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9.11.8 Ideal envelopes defined on resolutions491CF8180F3F

Multiplier ideals on smooth models give a singularity relation on $X$. Call two potentials $u,v\in \mathrm{PSH}(X,\theta )$ resolution-ideal equivalent if $u\circ \pi \sim _{\mathcal I}v\circ \pi $ on one resolution $\pi :Y\to X$ (the ideal type of section 5.33), that is, if $\mathcal I(k\, u\circ \pi )=\mathcal I(k\, v\circ \pi )$ for every real $k>0$. By lemma 5.43.24, applied on a common refinement, this does not depend on the resolution. The ideal projection of theorem 18.4.2 then defines an envelope in this class.

Proposition 9.11.9.C43A6C429C4D For $u\in \mathrm{PSH}(X,\theta )$, there is a unique potential $P_\theta [u]_{\mathcal I}^{\mathrm{res}}$ on $X$ satisfying

\[ P_\theta [u]_{\mathcal I}^{\mathrm{res}}\circ \pi =P_{\pi ^*\theta }[u\circ \pi ]_{\mathcal I} \]

on every resolution $\pi :Y\to X$, where the right-hand side is the multiplier-ideal envelope of definition 18.4.1 on $Y$. It preserves the resolution-ideal class, is idempotent, has supremum zero, and is independent of the resolution.

Proof. For a fixed resolution, the right-hand side exists by theorem 18.4.2 and descends by theorem 9.11.5. Under a further smooth modification, the competitor families of the ideal envelopes correspond by the descent theorem and lemma 5.43.24, and the regularized suprema commute by the argument of proposition 9.11.7. Thus two resolutions give the same pullback on a common refinement, and hence the same function on $X$. Idempotence, preservation of the class and the normalization follow from the corresponding assertions on $Y$ and surjectivity of $\pi $. $\square$

An intrinsic restricted ideal trace on a subspace contained in the singular locus would require a regularization compatible with restriction to that subspace; the resolution-ideal envelope provides neither the regularization nor the trace.

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