In this section the background is a Hermitian form $h$ on $X$ with $v_-(h)>0$, and the reference is zero; thus $\theta =h$ and $\varphi =0$ in the notation of the chapter, and $\mathcal E_{\mathrm{env}}(X,h,0)$ is the minimal envelope class. Let $0\leq f\in L^1(\mathrm{d}V)$ be nonzero. Recall from (15.28.1) that the subsolution threshold $b_ h(f)$ is the limit of the normalizing constants of the truncations $\min (f,k)$; by theorem 15.28.4, it is the largest $a\geq 0$ for which some $u\in \mathrm{PSH}(X,h)$ satisfies $\mathrm{MA}^{\mathrm{ext}}_ h(u)\geq af\, \mathrm{d}V$. We show that $b_ h(f)$ is exactly the scalar of the normalized equation in the minimal envelope class: scalar comparison bounds every subsolution coefficient by this scalar. Dominated continuity also identifies the equation satisfied by every limit of normalized greatest exponential solutions; whether these limits coincide is a question about the potentials.
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.
$\square$
By theorem 5.25.2, every sequence $\lambda _ j\downarrow 0$ has a subsequence along which $v_{\lambda _ j}$ converges in $L^1$. The corollary identifies the equation of every such limit; the examples below show that this equation alone need not determine the normalized potential.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.