Example 13.4.11E9D7DF5798B0

Example 13.4.11.E9D7DF5798B0 Let $\omega _{\mathrm{FS}}$ be the Fubini–Study form on $\mathbb P^1$, as in proposition 13.4.10. There is a nonpositive $\omega _{\mathrm{FS}}$-plurisubharmonic function $q$ on $\mathbb P^1$ such that

Let $\mu $ be the atom-free probability measure on the compact polar set $K$ of proposition 4.1.12, and let $q$ be its potential from proposition 13.4.10. Since $\mu $ has no atoms, all Lelong numbers of $q$ vanish, and corollary 6.9.3 gives the assertion on multiplier ideals. On the affine chart, $q$ differs from the function $p(z)=\int _ K\log |z-a|^2\, \mathrm d\mu (a)$ of proposition 4.1.12 by the smooth function $-\log (1+|z|^2)$ and a constant, and $p=-\infty $ on $K$; hence $q=-\infty $ on $K$. Applying proposition 13.4.7 to local potentials of $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q$, we obtain

because $\mu $ is carried by $K$. By contrast, the potential $0$ has the same multiplier ideals and non-pluripolar mass one. Thus exponential integrability at every scale does not determine the non-pluripolar mass.