leq f$ quasi-everywhere on $X$, so $\widetilde u$ is a competitor and $\widetilde u\leq u$. Therefore $\widetilde u=u$, and $\mu _ u=(\theta +\mathrm{dd}^{\mathrm{c}}\widetilde u)^ n$ vanishes on $B$.

The set $\{ u<f\} $ is open, because $f-u$ is lower semicontinuous. Covering $\{ u<f\} \setminus A$ by countably many balls as above, we obtain $\mu _ u(\{ u<f\} \setminus A)=0$. Moreover $u\leq f$ quasi-everywhere by proposition 16.11.4, so the set $\{ u>f\} $ is pluripolar. Since $\mu _ u$ gives no mass to $A$ or to $\{ u>f\} $, it is carried by $\{ u=f\} $. $\square$

Proof. Write $u=P_\theta (f)$, which is a potential by proposition 16.11.4, and let $A$ be the proper analytic subset of lemma 16.13.2: every potential with minimal singularities is locally bounded on $X\setminus A$, and $A$ is closed and pluripolar. After reducing to obstacles that are bounded above, we treat bounded obstacles by approximating them with lower semicontinuous ones (Step 1), and then obstacles that are only bounded above by truncating them from below (Step 2).

Choose a constant $C_0>\sup _ Xu$ and put $\widetilde f=\min \{ f,C_0\} $, a quasi-continuous Borel function that is bounded above (lemma 16.11.2(2)). By proposition 16.11.4, $u\leq f$ quasi-everywhere, and $u<C_0$ everywhere, so $u\in \mathcal A_\theta (\widetilde f)\subset \mathcal A_\theta (f)$. Since $u$ is the largest member of $\mathcal A_\theta (f)$, it follows that $P_\theta (\widetilde f)=u$; moreover $\{ u=\widetilde f\} =\{ u=f\} $, because $u<C_0$. Replacing $f$ by $\widetilde f$, we may therefore assume that $f$ is bounded above.

Step 1: bounded obstacles. Suppose first that $|f|\leq C'$ quasi-everywhere for a constant $C'$. Then $\{ |f|>C'\} $ is a pluripolar Borel set, and replacing $f$ by $0$ on it changes neither $P_\theta (f)$ (proposition 16.11.4) nor the quasi-continuity of $f$ (lemma 16.11.2(3)); it changes the contact set only on a pluripolar set, which $\mu _ u$ does not charge. After this modification and the addition of a constant, we may therefore assume that $-C\leq f\leq 0$ everywhere.

We first construct open sets of small capacity outside which $f$ is continuous. Fix finitely many coordinate balls $D_1,\ldots ,D_ N$, compact sets $K_ i\subset D_ i$ with $\bigcup _ iK_ i=X$, and open sets $D_ i'$ with $K_ i\subset D_ i'\Subset D_ i$. For all $i$ and $k\geq 1$, the quasi-continuity of $f$ on $D_ i$ gives an open set $G_{ik}\subset D_ i'$ with $\operatorname {Cap}_{D_ i}(G_{ik})<2^{-k}$ such that $f$ is continuous on $K_ i\setminus G_{ik}$; intersecting with $D_ i'$ the open set provided by definition 7.5.1 does not change $K_ i\setminus G_{ik}$. Put

These open sets decrease, and $X\setminus G_ j$ is the union of the finitely many closed sets $K_ i\setminus \bigcup _{i'}G_{i'k}$, $1\leq i\leq N$, $1\leq k\leq j$. Since $f$ is continuous on each of them, it is continuous on the closed set $X\setminus G_ j$. Moreover, if $D$ is a coordinate ball and $L\subset D$ is compact, then $\operatorname {Cap}_ D(G_ j\cap L)\to 0$. Indeed, $L\cap \overline{D_ i'}$ is a compact subset of $D\cap D_ i$, so lemma 16.11.1(1) gives a constant $c_ i$ with $\operatorname {Cap}_ D(E)\leq c_ i\operatorname {Cap}_{D_ i}(E)$ for all Borel sets $E\subset L\cap \overline{D_ i'}$. Since $G_ j\cap L\subset \bigcup _ i(G_{ij}\cap L)$ and $G_{ij}\subset D_ i'$, it follows that

Next we approximate $f$ by lower semicontinuous obstacles that agree with $f$ outside $G_ j$. Fix a metric on $X$ that induces its topology (lemma 5.2.7). By the Tietze extension theorem in the form of lemma 7.5.3, applied to the restriction of $f+C/2$ to the compact set $X\setminus G_ j$, which takes values in $[-C/2,C/2]$, the restriction of $f$ to $X\setminus G_ j$ extends to a continuous function $g_ j\colon X\to [-C,0]$. Put

