Theorem 15.16.2352C3567BAAD

Proof. Step 1: approximating singularity types. Apply theorem 5.36.5 to $V_\theta $. Set $\delta _ j=2^{-j}$ and pass to a subsequence whose curvature errors are at most $\delta _ j$. Denote its potentials by $\widetilde\psi _ j$; thus $\widetilde\psi _ j\downarrow V_\theta $ and $\theta _{\widetilde\psi _ j}\geq -\delta _ j\omega $. Set

By Hartogs' lemma, $\sup _ X\widetilde\psi _ j\to \sup _ XV_\theta =0$, so $a_ j\to 0$. A common resolution of the singularities of $\widetilde\psi _ j$ and $\rho $ shows that $\psi _ j$ has a smooth remainder after resolution. Moreover,

The upper inequality holds because $\psi _ j$ is $\theta $-plurisubharmonic and nonpositive. Subtracting $a_ j$ explicitly keeps the normalization separate from the fixed barrier $\rho $.

Step 2: uniform bounds. By lemma 15.16.1 there are $u_ j\sim \psi _ j$ satisfying

Each $u_ j$ is bounded below by $\rho $ minus a constant depending on $j$, so proposition 15.14.2 gives a uniform upper bound for $\sup _ Xu_ j$.

There is also a uniform lower bound. We apply lemma 15.14.1 with the barrier $\rho $ replaced by $\rho -1-\sup _ ja_ j$ and with the reference $\psi _ j-1$, which lies between this barrier and zero. The lemma gives $v_ j$ with $\psi _ j-1\leq v_ j\leq \psi _ j\leq 0$ and $\mathrm{MA}_\theta (v_ j)\geq mf\, \mathrm{d}V$, where $m>0$ is independent of $j$. Put $w_ j=v_ j+\lambda ^{-1}\log m$. Then

and the exponential order in corollary 15.13.3 gives $u_ j\geq w_ j$. Thus, for a uniform constant $C$,

After extraction, $u_ j$ converges in $L^1$ and almost everywhere to a $\theta $-plurisubharmonic function $u$, and the preceding bounds give $V_\theta -C\leq u\leq V_\theta +C$.

Step 3: the equation for the limit. We identify the equation using envelopes. Put $A_ j=\sup _{k\geq j}a_ k$ and, for $N\geq j$, define

Both functions lie above the $\theta $-plurisubharmonic function

which is a competitor for every $p_{j,N}$; in particular, no envelope is identically $-\infty $. All these functions are locally bounded on $\Omega $, uniformly in $j$ and $N$. Contact support (theorem 15.8.6), together with a partition of the contact set according to the first index of contact, gives

No factor $N$ appears here: on each piece the envelope equals a single $u_ k$, and ordered contact bounds its measure by the measure of that $u_ k$. Similarly, locality and ordered contact for a finite maximum give

Now let $N\to \infty $, and set $p_ j=\lim _ Np_{j,N}$ and $t_ j=(\lim _ Nt_{j,N})^*$. Continuity of local bounded products along monotone sequences passes both inequalities to these limits. Next, $p_ j$ increases to a regularized limit $p$, while $t_ j$ decreases to $u$ by Hartogs' lemma. The bound by $b_ j$ gives $p\sim V_\theta $, and $p\leq u$. A second application of the monotone continuity gives

These inequalities hold on $\Omega $, and they extend across $Z$ because no measure involved charges $Z$. Thus $\mathrm{MA}_\theta (p)\leq \mathrm{e}^{\lambda (p-u)}\mathrm{MA}_\theta (u)$, and the exponential order in corollary 15.13.3 gives $p\geq u$. Hence $p=u$, and the equation follows. Applying the same order in both directions proves uniqueness. Note that the Monge–Ampère operator was only ever applied along monotone sequences that are locally bounded on $\Omega $. $\square$