PomeD9BE589BBC6ECorollary 18.26.5.D9BE589BBC6E For every countable family of modifications $\pi _ j:Y_ j\to X$ between connected compact complex manifolds, there are pairwise ordinarily inequivalent ideal $\omega $-models $p_ a$ with analytic singularities and with pairwise distinct families of multiplier ideals, such that
Proof. For each $j$, choose a proper closed analytic subset $S_ j\subset X$ over whose complement $\pi _ j$ is an isomorphism. Each $S_ j$ is nowhere dense, so the Baire category theorem gives a point $x\notin \bigcup _ jS_ j$. Apply proposition 18.26.2 at this point.
Each $\pi _ j$ is biholomorphic near the unique pole $y_ j$ of $p_ a\circ \pi _ j$, and the pulled-back potential has the same logarithmic coefficient there. Its smooth background and positive curvature current satisfy the trace estimate in the proof of proposition 18.26.2, which does not use strict positivity of the background. Thus the coefficient measures have no mass at $y_ j$, and their restriction to the pole set is zero. The same proposition gives the ordinary inequivalence and the distinct multiplier ideals. $\square$
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