Example 5.37.8`E39EAA145A67`

Example 5.37.8.E39EAA145A67 Exact multiplier-ideal pullback can fail at a critical fiber. On a polydisk in $\mathbb C^ n$, $n\geq 1$, let $f(z_0,\ldots ,z_{n-1})=z_0^2$ and put $v(t)=\tfrac 12\log |t|^2$. At the origin,

\[ \mathcal I(v)=\mathcal O,\qquad \mathcal I(v\circ f)=(z_0) \ne \mathcal I(v)\mathcal O. \]

Proof. On the curve the coefficient is $1/2$, so the multiplier ideal is the zeroth power of the maximal ideal. Upstairs $v\circ f=\log |z_0|^2$ has coefficient one along the smooth divisor $\{ z_0=0\} $. Apply lemma 5.37.1 in codimension one to get its first ideal power. $\square$

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@misc{pome-tag-E39EAA145A67,
  author = {{Pome contributors}},
  title = {Example 5.37.8},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/E39EAA145A67/}
}

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Example 5.37.8E39EAA145A67

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Example 5.37.8`E39EAA145A67`