11.43.9 A family with nonanalytic isolated poles592E13B7D2ED

Proof. The function $p\mapsto \sum _ i\sqrt{p_ i}$ is concave, so $P_ a$ is convex. Set

Then $b+\delta <1$ and $\sum _ i\sqrt{p_{*,i}}=\sqrt{(n+1)/2}>1$. Adding a point of $\delta \Sigma $ preserves the last inequality and keeps the sum of the coordinates below one, so $p_*+\delta \Sigma $ lies in every $P_ a$.

All the standard basis vectors $e_ i$ belong to $P_ a$, and therefore

Hence the reference is locally bounded away from $o$. The Cauchy–Schwarz inequality gives $\sum _ ip_ i\geq a/n$ on $P_ a$. When all $x_ i<0$, this yields $h_{P_ a}(x)\leq (a/n)\max _ i x_ i$, which tends to $-\infty $ as $z\to o$, and lemma 11.43.2 gives the pole at $o$. Local boundedness outside one point implies that all generic Lelong numbers along prime divisors vanish, and pulling back to the blowup at a different point introduces no pole.

To test analyticity, fix positive $\lambda _1,\ldots ,\lambda _ n$ and approach $o$ along $|z_ i|^2=\mathrm{e}^{-R\lambda _ i}$, $R\to \infty $. Writing $y_ i=\sqrt{p_ i}$, the Cauchy–Schwarz inequality gives

where the minimizer satisfies $\sum _ ip_ i\leq a\leq 1$, so that it lies in $P_ a$. Since $L(-R\lambda )\to 0$ and $\Phi _{P_ a}-\psi _{P_ a}$ is bounded, $-\Phi _{P_ a}/R\to d_ a(\lambda )$ along these curves.

Suppose that the type were that of $c\log \sum _ k|f_ k|^2$ near $o$, with $c>0$ and holomorphic germs $f_ k=\sum _\alpha c_{k,\alpha }z^\alpha $. Exponentiating the two inequalities expressing this equivalence and averaging over the arguments of the $z_ i$ gives the positive series

whose exponential decay rate in $R$ must be $d_ a(\lambda )/c$. That rate is the minimum of $\alpha \cdot \lambda $ over the monomials with nonzero coefficients: a minimizing term gives one bound, and convergence at a fixed $R_0$ bounds the series above by a constant times $\mathrm{e}^{-(R-R_0)\min \alpha \cdot \lambda }$. Near any $\lambda $ with positive coordinates, only finitely many monomials can attain the minimum, because these coordinates have a common positive lower bound and any one nonzero coefficient bounds the degree of the minimizing monomials. The rate is therefore locally piecewise linear in $\lambda $. But

This contradiction proves nonanalyticity. The assertion persists on a blowup at a point different from $o$, which does not change a neighborhood of the pole.

The removed corner $C_ a=\{ p\in \mathbb R^ n:p_ i\geq 0,\ \sum _ i\sqrt{p_ i}<\sqrt a\} =aC_1$ lies in $\Sigma $. The substitution $p_ i=y_ i^2$ gives

The last equality follows by iterated integration of monomials over a simplex. Removing the corner from $\Sigma $, whose volume is $1/n!$, and applying (11.43.4) gives (11.43.11). Since $n\geq 2$, one has $\kappa _ n<1$. $\square$

Proof. If $p\in P_{a_ j}\setminus P_1$, then $u=(\sum _ i\sqrt{p_ i})^2\in [a_ j,1)$ and $p/u\in P_1$. Since $|p|\leq \sum _ ip_ i\leq u$, one has $|p/u-p|\leq 1-u\leq 1-a_ j$. Thus $d_ H(P_{a_ j},P_1)\leq 1-a_ j$, and the hypotheses of the theorem hold with the simplex found in the proof of proposition 11.43.10. The mass formula (11.43.11) and the derivative bound $|\mathrm{d}\log (1-\kappa _ n a^ n)/\mathrm{d}a|\leq n\kappa _ n/(1-\kappa _ n)$ give the estimate for the functionals. By (11.43.12), the directional coefficients of the $j$th reference are multiplied by $r_ ja_ j=1-(j+2)^{-2}$, which strictly increases with $j$, so the ordinary types are distinct, while the masses $r_ j^ n(1-\kappa _ n a_ j^ n)$ strictly decrease to $1-\kappa _ n>0$. $\square$