Corollary 15.37.32.74BE83BA167D Let $h$ be a Hermitian form on $X$ with $v_-(h)>0$, and let $0\leq f\in L^1(\mathrm{d}V)$ be nonzero. There is $v\in \mathcal E_{\mathrm{env}}(X,h,0)$ such that
\[ \mathrm{MA}^{\mathrm{ext}}_ h(v)=b_ h(f)f\, \mathrm{d}V. \]
The number $b_ h(f)>0$ is the unique positive scalar for which this equation has a solution in that envelope class.
For $\lambda >0$, let $u_\lambda $ be the greatest solution of $\mathrm{MA}^{\mathrm{ext}}_ h(u_\lambda )=\mathrm{e}^{\lambda u_\lambda }f\, \mathrm{d}V$ from theorem 15.35.2, and set $v_\lambda =u_\lambda -\sup _ Xu_\lambda $. Each $u_\lambda $ belongs to $\mathcal E_{\mathrm{env}}(X,h,0)$. If $\lambda _ j\downarrow 0$ and $v_{\lambda _ j}\to v$ in $L^1$, then the convergence holds in capacity, and $v$ belongs to the same envelope class and satisfies the equation above.
Proof.
By theorem 15.37.28, applied with $\theta =h$, there are $c>0$ and $w\in \mathcal E_{\mathrm{env}}(X,h,0)$ with $\mathrm{MA}^{\mathrm{ext}}_ h(w)=cf\, \mathrm{d}V$. If $s\in \mathrm{PSH}(X,h)$ satisfies $\mathrm{MA}^{\mathrm{ext}}_ h(s)\geq af\, \mathrm{d}V$ with $a>0$, then
\[ \mathrm{MA}^{\mathrm{ext}}_ h(w) \leq \frac ca\, \mathrm{MA}^{\mathrm{ext}}_ h(s). \]
Every $h$-plurisubharmonic $s$ satisfies $s\preceq 0$, so the scalar assertion of corollary 15.37.11, with reference zero, gives $c/a\geq 1$. Taking the maximum over all subsolution coefficients $a$, theorem 15.28.4 yields $b_ h(f)\leq c$. The reverse inequality holds because $w$ itself is a subsolution with coefficient $c$. The same argument identifies the scalar of every solution in this envelope class with $b_ h(f)$.
For $\lambda >0$, theorem 15.37.28 also gives a solution $s_\lambda $ of the exponential equation in the envelope class. Since $s_\lambda $ is in particular a subsolution, the greatest-solution property gives $s_\lambda \leq u_\lambda $. Upward stability (proposition 16.28.62) places $u_\lambda $ and its translate $v_\lambda $ in that class.
Write $c_\lambda =\exp (\lambda \sup _ Xu_\lambda )$. By corollary 15.35.3, $c_\lambda \to b_ h(f)>0$ and
\[ \mathrm{MA}^{\mathrm{ext}}_ h(v_\lambda ) =c_\lambda \mathrm{e}^{\lambda v_\lambda }f\, \mathrm{d}V \longrightarrow b_ h(f)f\, \mathrm{d}V \quad \text{in total variation}. \]
Since $v_\lambda \leq 0$ and $c_\lambda $ stays bounded for small $\lambda $, these measures are dominated by one finite non-pluripolar measure $C f\, \mathrm{d}V$. Applying theorem 15.37.19 to the given $L^1$-convergent sequence $v_{\lambda _ j}$, we obtain class membership, convergence in capacity and the equation for its limit.
$\square$
Pome
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