16.23 Rooftops and mass inequalities4CC29395A55C

Proof. Subtracting the same constant from $u$ and $v$ changes neither the hypothesis nor the conclusion, so we may assume that $u,v\leq 0$. Put $m=\int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle $ and $\phi =P_\theta [\max \{ u,v\} ]$. By monotonicity $\int _ X\langle \theta _ u^ n\rangle \leq m$ and $\int _ X\langle \theta _ v^ n\rangle \leq m$, so the strict inequality forces both masses to be positive. Hence $m>0$, and by theorem 16.22.11, $\phi $ is a model potential of mass $m$. Moreover $u,v\leq \max \{ u,v\} \leq \phi $, because the nonpositive potential $\max \{ u,v\} $ is a competitor for $\phi $.

Since $\mu _ u$ and $\mu _ v$ do not charge pole sets, $\mu _ u(\{ u>\phi -k\} )\to \int _ X\langle \theta _ u^ n\rangle $ and $\mu _ v(\{ v>\phi -k\} )\to \int _ X\langle \theta _ v^ n\rangle $ as $k\to \infty $. Choose $k>0$ so large that

For $C>k$ put

Since $\phi -C$ is a common minorant, all three potentials have the singularity type of $\phi $ and mass $m$. As $C\to \infty $, $w_ C$ decreases to $P_\theta (u,v)$ by proposition 16.11.6. Suppose that $P_\theta (u,v)=-\infty $, so that $w_ C$ decreases to $-\infty $ identically. Then $\sup _ Xw_ C\to -\infty $ by corollary 10.3.6, and lemma 16.22.2 gives $w_ C\leq \phi -k$ for all large $C$. On $\{ w_ C=u_ C\} $ we then have $u_ C\leq \phi -k$, and similarly for $v_ C$, so corollary 16.13.5 gives

For the equality we used that, since $C>k$, $u_ C=u$ on the plurifine open set $\{ u>\phi -k\} $, so $\mu _{u_ C}=\mu _ u$ there, and similarly for $v$. This contradiction shows that $P_\theta (u,v)\ne -\infty $. $\square$

Proof. Since $w\leq \phi $, monotonicity gives $P_\theta [w]\leq P_\theta [\phi ]=\phi $, and likewise $P_\theta [w]\leq \psi $. Thus $P_\theta [w]$ is a common minorant of $\phi $ and $\psi $, and $P_\theta [w]\leq w$. Conversely, $w\leq \phi \leq 0$, so $w$ is a competitor for $P_\theta [w]$. $\square$

Proof. Shifting $u$ and $v$ by constants affects neither the existence of a common minorant nor the mass of $P_\theta (u,v)$. Indeed, a common minorant can be shifted down by the largest absolute change, and by proposition 16.11.4(4) the rooftop envelopes of the original and of the shifted pair differ by a bounded amount, because their obstacles do. We may therefore assume that $u\leq \phi $ and $v\leq \psi $.

We first replace $\phi $ by $u$, and then $\psi $ by $v$; both steps combine extrapolation with proposition 16.23.1. Since $w\leq \phi $ and $w\leq \psi $, the masses of $\phi $ and $\psi $ are at least that of $w$, hence positive. For $a>1$ let $u_ a=E_ a(u,\phi )$. Since $u$ and $\phi $ have the same positive mass, theorem 16.22.4 shows that $u_ a$ also has this mass and that $a^{-1}u_ a+(1-a^{-1})\phi \leq u$; moreover $u_ a\leq u\leq \phi $, because the obstacle $u+(a-1)(u-\phi )$ of $u_ a$ is at most $u$. Since $\max \{ u_ a,w\} \leq \phi $ and $\int _ X\langle \theta _ w^ n\rangle >0$,

By proposition 16.23.1, the rooftop envelope $P_\theta (u_ a,w)$ is a potential. By associativity and $u_ a\leq \phi $, it equals $P_\theta (u_ a,\psi )$. The potential

lies below $a^{-1}u_ a+(1-a^{-1})\phi \leq u$ and below $\psi $, so it is a common minorant of $u$ and $\psi $. Hence $z=P_\theta (u,\psi )$ is a potential, and positivity and monotonicity of mass give $\int _ X\langle \theta _ z^ n\rangle \geq (1-a^{-1})^ n\int _ X\langle \theta _ w^ n\rangle $. On the other hand $z\leq P_\theta (\phi ,\psi )=w$, so the mass of $z$ is at most that of $w$. Letting $a\to \infty $, we see that $z$ and $w$ have the same positive mass.

