Example 6.18.3E35B96BEFDE2

Example 6.18.3.E35B96BEFDE2 Let $u=c\log |z|^2$ on a disc about $0\in \mathbb C$, with $c>0$. A germ $z^ kh$ with $h(0)\neq 0$ lies in $\mathcal I(mu)_0$ if and only if $|z|^{2k-2mc}$ is integrable near $0$, that is, $k>mc-1$; hence $\mathcal I(mu)$ is generated by $z^{\lfloor mc\rfloor }$ (example 6.11.8). A potential of type $(m^{-1},\mathcal I(mu))$, such as the Bergman potential $u_ m$ of theorem 6.15.4, is therefore $(\lfloor mc\rfloor /m)\log |z|^2+O(1)$ near $0$. Its coefficient $\lfloor mc\rfloor /m$ need not equal $c$ for any $m$, and then such an approximation does not reproduce the multiplier ideals of $u$ at a fixed scale. For instance, let $c$ be irrational, so that $\lfloor mc\rfloor /m<c$ for every $m$, and take the scale $1/c$. Since $u_ m/c$ differs by a bounded function from $c_ m\log |z|^2$ with $c_ m=\lfloor mc\rfloor /(mc)<1$, the same computation gives $\mathcal I(u_ m/c)_0=\mathcal O_0$ for every $m$, whereas $\mathcal I(u/c)_0=\mathcal I(\log |z|^2)_0$ is generated by $z$. This is why definition 6.17.5 compares two different scales $\lambda '>\lambda $ instead of requiring equality of multiplier ideals.