PomeLemma 11.46.2.3CE4F16CE935 The canonical $u_\alpha $ is $\kappa $-plurisubharmonic and smooth off $o$. Its non-pluripolar measure has a positive density $q_\alpha $ of mass one, in $L^1$ and in no $L^ p$, $p>1$. Distinct $\alpha $ give distinct ordinary types, although all their multiplier ideals are trivial.
11.46.1 The prescribed ordinary type31F950502C14
Let $k\geq 2$, let $\kappa $ be the Fubini–Study form on $\mathbb P^ k$ with $\int \kappa ^ k=1$, and put $o=[1:0:\cdots :0]$. On the chart $Z_0\ne 0$ write $z_ i=Z_ i/Z_0$ and set
\[ r=\sum _{j=1}^ k|z_ j|^2,\qquad P=\frac r{1+r},\qquad T=1-\log P, \qquad u_\alpha =1-T^\alpha \quad (0<\alpha <1). \]
The ratio $P=\sum _{i=1}^ k|Z_ i|^2/\sum _{i=0}^ k|Z_ i|^2$ is global. Give $u_\alpha $ its canonical value $-\infty $ at $o$ and value zero on the hyperplane $Z_0=0$. Primes denote differentiation in $r$, and $b_\alpha =k(1-\alpha )>0$ throughout this section.
Proof. Plurisubharmonicity, smoothness off the pole, unit mass and the $L^ p$ assertions are those of lemma 15.30.1. To fix the profile used in the inverse below, put
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11.46.3
\begin{equation} \Phi _\alpha =\log (1+r)+u_\alpha ,\qquad Q_\alpha =r\Phi _\alpha '=P+\alpha (1-P)T^{\alpha -1}. \end{equation}
The tangential and radial eigenvalues are $Q_\alpha /r$ and $Q_\alpha '$, both positive. Radial determinant integration, with the constant fixed by $\Phi =\log (1+r)$, gives
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11.46.4
\begin{equation} \int _{a<|z|^2<b}(\mathrm{dd}^{\mathrm{c}}\Phi )^ k=Q(b)^ k-Q(a)^ k, \qquad Q=r\Phi '. \end{equation}
Since $Q_\alpha (0)=0$ and $Q_\alpha (\infty )=1$, the density is
Smoothness on the hyperplane and the exclusion of the canonical pole in the product leave no further measure. Near $r=0$, with $L=1-\log r$, direct differentiation gives
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11.46.6
\begin{align} Q_\alpha & \sim \alpha L^{\alpha -1},& Q_\alpha '& \sim \frac{\alpha (1-\alpha )}rL^{\alpha -2},\\ (Q_\alpha ^ k)'& \sim \frac{k\alpha ^ k(1-\alpha )}rL^{-1-k(1-\alpha )}.\nonumber \end{align}
In particular $q_\alpha \sim \alpha ^ k(1-\alpha )r^{-k}L^{-1-b_\alpha }$; the radial volume has order $r^{k-1}\, \mathrm{d}r$, recovering these integrability assertions.
The types for different $\alpha $ are ordinarily distinct, since their difference is unbounded at $o$. Yet every $\mathcal I(tu_\alpha )$ is trivial for finite $t>0$: integrability of the constant function reduces near $o$ to $\int ^\infty \mathrm{e}^{-kL+tL^\alpha }\, \mathrm{d}L<\infty $. Ordinary type information here is strictly finer than this ideal information. $\square$
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