18.45 Newton selection and moment-shell limits5CEF27CB9AAA

Proof. In the charts at infinity one sees that every global section of $L^ m$ is a polynomial of degree at most $mk_ j$ in each variable $z_ j$. If $s\in V_ m$, the growth bound follows from the triangle inequality, since $|z^\alpha |^2=\mathrm e^{\alpha \cdot r}\leq \mathrm e^{mH_ Q(r)}$ for $\alpha \in mQ$. Conversely, averaging the growth bound over the angles gives $\sum _\alpha |c_\alpha |^2\mathrm e^{\alpha \cdot r} \leq C_ s\mathrm e^{mH_ Q(r)}$ for the coefficients $c_\alpha $ of $s$. If some $c_\alpha \neq 0$ had $\alpha \notin mQ$, a vector $u$ with $\alpha \cdot u>mH_ Q(u)$, which exists by separation, would contradict this bound along the ray $r=tu$, $t\to \infty $. Thus the growth condition controls each monomial separately, even though the monomials of $s$ may cancel pointwise.

For $p\in \operatorname {int}P$, put $\delta _ p=\min _ j(p_ j,k_ j-p_ j)>0$. Since $w$ is bounded,

By strict convexity, $F-p\cdot r$ therefore has a unique minimizer, at which $\nabla F=p$. Conversely, every gradient of $F$ lies in $P$: compare the supporting hyperplanes of the convex function $F$ with the asymptotic slopes of $\varphi _0$. By the inverse function theorem the image of $\nabla F$ is open, so it is contained in the interior of $P$. This proves that $\nabla F$ is a diffeomorphism onto $\operatorname {int}P$. On a compact neighborhood of $Q$ in $\operatorname {int}P$, the inverse gradient and the first and second derivatives of $F^*$ are bounded, and the Hessian of $F^*$ is positive definite.

We next show that $G_ Q-\varphi _0$ is a competitor. In a product chart in which some of the coordinates are inverted, the local potential of $G_ Q-\varphi _0$ is

All $c_ j$ are bounded below by a fixed positive number. The expression is locally bounded above, tends uniformly to $-\infty $ at the coordinate hyperplanes when the other coordinates stay bounded, and is locally integrable by comparison with one fixed summand. At points where it is finite, a maximizing summand gives the submean inequality for plurisubharmonic functions; at points where it equals $-\infty $, this inequality is automatic. These local expressions glue. Boundedness of $F^*$ on $Q$ gives $G_ Q-H_ Q=O(1)$, and Fenchel's inequality gives $G_ Q\leq F$, that is, $G_ Q-\varphi _0\leq w$. Thus $G_ Q-\varphi _0$, extended by $-\infty $ on $X\setminus T$, belongs to $\mathcal C_ Q$ and satisfies the obstacle condition.

Now let $v$ be any competitor. The obstacle condition holds everywhere: the exceptional pluripolar set has Lebesgue measure zero, and submean values over shrinking balls carry the inequality with the continuous obstacle to every point. Consider the maximum over the angles, $M(r)=\max _\tau \{ v(\mathrm e^{r/2}\tau )+\varphi _0(r)\} $, where $\mathrm e^{r/2}\tau $ denotes the point with coordinates $\mathrm e^{r_ j/2}\tau _ j$. By compactness, $M$ is upper semicontinuous. Rotating a point at which the maximum is attained gives the submean inequality for the invariant function $z\mapsto M(\log |z_1|^2,\ldots ,\log |z_ n|^2)$, which is therefore plurisubharmonic on $T$. It extends across the affine coordinate hyperplanes by the local upper bound. If $M(r)=-\infty $ for some $r$, successive one-variable maximum principles would force this function to be $-\infty $ on the corresponding polydisk, contradicting $v\not\equiv -\infty $. Thus $M$ is finite. In logarithmic coordinates this invariant function depends only on the real parts, so its positive complex Hessian shows that $M$ is convex.

