Theorem 18.37.4.D9B0D46D42B2 Assume that $B$ is Kähler and that there is $\psi \in \mathrm{PSH}(B,\theta )$ with positive non-pluripolar mass, locally bounded off $Z$, such that
\[ \psi (z)=a_ b\log |z-b|^2+O(1)\quad \hbox{near each }b\in Z \]
in local coordinates $z$ on $B$. Let $E(v)=E_{\theta ,B}[\psi ]_{\mathcal I}(v)$. Then, for the selecting potential $\varphi $ of (18.37.2),
\[ \beta _{X,m,v\circ \pi ,\varphi }^{(q)} \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E(v))\bigr ). \]
More precisely, the total variation distance between $\beta _{X,m,v\circ \pi ,\varphi }^{(q)}$ and the lift of the kernel measure on $B$ selected by $\psi $ is $O(m^{-1})$. The determinant functionals satisfy
\[ \mathcal F_{X,m}^{(q)}(v\circ \pi )\longrightarrow \int _0^1\! \left(\int _ B v\, \mathrm{d}\mathrm{MA}_\theta (E(tv))\right)\mathrm{d}t. \]
The normalized dimensions $q!N_ m/m^ q$ converge to the total mass of the limit measure, which is finite and positive. If $n>q$, the kernel measures with the normalization $n!/m^ n$ tend to zero in total variation.
Proof.
By lemma 5.37.1 in dimension $q$, the space of sections of $A^ m$ selected by $\psi $ on $B$ is the space $W_ m$ of proposition 18.37.3. Corresponding sections in $V_ m$ and on $X$ have equal norms, since their pointwise norms and the continuous weight are pulled back from $B$ and every fiber probability has mass one. As in proposition 18.36.1, pulling back an orthonormal basis shows that the kernel measure on $X$ is the lift of the kernel measure of $V_ m$ on $B$.
By lemma 18.37.1, proposition 18.37.3, the $q$-normalized kernel measures of $V_ m$ and $W_ m$ on $B$ differ by a positive measure of total mass $O(m^{-1})$, and so do their lifts. Apply theorem 18.35.5 on $B$, with the identity as Kähler modification; its hypotheses are among ours. It gives the convergence of the kernel measures of $W_ m$, and since the difference tends to zero, the asserted limit on $X$ follows.
By the same lemma, the determinant functionals of $V_ m$ and $W_ m$ differ by at most $O(m^{-1})\| v\| _\infty $. The isometric identification of the norms and corollary 18.35.6 on $B$ give the limit of the functionals. Testing the constant function one proves the assertion about the dimensions. Since the $q$-normalized masses are bounded, multiplying them by $(n!/q!)m^{q-n}$ proves the last assertion.
$\square$
Pome
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