On a curve, the polar curvature records every logarithmic pole. In higher dimension, the restriction of the curvature current to the pole set no longer does: it vanishes for an isolated logarithmic pole, although such a pole changes the multiplier ideals. The curve metric therefore cannot be extended simply by replacing the polar measure with this restricted current.
Here $X$ has dimension $n\geq 2$, and $\omega $ is a Hermitian form on $X$. For $p\in \mathrm{PSH}(X,\omega )$, the first polar current of $p$ is
\[ \operatorname {Pol}_1(p) =\mathbf1_{\{ p=-\infty \} }(\omega +\mathrm{dd}^{\mathrm{c}}p), \]
where the restriction to the pole set is applied to the coefficient measures of the positive current. In the notation of proposition 5.46.7, $\operatorname {Pol}_1(p)=\operatorname {Pol}_\omega (p)$; it is a closed positive current.
Proposition 18.26.2.088A7D40B1D1 For every $x\in X$, there is $a_0>0$ such that, for each rational $a\in (0,a_0)$, there is an ideal model $p_ a$ with analytic singularities and pole set $\{ x\} $ satisfying
\[ p_ a=a\log |z|^2+O(1)\quad \text{near }x,\qquad \operatorname {Pol}_1(p_ a)=0. \]
The models $p_ a$ are pairwise ordinarily inequivalent and have pairwise distinct families of multiplier ideals $\mathcal I(tp_ a)$, $t>0$, whereas also $\operatorname {Pol}_1(0)=0$. More precisely, with $\mathfrak m_ x$ the maximal ideal at $x$,
55D42653A018
18.26.3
\begin{equation} \mathcal I(tp_ a)_ x =\mathfrak m_ x^{\max \{ \lfloor ta\rfloor -n+1,0\} } \qquad (t>0). \end{equation}
Proof.
Choose the cutoff potentials $u_ a$ of lemma 5.37.2 for the single point $x$, with $a_0$ smaller than the reciprocal of the constant in that lemma, and put $p_ a=P_\omega [u_ a]$. By proposition 18.4.8, $p_ a$ is an ideal model with analytic singularities, ordinarily equivalent to $u_ a$. It is locally bounded away from $x$, and its only pole has the stated logarithmic coefficient. Distinct coefficients give unbounded differences near $x$. Formula (18.26.3) follows from lemma 5.37.1 with codimension $n$, and its exponents distinguish any two coefficients at sufficiently large weights.
It remains to show that the current has no coefficient mass at $x$. In coordinates centered at $x$, let $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$ and let
\[ \nu =(\omega +\mathrm{dd}^{\mathrm{c}}p_ a)\wedge \beta ^{n-1}. \]
This is a positive measure. Choose smooth cutoff functions $0\leq \chi _\varepsilon \leq 1$, equal to one on the ball of radius $\varepsilon $ and supported in the ball of radius $2\varepsilon $, with $-C\varepsilon ^{-2}\beta \leq \mathrm{dd}^{\mathrm{c}}\chi _\varepsilon \leq C\varepsilon ^{-2}\beta $; their complex Hessians are supported in the annulus between the two balls. Since $\beta $ is closed, distributional integration by parts gives
\[ \int \chi _\varepsilon \, \mathrm d\nu =\int \chi _\varepsilon \omega \wedge \beta ^{n-1} +\int p_ a\, \mathrm{dd}^{\mathrm{c}}\chi _\varepsilon \wedge \beta ^{n-1}. \]
The logarithmic bound for $p_ a$ on the annulus and the Euclidean volume of the annulus therefore give
\[ 0\leq \nu (\{ x\} )\leq \int \chi _\varepsilon \, \mathrm d\nu \leq C_ a\bigl (\varepsilon ^{2n} +\varepsilon ^{2n-2}(1+|\log \varepsilon |)\bigr ) \longrightarrow 0. \]
This is where $n\geq 2$ is used. By positivity, the coefficient measures of $\omega +\mathrm{dd}^{\mathrm{c}}p_ a$ are bounded by its trace measure $\nu $, so they have no mass at $x$. Hence the restriction to the pole set is zero, as asserted. The zero potential is an ordinary and an ideal model because $\omega >0$, and its pole set is empty.
$\square$
Consequently, no comparison that depends only on the fixed background and on $\operatorname {Pol}_1$ separates these analytic ideal models, and a complete topology in higher dimension must involve singularity data of higher codimension.
A modification can turn the isolated pole into a divisor. For a modification $\pi :Y\to X$, pullback maps $\mathrm{PSH}(X,\omega )$ to $\mathrm{PSH}(Y,\pi ^*\omega )$ by proposition 5.43.18, and we define
\[ \operatorname {Pol}_\pi (p) =\mathbf1_{\{ p\circ \pi =-\infty \} } (\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(p\circ \pi )). \]
Corollary 18.26.4.91E1AFBF43BE Fix $x$ and the models $p_ a$ of proposition 18.26.2. Let $\pi :Y\to X$ be the blowup at $x$, with exceptional divisor $E$, and let $\eta $ be a Hermitian form on $Y$. Then
\[ \operatorname {Pol}_\pi (p_ a)=a[E]. \]
For $\mu _ a=\operatorname {Pol}_\pi (p_ a)\wedge \eta ^{n-1}$ and any two admissible coefficients $a,b$,
\[ \| \mu _ a-\mu _ b\| _{\mathrm{TV}} =|a-b|\int _ E\eta ^{n-1}. \]
Proof.
On relatively compact blowup charts, with $z=(s,sw)$, one has $p_ a\circ \pi =a\log |s|^2+O(1)$ along $E=\{ s=0\} $. Thus the pole set is $E$ and the generic Lelong number along it is $a$. Add a local smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \pi ^*\omega $. By the argument of lemma 5.38.1, the restriction of the closed positive current $\mathrm{dd}^{\mathrm{c}}(p_ a\circ \pi +\rho )$ to $E$ is $a[E]$, and subtracting the smooth correction does not change the coefficient mass on $E$. Consequently $\mu _ a=a[E]\wedge \eta ^{n-1}$. The positive measure $[E]\wedge \eta ^{n-1}$ has mass $\int _ E\eta ^{n-1}>0$, which proves the formula for the total variation.
$\square$
A fixed modification detects isolated poles only over the set where it is not an isomorphism, so countably many modifications still miss the poles at all points outside a countable union of nowhere dense analytic sets.
Corollary 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
\[ \operatorname {Pol}_{\pi _ j}(p_ a)=0 \qquad \text{for every }j\text{ and every }a. \]
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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