Example 11.42.3C6588E30CFF6

Example 11.42.3.C6588E30CFF6 Let $X=\mathbb P^ n$ with $n\geq 1$, let $\theta $ be the Fubini–Study form, normalized by $\int _ X\theta ^ n=1$, and let $D$ be a hyperplane. Let $s_ D$ be a section of $\mathcal O(1)$ defining $D$, measured with the Fubini–Study metric and rescaled so that $a_ D=\log |s_ D|^2\leq 0$, and choose $t\in (0,1)$. Then

Put $\varphi _{t,D}=P_\theta [ta_ D]$. By proposition 18.4.8, this envelope fixed point has the ordinary type of $ta_ D$. Moreover $ta_ D\in \mathrm{PSH}(X,t\theta )$ and $ta_ D\leq \varphi _{t,D}$, so $\varphi _{t,D}$ satisfies (16.28.67) with $\varepsilon =t$. Off $D$, the potential $ta_ D$ has Monge–Ampère measure $(1-t)^ n\theta ^ n$, and its non-pluripolar measure puts no mass on $D$.

Since mass is invariant under bounded changes of the potential in a closed background,

Thus the constant in proposition 11.39.1 is sharp.

Now let $n=1$, and take for $D$ two distinct points $p,q\in \mathbb P^1$. The corresponding references have distance zero, although $\varphi _{t,p}$ is unbounded near $p$ and bounded near $q$, while $\varphi _{t,q}$ behaves in the opposite way; in particular they are not ordinarily equivalent. For every positive integer $m$,

by lemma 5.37.1, and for $mt\geq 1$ these ideals differ. Thus the distance $\Delta $ of (11.39.3) does not even separate multiplier-ideal types, and it determines neither the ordinary nor the ideal singularity type.

Fix a smooth real function $F$ on $\mathbb P^1$, and let $w_ F$ be the solution of $\theta +\mathrm{dd}^{\mathrm{c}}w_ F=\mathrm{e}^ F\mathrm{d}V/Z_ F$ with $\int _ Xw_ F\, \mathrm{d}V=0$. The normalized solutions for $D=p$ and $D=q$ are

Their difference satisfies $\mathrm{dd}^{\mathrm{c}}(u_{t,p}-u_{t,q})=t([p]-[q])\ne 0$. Hence the sequence of solutions obtained by alternating between $D=p$ and $D=q$ has two different $L^1$ cluster points, although the scalar functional and all its supporting sets are the same at every step. Within this one-parameter family, the functional does recover $t$, since $t\mapsto (1-t)^ n$ is injective.