We have $M\leq F$ and $M\leq H_ Q+C_ v$. Comparing along rays shows that every supporting slope of $M$ lies in $Q$. At any $r$, choose such a slope $p$. Since $M\leq F$ gives $M^*\geq F^*$, convex duality yields $M(r)=p\cdot r-M^*(p)\leq p\cdot r-F^*(p)\leq G_ Q(r)$. Hence $v\leq G_ Q-\varphi _0$ on $T$. Submean values over shrinking balls and the uniform decay of $G_ Q-\varphi _0$ near $X\setminus T$ show that $v=-\infty $ on $X\setminus T$. Since $G_ Q-\varphi _0$, extended by $-\infty $, is already upper semicontinuous, this proves the identity for the envelope after upper regularization. $\square$

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:

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

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

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. $\square$

Proof. For $\alpha \in mQ\cap \mathbb Z^ n$ and $q=\alpha /m$, the normalized monomial $z^\alpha e^ m$ gives the probability measure $\mathrm dP_{m,q}=A_ m(\alpha )^{-1} \mathrm e^{m(q\cdot r-F(r))}a(r)\, \mathrm dr\, \mathrm d\tau $ on $T$. For every $\eta >0$ there is $c_\eta >0$ such that $F(r)-q\cdot r+F^*(q)\geq c_\eta $ whenever $|r-r(q)|\geq \eta $, uniformly for $q\in Q$: on a large compact set this follows from the uniqueness of the minima, and outside it from (18.45.5), which is uniform on $Q$. Together with (18.45.9), this gives $P_{m,q}\{ |r-r(q)|\geq \eta \} \leq C m^{n/2}\mathrm e^{-mc_\eta }$. Hence, for every continuous function on $X$, its integral against $P_{m,q}$ differs from its average over the torus $\{ r=r(q)\} $ by $o(1)$, uniformly in $q\in Q$. The identity $d!m^{-d}B_ m\nu =d!m^{-d}\sum _\alpha P_{m,\alpha /m}$ and the lattice Riemann sums prove (18.45.12). When $d<n$, the measure $\mu _ Q$ is carried by the moment shell, which has real dimension $n+d<2n$, so it is singular with respect to smooth volume forms.

For the curvature, note that $\theta +\mathrm{dd}^{\mathrm{c}}E_ Q=\mathrm{dd}^{\mathrm{c}}G_ Q$ on $T$. By uniform strong convexity of $F^*$ on $Q$, the slope $q(r)$ maximizing $q\cdot r-F^*(q)$ over $Q$ is unique. Adding the variational inequalities at $r$ and $s$ gives $(r-s)\cdot (q(r)-q(s))\geq c|q(r)-q(s)|^2$. Thus $q$ is Lipschitz, and comparison of the maximal values gives $\nabla G_ Q=q$. Hence $G_ Q$ is $C^{1,1}$, and its Hessian has rank at most $d$ almost everywhere, since $q(r)\in b+W$. In logarithmic coordinates, convex mollifications of $G_ Q$ converge in $C^1$, their Hessians stay bounded and converge in $L^ p_{\mathrm{loc}}$ for every finite $p$, and their minors converge in $L^1_{\mathrm{loc}}$. Induction on the degree in the Bedford–Taylor definition of products of bounded plurisubharmonic functions, together with integration by parts, identifies the powers of $\mathrm{dd}^{\mathrm{c}}G_ Q$ on $T$ with the measures given by these minors. The non-pluripolar product agrees with them on $T$ and puts no mass on the pluripolar set $X\setminus T$. The bound on the rank therefore proves the vanishing assertion.

When $d=n$, a maximizer in the interior of $Q$ satisfies $r=\nabla F^*(q)$, so $G_ Q=F$ on the set where $\nabla F(r)\in \operatorname {int}Q$. Elsewhere the maximizing slope lies in one of the finitely many facets of $Q$. On the set where it lies in a given facet, its component normal to that facet is constant, so the derivative of this component vanishes at almost every point of the set: at a point of differentiability and of density, a nonzero derivative would force a cone of positive density outside the set. Consequently $\det D^2G_ Q=\boldsymbol 1_{\{ \nabla F(r)\in Q\} }\det D^2F$ almost everywhere. In our normalization, $(\mathrm{dd}^{\mathrm{c}}F)^ n=n!\det D^2F(r)\, \mathrm dr\, \mathrm d\tau $. The change of variables $q=\nabla F(r)$ gives the asserted identity for the top product. $\square$