13.14.5 Contact sets900BB92BD0E3

Proof. Fix a coordinate ball with a constant positive form $\kappa $, a level $t>0$, and $\epsilon >0$. On smaller neighborhoods, freeze $\theta $ to a constant form $\theta '$ with $-\epsilon \kappa \leq \theta -\theta '\leq \epsilon \kappa $, and choose a quadratic function $r$ with $\mathrm{dd}^{\mathrm{c}}r=\theta '+\epsilon \kappa $. Then $u+r$ and $v+r$ are plurisubharmonic. For a large integer $k$ put

These functions are bounded on relatively compact neighborhoods, satisfy $a_ k\leq b_ k$, and

with the same inequality for $b_ k$. We may therefore apply the top-degree case of lemma 5.28.3 with background $\theta +2\epsilon \kappa $.

On $D_ t=\{ u>-t\} $, both $u$ and $v$ are larger than $-t$. Taking $k>t+\sup |r|$ makes $a_ k=u$ and $b_ k=v$ on $D_ t$. Restricting the bounded inequality to $\{ u=v\} \cap D_ t$ and using locality, we obtain the contact inequality for the background $\theta +2\epsilon \kappa $ on that set. The smaller neighborhoods cover the original ball, so the restricted inequality holds throughout it.

Choose a Hermitian form $\Omega \geq \theta $. The truncations $u_ t=\max (u,-t)$ and $v_ t=\max (v,-t)$ are bounded and $\Omega $-plurisubharmonic, so their signed products with background $\theta $ are defined. On $D_ t$, the products used above are the restrictions of $(\theta +\mathrm{dd}^{\mathrm{c}}u_ t+2\epsilon \kappa )^ n$ and $(\theta +\mathrm{dd}^{\mathrm{c}}v_ t+2\epsilon \kappa )^ n$. These are polynomials in $\epsilon $ whose coefficients have locally finite variation, so they converge in local total variation as $\epsilon \downarrow 0$, also after restriction to $\{ u=v\} \cap D_ t$. This proves the inequality for the background $\theta $ on each finite level. Finally let $t\to +\infty $: the common pole set has zero extended mass, and the sets $\{ u=v\} \cap D_ t$ exhaust its complement in the contact set. $\square$