Now let $v_ a=E_ a(v,\psi )$. In the same way, $v_ a\leq v\leq \psi $, the mass of $v_ a$ is that of $\psi $, and $a^{-1}v_ a+(1-a^{-1})\psi \leq v$. Since $\max \{ v_ a,z\} \leq \psi $ and $z$ has positive mass, we have

so proposition 16.23.1 shows that $P_\theta (z,v_ a)$ is a potential, and by associativity and $v_ a\leq \psi $ it equals $P_\theta (u,v_ a)$. The potential

lies below $u$ and below $a^{-1}v_ a+(1-a^{-1})\psi \leq v$, and its mass is at least $(1-a^{-1})^ n\int _ X\langle \theta _ z^ n\rangle $. Hence $P_\theta (u,v)$ is a potential whose mass is at least $(1-a^{-1})^ n\int _ X\langle \theta _ w^ n\rangle $, for every $a>1$. Since $P_\theta (u,v)\leq w$, its mass is at most that of $w$, and letting $a\to \infty $ completes the proof. $\square$

Proof. Put $\lambda _ j=\mathbf1_{\{ q_ j=0\} }\mu _{q_ j}$. Since $q_ j\leq V_\theta \leq 0$, the set $\{ q_ j=0\} $ is contained in $\{ q_ j=V_\theta \} $, and lemma 16.13.1 gives $\lambda _ j\leq \mu _{V_\theta }$. Let $A$ be the proper analytic subset of lemma 16.13.2: potentials with minimal singularities are locally bounded on $X\setminus A$, and $\mu _{V_\theta }(A)=0$.

For $t>0$ put $b_ j=\max \{ 1+tq_ j,0\} $ and $b=\max \{ 1+tq,0\} $, with value zero at the poles. Fix a coordinate ball $D\Subset X\setminus A$ with a smooth potential $\rho $ of $\theta $ near $\overline D$ (lemma 5.4.2) and a number $L>1/t$, and put $\widetilde q_ j=\max \{ q_ j,V_\theta -L\} $ and $\widetilde q=\max \{ q,V_\theta -L\} $. On $D$ the functions $\widetilde q_ j$ are uniformly bounded and decrease to $\widetilde q$, so they converge in capacity (proposition 7.5.6). Since $V_\theta -L<-1/t$, the weight $b_ j$ vanishes on $\{ q_ j\leq V_\theta -L\} $, and on the complementary plurifine open set $\widetilde q_ j=q_ j$; hence $b_ j\mu _{q_ j}=b_ j\mu _{\widetilde q_ j}$ by plurifine locality, and likewise $b\mu _ q=b\mu _{\widetilde q}$. Moreover $b_ j=\max \{ 1+t\widetilde q_ j,0\} $ and $b=\max \{ 1+t\widetilde q,0\} $ on $D$, so the weights are bounded and quasi-continuous and converge in capacity to $b$. The weighted form of theorem 7.5.8, applied to the bounded plurisubharmonic functions $\rho +\widetilde q_ j$, therefore gives

Let $\delta >0$. Since $\mu _{V_\theta }(A)=0$, lemma 16.12.1(1) gives a compact set $K\subset X\setminus A$ with $\mu _{V_\theta }(X\setminus K)<\delta $; choose a continuous function $0\leq \zeta \leq 1$ with compact support in $X\setminus A$ and $\zeta =1$ on $K$ (proposition 5.2.6). Since $b_ j=1$ on $\{ q_ j=0\} $ and $\lambda _ j\leq \mu _{V_\theta }$,