These functions are lower semicontinuous, take values in $[-C,0]$, and decrease with $j$. Since $G_ k\subset G_ j$ for $k\geq j$, we have $h_ j=f$ on $X\setminus G_ j$. The intersection $\bigcap _ jG_ j$ is pluripolar: if $D$ is a coordinate ball and $U\Subset D$ is open, then $U\cap \bigcap _ jG_ j$ is contained in the open sets $U\cap G_ j$, whose capacities $\operatorname {Cap}_ D$ tend to zero, so it has outer capacity zero and lies in the pole set of a plurisubharmonic function on $D$ by theorem 5.47.8. Outside $\bigcap _ jG_ j$ we have $h_ j=f$ for all large $j$. Hence the limit $h$ of the $h_ j$ equals $f$ quasi-everywhere.

Finally we pass to the limit in the contact property of the envelopes $w_ j=P_\theta (h_ j)$. By proposition 16.11.4, $w_ j$ decreases to $P_\theta (h)=P_\theta (f)=u$. Since the obstacles $h_ j$ and $f$ take values in $[-C,0]$, the potential $V_\theta -C$ is a competitor for each of them, and every competitor is nonpositive, hence at most $V_\theta $; thus $V_\theta -C\leq w_ j\leq V_\theta $ and $V_\theta -C\leq u\leq V_\theta $. Fix a coordinate ball $D\Subset X\setminus A$ with a smooth potential $\rho $ of $\theta $ near $\overline D$ (such balls exist around every point of $X\setminus A$, by lemma 5.4.2), and a continuous function $\chi \geq 0$ with compact support in $D$. On $D$ the functions $w_ j$ and $u$ are bounded uniformly in $j$. By lemma 16.13.3, $\mu _{w_ j}$ is carried by $\{ w_ j=h_ j\} $, so

The integrands with $h_ j$ and with $f$ are bounded uniformly in $j$ on the support of $\chi $, and they coincide outside $G_ j$. By lemma 7.5.7, applied to the bounded plurisubharmonic functions $\rho +w_ j$, the measures $\mu _{w_ j}$ of $G_ j\cap \operatorname{Supp}\chi $ are at most a fixed constant times $\operatorname {Cap}_ D(G_ j\cap \operatorname{Supp}\chi )$, which tends to zero, as shown above. Consequently

The weights $1-\mathrm e^{w_ j-f}$ are bounded and quasi-continuous on $D$, and they converge in capacity to $1-\mathrm e^{u-f}$. Indeed, the uniformly bounded potentials $w_ j$ decrease to $u$ and therefore converge in capacity (proposition 7.5.6), and $|\mathrm e^{w_ j-f}-\mathrm e^{u-f}|\leq \mathrm e^{C}|w_ j-u|$, because $w_ j-f$ and $u-f$ are at most $C$. The weighted form of theorem 7.5.8, applied to the bounded plurisubharmonic functions $\rho +w_ j$ on $D$, therefore gives

The integrand is nonnegative outside the pluripolar set $\{ u>f\} $, which $\mu _ u$ does not charge, and it is positive exactly at the points of $D$ where $u<f$. Since $\chi $ is arbitrary, lemma 16.12.1(2), applied on $W=D$ to the positive measure $\max \{ 1-\mathrm e^{u-f},0\} \mu _ u$ and the zero measure, shows that this measure vanishes on $D$; hence $\mu _ u(\{ u<f\} \cap D)=0$. Covering $X\setminus A$ by such balls, and recalling that $\mu _ u$ does not charge the pluripolar sets $A$ and $\{ u>f\} $, we conclude that $\mu _ u$ is carried by $\{ u=f\} $.

Step 2: obstacles bounded above. Adding a constant, we may assume that $f\leq 0$. Put $f_ j=\max \{ f,-j\} $ and $u_ j=P_\theta (f_ j)$. The obstacles $f_ j$ are quasi-continuous by lemma 16.11.2(2), satisfy $-j\leq f_ j\leq 0$, and decrease to $f$; their families of competitors contain $V_\theta -j$ and are bounded above by $0$. By proposition 16.11.4, $u_ j$ decreases to the potential $u$, and by Step 1, $\mu _{u_ j}$ is carried by $\{ u_ j=f_ j\} $. Moreover $V_\theta -j\leq u_ j\leq V_\theta $. The potentials $u_ j$ are not uniformly bounded below, so we pass to the limit on a fixed ball with truncated potentials, and with weights that vanish where the truncation is active.

