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 $n!N_ m/m^ n\to M$.
$\square$
Pome
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