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The growth degree $\lim _ m\log N_ m/\log m$ of the selected spaces does not by itself determine a limit of the kernel measures. At a critical logarithmic coefficient, the rounding in multiplier ideals can produce different leading section counts in even and odd degrees, while the potentials still converge to the ideal envelope. In the following product example the growth degree is positive.
Fix $q\geq 1$ and put $X=\mathbb P^1\times Y$, where $Y=(\mathbb P^1)^ q$. Write $p_0$ for the projection onto the first factor. For $0\leq j\leq q$, let $\sigma _ j$ be the Fubini–Study form on the $j$th factor, normalized to have integral one, where $j=0$ refers to the first factor. Use
\[ L=\mathcal O(1,\ldots ,1),\qquad \theta =c_1(L,h)=\sigma _0+\theta _ Y,\qquad \theta _ Y=\sum _{j=1}^ q\sigma _ j, \]
where $h$ is the product Fubini–Study metric and the forms $\sigma _ j$ are pulled back to $X$. We use the smooth positive probability measure $\nu =\sigma _0\otimes \nu _ Y$, where $\nu _ Y=\sigma _1\wedge \cdots \wedge \sigma _ q=\theta _ Y^ q/q!$, and the continuous weight zero.
On the affine line of the first factor define
\[ \psi _0(z)=\tfrac 12\log |z|^2-\log (1+|z|^2),\qquad \psi =\psi _0\circ p_0. \]
In the coordinate $1/z$ at infinity, $\psi _0$ has the same expression. It equals $-\infty $ at $0$ and $\infty $, and
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18.46.1
\begin{equation} \theta +\mathrm{dd}^{\mathrm{c}}\psi =\tfrac 12[D_0]+\tfrac 12[D_\infty ]+\theta _ Y\geq 0, \qquad D_0=\{ 0\} \times Y,\quad D_\infty =\{ \infty \} \times Y. \end{equation}
We use $\psi $ as selecting potential, so that the selected spaces are $V_ m=H^0(X,L^ m\otimes \mathcal I(m\psi ))$, with the norm $\int _ X|s|_{h^ m}^2\, \mathrm d\nu $. For an orthonormal basis $(s_{m,j})$, write $\mathcal B_ m=\sum _ j|s_{m,j}|_{h^ m}^2$ and $U_ m=m^{-1}\log \mathcal B_ m$. Let $\varsigma $ be the Haar probability measure on the circle $\{ |z|=1\} \subset \mathbb P^1$.
Theorem 18.46.2.318BC0622217 Put $a_ m=1$ for even $m$ and $a_ m=2$ for odd $m$. Then
\[ N_ m=\dim V_ m=a_ m(m+1)^ q,\qquad \lim _{m\to \infty }\frac{\log N_ m}{\log m}=q. \]
The partial envelope (18.32.1) with $K=X$ and obstacle zero is
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18.46.3
\begin{equation} E_{\theta ,X}[\psi ]_{\mathcal I}(0)=E=\psi +\log 2. \end{equation}
The norms satisfy the Bernstein–Markov inequality, and $U_ m\to E$ locally uniformly on $X\setminus (D_0\cup D_\infty )$ and in $L^ p$ for every finite $p$ with respect to smooth volume forms; moreover, the currents $\theta +\mathrm{dd}^{\mathrm{c}}U_ m$ converge weakly to $\theta +\mathrm{dd}^{\mathrm{c}}E$. However,
\begin{align*} \frac{q!}{(2k)^ q}\mathcal B_{2k}\nu & \rightharpoonup q!\, \varsigma \otimes \nu _ Y,\\ \frac{q!}{(2k+1)^ q}\mathcal B_{2k+1}\nu & \rightharpoonup 2q!\, \varsigma \otimes \nu _ Y. \end{align*}
Thus the kernel measures normalized by $q!/m^ q$ do not converge as $m\to \infty $ through all integers. Dividing instead by the dimension gives $N_ m^{-1}\mathcal B_ m\nu \rightharpoonup \varsigma \otimes \nu _ Y$. The envelope has zero non-pluripolar top mass: $\langle (\theta +\mathrm{dd}^{\mathrm{c}}E)^{q+1}\rangle =0$.
Proof.
By lemma 5.37.1,
\[ \mathcal I(m\psi )= \mathcal O_ X(-r_ mD_0-r_ mD_\infty ),\qquad r_ m=\lfloor m/2\rfloor . \]
In the product affine frame, global sections are polynomials of degree at most $m$ in each variable. A global section belongs to $V_ m$ exactly when each of its monomials vanishes to order at least $r_ m$ along $D_0$ and along $D_\infty $, that is, when the exponent $\alpha _0$ of the first variable satisfies $r_ m\leq \alpha _0\leq m-r_ m$; the other exponents range from $0$ to $m$. There is one admissible value of $\alpha _0$ when $m$ is even and two when $m$ is odd. This proves the dimension formula.
To identify the envelope, let $v\in \mathrm{PSH}(X,\theta )$ be a competitor, that is, $v\preceq _{\mathcal I}\psi $ in the multiplier-ideal order of section 5.33 and $v\leq 0$ quasi-everywhere. By theorem 5.37.3, the Lelong numbers of $v$ at the points of $D_0$ and $D_\infty $ are at least $1/2$. Fix $y\in Y$ and restrict $v$ to the line $\mathbb P^1\times \{ y\} $. The restriction is either identically $-\infty $ or $\sigma _0$-plurisubharmonic. In the latter case, its Lelong numbers at the two points over $0$ and $\infty $ are at least those of $v$: the supremum over a ball in the line is at most the supremum over the corresponding ambient ball, and division by $\log r^2<0$ reverses the inequality. By the identification of Lelong numbers with atoms in lemma 5.37.18, the curvature measure of the restriction has atoms of mass at least $1/2$ at these two points. By Stokes' theorem its total mass is one, so these atoms exhaust the measure. The difference between the restriction and $\psi _0$ is therefore harmonic on the compact projective line, hence constant.
