PomeTheorem 18.36.3.87C196C13DDE Under these hypotheses,
\[ \frac{q!}{m^ q}B_{X,m}\, \mu _ X \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E)\bigr ). \]
If $n>q$, then $(n!/m^ n)B_{X,m}\, \mu _ X\to 0$ in total variation.
18.36.2 The quotient equilibrium measure29138044FA48
On a Kähler base, the partial kernel measures converge to the Monge–Ampère measure of the partial envelope. Since lifting is continuous, the kernel measures on the total space converge to the lift of this measure, with the normalization given by the dimension of the base.
Keep the submersion and the weighted compact set $(K,w,\mu )$ above. Assume that $B$ is Kähler, put $\theta =c_1(A,h_ A)$, and take $\psi \in \mathrm{PSH}(B,\theta )$ with positive non-pluripolar Monge–Ampère mass. Write $B_{B,m}$ and $B_{X,m}$ for the partial Bergman kernels on $K$ and $K_ X$, put $\mu _ X=\mathcal L\mu $, and let
\[ E=E_{\theta ,K}[\psi ]_{\mathcal I}(w) \]
be the partial envelope of section 18.32. We call $\mathcal L(\mathrm{MA}_\theta (E))$ the quotient equilibrium measure of these data on $X$.
Proof. Apply [Theorem 14.3.1, SGPT] on the compact Kähler manifold $B$, with trivial auxiliary twisting line bundle. Its hypotheses are exactly our assumptions: $\psi $ has positive mass, $K$ is closed and non-pluripolar, $w$ is continuous, and $\mu $ is a Bernstein–Markov probability measure. It gives
\[ \frac{q!}{m^ q}B_{B,m}\, \mu \rightharpoonup \mathrm{MA}_\theta (E). \]
By proposition 18.36.1, the corresponding measures on $X$ are the lifts of these measures, and lifting preserves weak convergence; this proves the first assertion. In particular the $q$-normalized total masses are bounded, and multiplying them by $(n!/q!)m^{q-n}\to 0$ proves the last assertion when $n>q$. $\square$
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