Theorem 15.3.5A972C11565ED

Proof. Step 1: approximating equations and their constants. For integers $j\geq 1$, put $\omega _ j=\theta +2^{-j}\omega $, and choose continuous normalized solutions

We first bound $c_ j$. Fix a Gauduchon metric $\gamma _0$; when $n=1$ any Hermitian form will do, and the calculation below reduces to Stokes' theorem. The smooth approximation in the proof of theorem 14.19.2, together with the Gauduchon trace, gives a common upper bound: the numerator $\int _ X\omega _ j\wedge \gamma _0^{n-1}$ is bounded above by $\int _ X(\theta +\omega )\wedge \gamma _0^{n-1}$, and the root integral of $f$ is positive and fixed. Here the estimate is applied to the smooth approximating equations before passing to their continuous limits.

For a lower bound, take $s,b$ from lemma 15.3.1. If $c_ j<b$, then

for every constant $A$, and strict domination in the Hermitian background $\omega _ j$ (theorem 14.19.10) would give $\varphi _ j\geq s+A$ for every $A$, which is absurd. Thus $c_ j\geq b$. In fact the constants decrease with $j$: if $k>j$, positivity gives $\mathrm{MA}_{\omega _ j}(\varphi _ k)\geq c_ kf\, \mathrm{d}V$, and the same argument excludes $c_ j<c_ k$. Hence $c_ j\downarrow c>0$.

Step 2: a uniform bound for the potentials. For the Hermitian form $\Omega =\theta +\omega $, all $\varphi _ j$ are normalized $\Omega $-plurisubharmonic functions. Using uniform exponential integrability (lemma 5.26.2) and Hölder's inequality, choose $\gamma >0$ small and $1<q<p$ such that

is bounded in $L^ q$. Since $\varphi _ j\leq 0$, we have $g_ j\geq f$. Theorem 15.3.3, applied with integrability exponent $q$ and exponential exponent $\gamma $, therefore gives uniformly bounded $w_ j\in \mathrm{PSH}(X,\theta )$ with

Choose $C_0\geq \max \{ 1,\sup _ jc_ j\} $ and set $z_ j=w_ j-\gamma ^{-1}\log C_0$. Then

and exponential comparison gives $z_ j\leq \varphi _ j\leq 0$. Thus all $\varphi _ j$ take values in a common bounded interval.

Step 3: the upper envelopes. Extract a subsequence converging in $L^1$ and almost everywhere to the canonical $\theta $-plurisubharmonic representative $\varphi $. We keep the indices $j$; the coefficients of $\omega $ in $\omega _ j$ still decrease to zero. Define

All these functions take values in a common bounded interval. The upper envelopes are $\omega _ j$-plurisubharmonic and decrease to $\varphi $, since the regularized upper envelope agrees almost everywhere with the $L^1$ limit; this also follows from the local Hartogs lemma applied to the bounded tails.

For $\ell \geq j$, positivity gives $\mathrm{MA}_{\omega _ j}(\varphi _\ell )\geq c_\ell f\, \mathrm{d}V\geq cf\, \mathrm{d}V$. By ordered contact (lemma 5.28.3), finite maxima of these functions satisfy the same lower bound. Continuity of bounded products along increasing sequences, followed by the decreasing limit in $j$ and the expansion in the smooth background, gives

Step 4: the lower envelopes. For the reverse inequality we use lower envelopes, which carry a useful exponential weight. For $N\geq j$ put

The obstacle is continuous and bounded, so the measure of $p_{j,N}$ is carried by its contact set. Partition this set into disjoint Borel pieces according to the first index $\ell $ at which $p_{j,N}=\varphi _\ell $. On such a piece, positivity and ordered contact in the background $\omega _\ell $ give

Moreover $p_{j,N}=\varphi _\ell \geq h_ j$ there, and $c_\ell \leq c_ j$. Summing over the pieces therefore yields the global inequality

As $N\to \infty $, the envelopes decrease to $p_ j$. Indeed, their limit lies below each tail obstacle quasi-everywhere and is thus a competitor for $P_\theta (h_ j)$, while the reverse inequality is immediate. Continuity of bounded products along monotone sequences and dominated convergence give

The functions $p_ j$ increase. Let $\varphi '=(\lim _ jp_ j)^*$. Since $h_ j\to \varphi $ almost everywhere, we have $\varphi '\leq \varphi $ for the canonical representatives. Passing to the increasing limit and using (15.3.6), we obtain

Bounded exponential comparison forces $\varphi '\geq \varphi $. Thus $\varphi '=\varphi $, and the last inequality, together with (15.3.6), proves the equation. If necessary, subtracting $\sup _ X\varphi $ gives the required normalization.

Step 5: uniformity and uniqueness of $c$. All bounds used above depend only on the two stated density bounds and the fixed data, which proves the uniform assertion for the constructed solution. Finally, two different positive constants for the same $f$ would contradict strict domination after an arbitrary translation of one of the bounded solutions, exactly as in the lower bound for $c_ j$. This proves the uniqueness of $c$. $\square$