It follows that $v(z,y)=\psi _0(z)+c(y)$ off $D_0\cup D_\infty $, where $c(y)=-\infty $ on lines along which $v$ is identically $-\infty $. Restriction to $z=1$ shows that $c(y)=v(1,y)+\log 2$ is a $\theta _ Y$-plurisubharmonic function; it is not identically $-\infty $, since otherwise $v$ would be. Submean values over shrinking balls upgrade the obstacle condition to $v\leq 0$ everywhere. Since $\max \psi _0=-\log 2$, we get $c\leq \log 2$, and hence $v\leq E$. On $D_0\cup D_\infty $ both $v$ and $E$ equal $-\infty $. Conversely, $E\leq 0$ and $E$ differs from $\psi $ by a constant, so $E$ is itself a competitor. This proves (18.46.3).
The norms of the monomials in one factor are beta integrals. With $x=|z|^2/(1+|z|^2)$, they are
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18.46.4
\begin{equation} \int _{\mathbb P^1}|z^\alpha e^ m|_{h^ m}^2\, \sigma _0 =\int _0^1x^\alpha (1-x)^{m-\alpha }\, \mathrm dx =\frac{\alpha !(m-\alpha )!}{(m+1)!}. \end{equation}
Orthogonality under the rotations and integration over the product diagonalize the global norm. Summing over all monomials in one factor gives the constant kernel $m+1$, by the binomial formula. Therefore the full kernel on $Y$ is $(m+1)^ q$. With $C_ k=(2k+1)\binom {2k}{k}$, the selected kernels are
27.125
\begin{align} \mathcal B_{2k}& =(2k+1)^ q C_ k\mathrm e^{2k\psi _0},\notag \\ \mathcal B_{2k+1}& =2(2k+2)^ q C_ k\mathrm e^{2k\psi _0}. \tag{18.46.5} \end{align}
We use the even formula for $k\geq 1$; the odd formula also holds for $k=0$. In the odd case the two monomials in the first factor have equal norms, and the sum of their squared moduli contributes the factor $1+|z|^2$, which gives the second identity. The full kernel on $X$ is $(m+1)^{q+1}$, so evaluation gives $\sup _ X|s|_{h^ m}\leq (m+1)^{(q+1)/2}\| s\| _ m$ for every selected section. This polynomial bound proves the Bernstein–Markov inequality.
The central binomial coefficient is the largest in its row. Hence $4^ k\leq C_ k\leq (2k+1)4^ k$, and $\log C_ k=2k\log 2+O(\log (k+1))$. Taking logarithms in (18.46.5) gives
\[ |U_ m-E|\leq \frac{C_ q}{m} \bigl (1+|\log |z|^2|+\log m\bigr ) \quad \text{off }D_0\cup D_\infty , \]
because $|\psi _0|=\log (2\cosh (\tfrac 12\log |z|^2))$ is bounded by $\log 2+\tfrac 12|\log |z|^2|$. The function $\log |z|^2$ belongs to $L^ p$ for every finite $p$ with respect to smooth volume forms. This proves the convergence of the potentials, and differentiating the $L^1$ limit proves the convergence of the currents. For every $m\geq 2$, both $U_ m$ and $E$ are $-\infty $ on $D_0\cup D_\infty $.
Let $P_ k$ be the probability measure on the first factor with density $C_ k\mathrm e^{2k\psi _0}$ with respect to $\sigma _0$. Under $P_ k$ the angle of $z$ is uniformly distributed, and $x$ has the beta distribution with parameters $(k+1,k+1)$, whose mean is $1/2$ and whose variance is $1/[4(2k+3)]$. Chebyshev's inequality and continuity of the test functions in the angle therefore give $P_ k\rightharpoonup \varsigma $. The measure identities
\[ \mathcal B_{2k}\nu =(2k+1)^ q P_ k\otimes \nu _ Y, \qquad \mathcal B_{2k+1}\nu =2(2k+2)^ q P_ k\otimes \nu _ Y \]
prove both limits along even and odd degrees and the limit after division by $N_ m$. Since the two limits have different total masses, the $q$-normalized family does not converge. Normalization by the ambient degree, $(q+1)!/m^{q+1}$, instead gives zero in total variation.
Finally, off $D_0\cup D_\infty $ the current in (18.46.1) is $\theta _ Y$, whose rank is $q$. The non-pluripolar product charges neither divisor, so the top product is zero.
$\square$
Unlike the Newton family of section 18.45, this family of multiplier-ideal spaces is not multiplicative. For two odd degrees, the monomials of minimal order along $D_0$ are selected, but their product misses the condition in the even sum degree by one order. The Newton family has a leading term given by lattice counting in every sufficiently large degree; here the ideal envelope is identified, but the leading term oscillates. Testing against the constant function one shows that convergence of $q!N_ m/m^ q$ is necessary for the $q$-normalized kernel measures to converge. Since the envelope has zero non-pluripolar top mass, theorem 18.35.5, whose hypotheses require positive mass, does not apply to this selecting potential.
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