Corollary 18.39.5.115614FB1F9D The total variation distance between the kernel measure $\beta _{X,m,v\circ \pi ,\varphi }^{(q)}$ selected by $\varphi $ on $X$ and the lift of the kernel measure selected by $\psi $ on $B$ is $O(m^{-1})$. In particular,
\[ \beta _{X,m,v\circ \pi ,\varphi }^{(q)} \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E(v))\bigr ). \]
The determinant functionals, normalized to vanish at the zero weight, satisfy
\[ \mathcal F_{X,m}^{(q)}(v\circ \pi )\longrightarrow \int _0^1\left(\int _ Bv\, \mathrm{d}\mathrm{MA}_\theta (E(tv))\right)\mathrm{d}t. \]
With the normalization $(q+r)!/m^{q+r}$ given by the dimension of $X$, the kernel measures tend to zero in total variation.
Proof.
As in proposition 18.36.1, fiber integration identifies the norm on $X$ with the norm on $V_ m\subseteq H^0(B,A^ m)$, so the kernel measure on $X$ is the lift of the kernel measure of $V_ m$ on $B$. By lemma 18.37.1, theorem 18.39.3, the latter differs from the kernel measure of $W_ m$ by a positive measure of total mass $q!(\dim V_ m-\dim W_ m)/m^ q=O(m^{-1})$, and so do the lifts. Since $B$ is Kähler and $\psi $ has positive non-pluripolar mass, the equilibrium theorem used in the proof of theorem 18.36.3 applies to $W_ m$ and gives the weak limit. By lemma 18.37.1, the determinant functionals of $V_ m$ and $W_ m$ differ by at most $O(m^{-1})\| v\| _\infty $, and corollary 18.35.6, applied on $B$ with the identity modification, gives the limit of the functionals; its dominated convergence bound is $\| v\| _\infty $ times the bounded sequence of $q$-normalized dimensions. Finally, the $q$-normalized masses are bounded, so multiplying them by $(q+r)!m^{-r}/q!$ proves the last assertion.
$\square$
Pome
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