2502C12D48F4
The Bernstein–Markov inequality keeps the partial Fubini–Study potentials below the weight on $K$, up to an error that becomes arbitrarily small as the degree grows. The multiplier-ideal constraint survives in the limit because, on a fixed local modification, the Lelong numbers of the potentials fall short of those of the selecting potential by at most $O(m^{-1})$. To pass the obstacle condition to $L^1$ limits when $K$ has zero smooth volume, we compare the potentials with the envelope of the obstacle alone, which is defined on all of $X$.
Now take $\theta =c_1(L,h)$, assume $\varphi \in \mathrm{PSH}(X,\theta )$, and let $\mu $ be a Bernstein–Markov probability measure for $(K,w)$, with $K$ and $w$ as in section 18.32. Use the selected spaces $V_ m$ and their dimensions $N_ m$ from section 18.30. Choose an orthonormal basis $s_1,\ldots ,s_{N_ m}$ of $V_ m$ for the norm $\| \cdot \| _{m,w,\mu }$ of section 18.31, and define the partial Fubini–Study potential
\[ U_ m=\frac1m\log \sum _{j=1}^{N_ m}|s_ j|_{h^ m}^2 \quad (N_ m>0). \]
Set $U_ m\equiv -\infty $ if $N_ m=0$. When $N_ m>0$, the local expression in a holomorphic frame, as in the proof of proposition 19.8.3, shows that $U_ m\in \mathrm{PSH}(X,\theta )$ and that $U_ m$ has analytic singularities. On $K$ it satisfies $\mathrm{e}^{m(U_ m-w)}=B_{m,w,\varphi ,\mu }$.
Let $\pi :Y\to W$ be a local smooth modification of an open set $W\subseteq X$, let $y\in Y$, and write $J_\pi $ for the local holomorphic Jacobian determinant of $\pi $. Each $s_ j$ lies in $V_ m$, so lemma 5.38.10 applies to its local coefficient. Since the Lelong number of the logarithm of a sum of squared moduli of holomorphic functions is the minimum of their vanishing orders, and the smooth metric contributes nothing, we obtain
\[ \nu (U_ m\circ \pi ,y) \geq \nu (\varphi \circ \pi ,y) -\frac{n+\operatorname {ord}_ yJ_\pi }{m} \qquad (N_ m>0). \]
Theorem 18.33.1.336ACC796329 The potentials $U_ m$ with $N_ m>0$ form a relatively compact subset of $L^1(X)$. Let $U$ be the $L^1$ limit of a sequence $U_{m_ j}$ with $N_{m_ j}>0$ and $m_ j\to \infty $. Then $U\in \mathrm{PSH}(X,\theta )$,
\[ U\preceq _{\mathcal I}\varphi , \]
and $U\leq w$ quasi-everywhere on $K$. Consequently
\[ \left(\limsup _{m\to \infty }U_ m\right)^* \leq E_{\theta ,K}[\varphi ]_{\mathcal I}(w). \]
If only finitely many selected spaces are nonzero, the left-hand side is identically $-\infty $.
Proof.
Step 1: compactness. The extremal formula for the kernel and the Bernstein–Markov inequality give, for every $\epsilon >0$ and with $C_\epsilon \geq 1$,
\[ U_ m\leq w+\delta _{m,\epsilon }\quad \hbox{on }K, \qquad \delta _{m,\epsilon }=2\epsilon +\frac{2\log C_\epsilon }{m}. \]
No factor depending on $N_ m$ occurs, because the extremal formula is a supremum over single sections of unit norm. Fix one $\epsilon >0$. Since $\delta _{m,\epsilon }\leq 2\epsilon +2\log C_\epsilon $, lemma 5.40.1 gives an upper bound for $\sup _ XU_ m$ that is uniform over all $m$ with $N_ m>0$. On the other hand, the total mass identity gives
\[ \int _ K \mathrm{e}^{m(U_ m-w)}\, \mathrm{d}\mu =N_ m\geq 1. \]
Since $\mu $ is a probability measure, $U_ m-w\geq 0$ somewhere on $K$, and therefore $\sup _ XU_ m\geq \min _ Kw$. The normalized compactness theorem (theorem 5.25.2) now gives relative compactness in $L^1(X)$, with limits in $\mathrm{PSH}(X,\theta )$.
Step 2: the multiplier-ideal order. Let $U_{m_ j}\to U$ in $L^1(X)$, with $m_ j\to \infty $. Fix a local smooth modification $\pi :Y\to W$ of an open set $W\subseteq X$, and a point $y\in Y$. By lemma 5.43.19, $U_{m_ j}\circ \pi \to U\circ \pi $ in $L^1_{\mathrm{loc}}$. Upper semicontinuity of Lelong numbers (lemma 5.38.7) and the order bound before the theorem give
\[ \nu (U\circ \pi ,y) \geq \limsup _ j\nu (U_{m_ j}\circ \pi ,y) \geq \nu (\varphi \circ \pi ,y). \]
Since $\pi $ and $y$ are arbitrary, the local valuative characterization in theorem 5.37.3 gives $U\preceq _{\mathcal I}\varphi $. Note that the modification and the point are fixed before passing to the limit; the same convergent sequence serves for all of them.
Step 3: the obstacle. Since $K$ may have zero smooth volume, the inequality on $K$ in Step 1 does not pass directly to $L^1$ limits. We therefore use the envelope of the obstacle alone. Set
\[ G=E_{\theta ,K}[V_\theta ]_{\mathcal I}(w). \]
Every $a\in \mathrm{PSH}(X,\theta )$ satisfies $a\leq V_\theta +\sup _ Xa$, and hence $a\preceq _{\mathcal I}V_\theta $. Thus the multiplier-ideal constraint in the definition of $G$ is no restriction, and $G$ is the largest $\theta $-plurisubharmonic function lying below $w$ quasi-everywhere on $K$. Since $U_ m-\delta _{m,\epsilon }$ is such a function, Step 1 gives, on all of $X$,
\[ U_ m\leq G+\delta _{m,\epsilon }. \]
Passing to the $L^1$ limit gives $U\leq G+2\epsilon $ almost everywhere, and hence everywhere by lemma 5.24.3. Letting $\epsilon \downarrow 0$ gives $U\leq G$. Since $G\leq w$ quasi-everywhere on $K$ by theorem 18.32.2, the same holds for $U$. Thus $U$ satisfies both defining conditions of the partial envelope, and $U\leq E_{\theta ,K}[\varphi ]_{\mathcal I}(w)$.
Step 4: the upper limit. By the local Hartogs lemma, if $U_{m_ j}\to U$ in $L^1$ and $x_ j\to x$, then $\limsup _ jU_{m_ j}(x_ j)\leq U(x)$. Given a point $x$, a sequence of indices $m_ j\to \infty $ with $N_{m_ j}>0$ and points $x_ j\to x$, pass to a subsequence along which $U_{m_ j}(x_ j)$ converges to $\limsup _ jU_{m_ j}(x_ j)$ and $U_{m_ j}$ converges in $L^1$. By Steps 2 and 3 the $L^1$ limit is at most the partial envelope, so $\limsup _ jU_{m_ j}(x_ j)\leq E_{\theta ,K}[\varphi ]_{\mathcal I}(w)(x)$. This proves the inequality for the regularized upper limit, also at the poles of the envelope. Indices with $N_ m=0$ contribute only $-\infty $.
$\square$
A matching lower bound would require global sections approximating the envelope; for the projective models later in this chapter such sections are constructed explicitly. Before turning to limits of measures, we show that the partial kernel measures are the derivatives of a determinant functional.
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