Through the Jacobian isometry, the equilibrium theorem on a compact Kähler manifold can be pushed down along a modification. The limit does not depend on the modification, because both the partial envelope and its non-pluripolar Monge–Ampère measure descend.
Theorem 18.35.5.B187DFAF6D97 Suppose that $X$ admits a modification from a compact Kähler manifold. Let $L\to X$ be a holomorphic line bundle with a smooth Hermitian metric $h$, put $\theta =c_1(L,h)$, and let $\varphi \in \mathrm{PSH}(X,\theta )$ satisfy
\[ \mathrm{MA}_\theta (\varphi )(X)>0. \]
Then, for every $v\in C(X)$ and every smooth positive probability volume $\nu _ X$, the partial kernel measures on $X$ satisfy
\[ \frac{n!}{m^ n}B_{m,v,\varphi ,\nu _ X}\, \nu _ X \rightharpoonup \mathrm{MA}_\theta (E_ X(v)). \]
The limit measure is finite and nonzero. Its total mass is
\[ M=\lim _{m\to \infty }\frac{n!}{m^ n} h^0(X,L^ m\otimes \mathcal I(m\varphi )), \]
which does not depend on $v$ or $\nu _ X$. The limit does not depend on the choice of Kähler modification.
Proof.
Choose a modification $\pi :Y\to X$ from a compact Kähler manifold $Y$, and a smooth positive probability volume $\nu _ Y$ on $Y$. The pullback $\varphi \circ \pi $ belongs to $\mathrm{PSH}(Y,\pi ^*\theta )$, and by proposition 18.35.1 its non-pluripolar mass equals the given positive mass of $\varphi $. This mass is finite, by the theory of closed classes on compact Kähler manifolds.
Apply [Theorem 14.3.1, SGPT] on $Y$, with the line bundle $\pi ^*L$, the auxiliary twisting line bundle $R_\pi $, the continuous obstacle $v\circ \pi $ on $Y$ and the probability volume $\nu _ Y$. Its Bernstein–Markov hypothesis follows from the proof of proposition 18.31.1, since a fixed smooth auxiliary bundle only changes the local constants. The multiplier ideals and the continuous weighted norms in that theorem agree exactly with those of proposition 18.35.3. The theorem gives
\[ \beta _{Y,m}^{(n)}\rightharpoonup \mathrm{MA}_{\pi ^*\theta } \bigl (E_{\pi ^*\theta ,Y}[\varphi \circ \pi ]_{\mathcal I}(v\circ \pi )\bigr ), \]
and the limit measure has finite positive mass.
Now push this convergence forward to $X$, testing against continuous functions on $X$. By the kernel identity of proposition 18.35.3 and the envelope and pushforward identities of proposition 18.35.1, this gives the asserted limit on $X$. Testing the constant function one shows that $n!N_ m/m^ n$ converges to the total mass of the limit, which proves the formula for $M$; the dimensions $N_ m$ involve neither $v$ nor the volumes. Finally, the limit is expressed entirely on $X$, so it does not depend on the modification.
$\square$
Since $M>0$, the sections of $L$ grow at the maximal rate $m^ n$, so $X$ is Moishezon by corollary 19.9.17. A Kähler modification also forces the bounded mass property, by lemma 19.9.12; the Jacobian identities, by contrast, hold for every modification of compact complex manifolds. When the section spaces grow more slowly than $m^ n$, the total mass identity shows that the kernel measures must instead be normalized by the growth degree of $N_ m$, as in section 18.36.
Corollary 18.35.6.9C43863CE5F1 Under the hypotheses of theorem 18.35.5, let $M$ be the limit of the normalized dimensions in that theorem. The determinant functionals $\mathcal F_ m^{(n)}$ of section 18.34 converge pointwise on $C(X)$ to
\[ \mathcal F(v)=\int _0^1\! \left(\int _ X v\, \mathrm{d}\mathrm{MA}_\theta (E_ X(tv))\right)\mathrm{d}t. \]
This functional is concave and vanishes at zero, and for $v,w\in C(X,\mathbb R)$ and $c\in \mathbb R$,
\[ \mathcal F(v+c)=\mathcal F(v)+Mc,\qquad |\mathcal F(v)-\mathcal F(w)|\leq M\| v-w\| _\infty . \]
Proof.
Here $N_ m=\dim H^0(X,L^ m\otimes \mathcal I(m\varphi ))$. Integrate the first-variation formula of proposition 18.34.1 along the segment $t\mapsto tv$, $0\leq t\leq 1$. For each fixed $t$, the integrand converges by the theorem, and its absolute value is bounded by
\[ \| v\| _\infty \frac{n!N_ m}{m^ n}, \]
which is bounded in $m$ because it converges to $M\| v\| _\infty $. Dominated convergence on $[0,1]$ gives the formula; the limit integrand is measurable in $t$ as the pointwise limit of continuous functions. Concavity, the translation identity and the Lipschitz bound pass from $\mathcal F_ m^{(n)}$ to the pointwise limit, since $n!N_ m/m^ n\to M$.
$\square$
This argument uses only weak convergence of measures along a segment of weights. It does not give a lower bound for the partial Fubini–Study potentials complementing theorem 18.33.1; such a bound would require the construction of sections approximating the envelope.
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