Theorem 18.45.7.2EC775894117 The norms are nondegenerate and satisfy the Bernstein–Markov inequality of section 18.31 for the weight $w$. Writing $N_ m=\dim V_ m$, we have
\[ N_ m=m^ d\operatorname {vol}_\Lambda (Q)+O(m^{d-1})\quad (d\geq 1), \qquad N_ m=1\quad (d=0). \]
For all sufficiently large $m$,
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18.45.8
\begin{equation} |U_ m-E_ Q|\leq \frac{C}{m} \left(1+\sum _ j|r_ j|+\log m\right)\quad \text{on }T. \end{equation}
Consequently $U_ m\to E_ Q$ locally uniformly on $T$ and in $L^ p$ for every finite $p$ with respect to smooth volume forms on $X$, and the closed positive currents $\theta +\mathrm{dd}^{\mathrm{c}}U_ m$ converge weakly to $\theta +\mathrm{dd}^{\mathrm{c}}E_ Q$. Both $U_ m$ and $E_ Q$ equal $-\infty $ on $X\setminus T$.
Proof.
Since the volume form is smooth and positive, the norm is positive on every nonzero section. Orthogonality of the angular Fourier modes diagonalizes it:
\begin{align*} A_ m(\alpha )& =\| z^\alpha e^ m\| _ m^2 =\int _{\mathbb R^ n}\mathrm e^{\alpha \cdot r-mF(r)}a(r)\, \mathrm dr,\\ \mathcal B_ m(r)& =\sum _{\alpha \in mQ\cap \mathbb Z^ n} \frac{\mathrm e^{\alpha \cdot r-m\varphi _0(r)}}{A_ m(\alpha )}. \end{align*}
For $q=\alpha /m$, the function $F(r)-q\cdot r$ has the unique minimum value $-F^*(q)$, attained at $r(q)$. Taylor's formula on a ball of radius $m^{-1/2}$ about $r(q)$, with uniform bounds on a compact neighborhood of $r(Q)$, yields
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18.45.9
\begin{equation} c m^{-n/2}\mathrm e^{mF^*(q)}\leq A_ m(\alpha ) \leq \mathrm e^{mF^*(q)}. \end{equation}
The upper bound uses only the minimum and $\int a\, \mathrm dr=1$.
Since $b\in \mathbb Z^ n$, we have $(m\operatorname {aff}Q)\cap \mathbb Z^ n=mb+\Lambda $. Counting fundamental cells of $\Lambda $ in $m(Q-b)$ gives the dimension estimate: the cells meeting the relative boundary of $m(Q-b)$ lie in a boundary layer of fixed width, and there are $O(m^{d-1})$ of them. The same cells give convergence of the lattice Riemann sums on $Q$. Every $q\in Q$ lies within $O(m^{-1})$ of $Q_ m=Q\cap (b+m^{-1}\Lambda )$: move $q$ by $O(m^{-1})$ towards a fixed relative interior point so that a ball larger than the covering radius of the lattice fits inside $Q$, and then round to a lattice point inside that ball. This works for all sufficiently large $m$, and for a singleton it is exact.
Put $G_{Q,m}=\max _{q\in Q_ m}(q\cdot r-F^*(q))$. The approximation by the grid gives $0\leq G_ Q-G_{Q,m}\leq C(1+\sum _ j|r_ j|)/m$. The formula for $\mathcal B_ m$ and (18.45.9) give
\[ G_{Q,m}-\varphi _0\leq U_ m\leq G_{Q,m}-\varphi _0+ \frac{(n/2)\log m+\log N_ m-\log c}{m}. \]
This proves (18.45.8). The selected monomials vanish on every component of $X\setminus T$. The functions $\log |z_ j|^2$ have finite $L^ p$ norms for each finite $p$ with respect to smooth volume forms, so (18.45.8) gives the global convergence, and differentiation in the sense of distributions gives the convergence of the currents.
Fenchel's inequality $\alpha \cdot r-mF(r)\leq mF^*(\alpha /m)$ and (18.45.9) also give $\sup _ X B_ m\leq C N_ m m^{n/2}$. By the Cauchy–Schwarz inequality in the orthonormal basis, $\sup _ X|s|_{h^ m}\mathrm e^{-mw/2} \leq C m^{d/2+n/4}\| s\| _ m$ for $s\in V_ m$. The finitely many initial degrees can be absorbed in $C$, and a polynomial bound implies the Bernstein–Markov inequality with any positive exponential loss.
$\square$
Pome
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