Cover the support of $\zeta $ by finitely many balls $D_ l$ as above, and write $\zeta =\sum _ l\zeta _ l$ with continuous functions $\zeta _ l\geq 0$ with compact support in $D_ l$ (for instance $\zeta _ l=\zeta \psi _ l/\sum _ k\psi _ k$, where $\psi _ l\geq 0$ is continuous with compact support in $D_ l$ and $\sum _ k\psi _ k>0$ on the support of $\zeta $). The weak convergence on each $D_ l$ gives $\int _ X\zeta b_ j\, \mu _{q_ j}\to \int _ X\zeta b\, \mu _ q\leq \int _ Xb\, \mu _ q$, and letting then $\delta \downarrow 0$ gives $\varlimsup _ j\lambda _ j(X)\leq \int _ Xb\, \mu _ q$. Finally, as $t\to \infty $ the weights $b$, which depend on $t$, decrease to $\mathbf1_{\{ q=0\} }$, because $q\leq 0$; dominated convergence for the finite measure $\mu _ q$ proves the lemma. $\square$

Proof. We first reduce to model potentials, and then apply contact concentration to the rooftop envelopes of their truncations.

If $u$ has mass zero, the inequality follows from the monotonicity of mass, $\int _ X\langle \theta _ v^ n\rangle \leq \int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle $, and similarly if $v$ has mass zero. Otherwise, subtracting the same constant from $u$ and $v$, which subtracts it from $\max \{ u,v\} $ and $w$ and changes no mass, we may assume that $u,v\leq 0$. Put $\phi =P_\theta [u]$ and $\psi =P_\theta [v]$; by theorem 16.22.11, these are model potentials with the same masses as $u$ and $v$, respectively, and $u\leq \phi $, $v\leq \psi $, since $u$ and $v$ are competitors for their own singularity envelopes. Put also $\gamma =\max \{ \phi ,\psi \} $ and $r=P_\theta (\phi ,\psi )$.

The four masses in the inequality do not change when $u,v$ are replaced by $\phi ,\psi $. Indeed, for every integer $C$ the potential $\max \{ P_\theta (u+C,V_\theta ),P_\theta (v+C,V_\theta )\} $ has the singularity type of $\max \{ u,v\} $, by proposition 16.11.6, and by proposition 16.21.2(2) these potentials increase almost everywhere to $\gamma $ as $C\to \infty $; so $\gamma $ has the same mass as $\max \{ u,v\} $, by corollary 13.8.9. Moreover $w$ is a common minorant of $\phi $ and $\psi $, so $r\geq w$ is a potential. If $r$ has mass zero, then so has $w$, by monotonicity. If the mass of $r$ is positive, then $w$ has the same mass as $r$ by proposition 16.23.3, applied to $u\in \mathcal E(X,\theta ,\phi )$ and $v\in \mathcal E(X,\theta ,\psi )$. It therefore suffices to prove the theorem for the model potentials $\phi $ and $\psi $, whose maximum is $\gamma $ and whose rooftop envelope is $r$.

Put $m=\int _ X\langle \theta _\gamma ^ n\rangle $, and for $C>0$ let

These potentials have the singularity type of $\gamma $, hence mass $m$, and $r_ C$ decreases to $r$ as $C\to \infty $. By proposition 16.21.3, the measure $\mu _\phi $ is carried by $\{ \phi =0\} $, which is contained in the plurifine open set $\{ \phi >\gamma -C\} $, where $\phi _ C=\phi $. Plurifine locality therefore gives

and the last summand has mass $m-\int _ X\langle \theta _\phi ^ n\rangle $; the analogous decomposition holds for $\psi _ C$. By corollary 16.13.5, $\mu _{r_ C}\leq \mathbf1_{\{ r_ C=\phi _ C\} }\mu _{\phi _ C}+\mathbf1_{\{ r_ C=\psi _ C\} }\mu _{\psi _ C}$. On the set $\{ r_ C<0\} $, the summand $\mu _\phi $ of $\mu _{\phi _ C}$ contributes nothing to this bound, because at a point of $\{ \phi =0\} \cap \{ r_ C=\phi _ C\} $ we have $r_ C=\phi _ C=0$; the same holds for $\mu _\psi $. Consequently

Since $r_ C\leq 0$ and $\int _ X\langle \theta _{r_ C}^ n\rangle =m$, it follows that

Applying lemma 16.23.4 along the integers $C\to \infty $ to the potentials $r_ C$, which decrease to $r$, we find that

which is the assertion for $\phi $ and $\psi $. $\square$