Theorem 18.40.2.ED8E35B86C82 The envelope satisfies $\mathcal P(\varphi )=u_ c\circ \pi $. With $\ell _ m=\lfloor mc\rfloor $ and $t_ m=\max \{ \ell _ m-r,0\} $,
\[ N_ m=\dim V_ m=\binom {km-t_ m+q}{q} =C_ cm^ q+O(m^{q-1}). \]
Thus the section spaces grow like $m^ q$, and $q<\dim X$. For every $m\geq 1$, in the extended-value sense,
0AED85F8B0A4
18.40.3
\begin{equation} \frac1m\log B_ m^ X\geq \mathcal P(\varphi )-\frac{\log M}{m}. \end{equation}
The functions $m^{-1}\log B_ m^ X$ converge to $\mathcal P(\varphi )$ in $L^1$ for every smooth positive volume on $X$, and locally uniformly on $X\setminus \pi ^{-1}(Z)$. In the normalization of section 18.30, the kernel measures converge in total variation:
2E8D8F74367A
18.40.4
\begin{equation} \begin{split} \frac{q!}{m^ q}B_ m^ B\nu _ B& \longrightarrow \mu _ c:=\mathbf1_{D_ c}\theta ^ q,\\ \frac{q!}{m^ q}B_ m^ X\nu _ X& \longrightarrow \Lambda _{\nu _ X}:=\frac{k^ q\mathbf1_{D_ c}\circ \pi }{w\circ \pi }\nu _ X. \end{split} \end{equation}
Both limits have mass $(k-c)^ q$, and $\pi _*\Lambda _{\nu _ X}=\mu _ c$. Dividing the unnormalized kernel measures by $N_ m$ instead gives the probability limits $\mu _ c/(k-c)^ q$ and $\Lambda _{\nu _ X}/(k-c)^ q$, while normalization by $m^{q+r}$ gives zero in total variation.
Let $W_ m$ be the reference spaces of theorem 18.39.3. For every sequence of intermediate spaces $W_ m\subseteq U_ m\subseteq V_ m$, equipped with the restricted norms, the lower bound and all these convergences remain valid. The dimensions of these spaces are $C_ cm^ q+O(m^{q-1})$, and the dimension normalization then uses $\dim U_ m$.
Proof of theorem 18.40.2.
The identification of the selected spaces is theorem 18.39.3, and the dimension formula follows from the monomial count. The envelope computation and the kernel calculations for the two endpoint spaces $W_ m,V_ m$ prove the assertions on the base. Isometric descent gives $B_ m^ X=B_ m^ B\circ \pi $. For $f_ c=k^ q\mathbf1_{D_ c}/w$, fiber integration gives
\[ \int _ X\left|\frac{q!}{m^ q}B_ m^ X-f_ c\circ \pi \right|\nu _ X =\int _ B\left|\frac{q!}{m^ q}B_ m^ B-f_ c\right|\nu _ B. \]
This proves the total-variation limit on $X$ and the identity $\pi _*\Lambda _{\nu _ X}=\mu _ c$. The same fiber integration transfers the $L^1$ convergence, and every other smooth positive volume on the compact manifold $X$ is comparable to $\nu _ X$. Local uniform convergence pulls back to $X\setminus \pi ^{-1}(Z)$. The dimension asymptotics give the probability limits, and $N_ m/m^{q+r}=O(m^{-r})$ proves the zero limit in the ambient degree.
For $W_ m\subseteq U_ m\subseteq V_ m$, monotonicity of the kernels gives $B_{W,m}\leq B_{U,m}\leq B_{V,m}$. By lemma 18.37.1, the positive difference of the $q$-normalized $U_ m$ and $W_ m$ measures has total mass at most $q!(\dim V_ m-\dim W_ m)/m^ q=O(m^{-1})$. The lower bound and the local uniform convergence for $U_ m$ follow from those for the two endpoint spaces by sandwiching. The $L^1$ errors of the endpoint functions bound the error of the intermediate one, which proves global convergence as well.
$\square$
Pome
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