Fix a coordinate ball $D\Subset X\setminus A$ as in Step 1 and constants $M,\varepsilon >0$, and put

and define $v^ M$ and $a$ in the same way with $u$ in place of $u_ j$. On $D$ these truncated potentials are bounded uniformly in $j$ and decrease to $v^ M$, so they converge in capacity (proposition 7.5.6). The weights satisfy $0\leq a_ j\leq 1$ and $a_ j=0$ on $\{ u_ j\leq V_\theta -M\} $. They are quasi-continuous, and they converge in capacity to $a$, since $s\mapsto (s+M)/(s+M+\varepsilon )$ is Lipschitz on $[-M,0]$.

Next put $b_ j=\mathrm e^{u_ j-f_ j}$ at the points where $u_ j$ and $f_ j$ are finite and $u_ j\leq f_ j$, and $b_ j=1$ elsewhere; define $b$ from $u$ and $f$ in the same way. Since $f\leq 0$ everywhere, $u\leq f$ and $u_ j\leq f_ j$ quasi-everywhere, and $u_ j$ and $f_ j$ are finite on $D$, the set of points of $D$ at which some $b_ j$ or $b$ is given the value $1$ is contained in the union of the pole set of $u$ and countably many pluripolar sets, so it is pluripolar.

We claim that $b_ j\to b$ in capacity on compact subsets of $D$ and that $b$ is quasi-continuous. Let $K\subset D$ be compact and $\varepsilon '>0$. Cover the exceptional pluripolar set by an open set of capacity less than $\varepsilon '$, which is possible by theorem 5.47.8, and enlarge it, using the quasi-continuity of $u$ (lemma 16.11.2(1)) and of $f$, to an open set $G$ of capacity less than $2\varepsilon '$ such that $u$ and $f$ are continuous and finite on the compact set $F=K\setminus G$. Then $u,f\geq -l$ on $F$ for some integer $l$, so $f_ j=f$ on $F$ for $j\geq l$. Since the exponential is $1$-Lipschitz on $(-\infty ,0]$, we get $|b_ j-b|\leq |u_ j-u|$ on $F$ for $j\geq l$. As $u\leq u_ j$ and $u\geq -l$ on $F$, we have $u_ j-u=\max \{ u_ j,-l\} -\max \{ u,-l\} $ there, and these truncations converge in capacity because they decrease. Hence, for every $\delta >0$, the outer capacity of $K\cap \{ |b_ j-b|>\delta \} $ is at most $2\varepsilon '$ plus a quantity that tends to zero, and $b$ is continuous on $F$. This proves the claim.

Since $a_ j$ vanishes outside the plurifine open set $\{ u_ j>V_\theta -M\} $, on which $v_ j^ M=u_ j$, and since $\mu _{u_ j}$ is carried by $\{ u_ j=f_ j\} $, where $b_ j=1$ up to a pluripolar set, we have

The weights $a_ j(1-b_ j)$ are bounded and converge in capacity to the quasi-continuous function $a(1-b)$. The weighted form of theorem 7.5.8, applied to the bounded plurisubharmonic functions $\rho +v_ j^ M$, gives $a(1-b)\mu _{v^ M}=0$ on $D$, and plurifine locality turns this into $a(1-b)\mu _ u=0$ on $D$. Let $\varepsilon \downarrow 0$ and then $M\to \infty $. Since $V_\theta $ is finite on $D$, the weights $a$ tend to $1$ at every point of $D$ where $u$ is finite. They are bounded by $1$, and $\mu _ u$ does not charge the pole set of $u$, so dominated convergence gives $(1-b)\mu _ u=0$ on $D$. Thus $u=f$ holds $\mu _ u$-almost everywhere on $D$. Covering $X\setminus A$ by such balls and using $\mu _ u(A)=0$ completes the proof. $\square$

Proof. By lemma 16.11.2, potentials are quasi-continuous, and hence so is the obstacle $\min \{ u,v\} $. Its family of competitors is nonempty, because $w\ne -\infty $, and bounded above by $\sup _ Xu$. By theorem 16.13.4, $\mu _ w$ is carried by $\{ w=\min \{ u,v\} \} \subset \{ w=u\} \cup \{ w=v\} $. Since $w\leq u$ and $w\leq v$ everywhere, lemma 16.13.1 gives $\mathbf1_{\{ w=u\} }\mu _ w\leq \mathbf1_{\{ w=u\} }\mu _ u$ and $\mathbf1_{\{ w=v\} }\mu _ w\leq \mathbf1_{\{ w=v\} }\mu _ v$. Adding these inequalities proves the first assertion; the intersection of the two contact sets is counted twice, which is harmless for an upper bound. The same argument, applied to the obstacle $\min _ ju_ j$, proves the last assertion. $\square$