18.6 geometry of ideal classes3D693DBCCBE2

Proof. The ideal envelopes are integrable and depend only on the ideal classes, so the formula is well defined and finite; symmetry and the triangle inequality are those of the $L^1$ norm. If the distance is zero, the two envelopes agree almost everywhere, hence everywhere by lemma 5.24.3, and projection shows that the two ideal classes coincide. Conversely, equal ideal classes have the same envelope.

By the normalized compactness theorem, $\mathfrak M_{\mathcal I}$ lies in the compact space $\mathcal K_\theta $. Every subset of a compact metric space is totally bounded, and its completion is its closure in that space. Finally, the ratio of two smooth positive volume forms on the compact manifold $X$ is bounded above and below by positive constants, and multiplying the integral by these bounds proves the last assertion. $\square$

Proof. We first show that the normalized Green potentials of finite atomic probability measures are ideal models, and then approximate the curvature measure of an arbitrary $u\in \mathcal K_\theta $ by such measures.

Let $\delta _ p$ denote the Dirac mass at $p$. For distinct points $p_1,\ldots ,p_ N\in Y$, consider a finitely supported probability measure

The Green-kernel construction gives a potential $q_\nu $ with

Indeed, choose a kernel with $\mathrm{dd}^{\mathrm{c}}_ xG(x,p)=\delta _ p-\theta $. The sum $\sum _ j a_ jG(x,p_ j)$ has curvature $\nu $, and subtracting its finite supremum gives $q_\nu $. In a coordinate $z$ centered at $p_ j$, the function $q_\nu $ is $a_ j\log |z|^2$ plus a smooth function, and it is smooth away from the finite support of $\nu $. Since all of its curvature lies on its pole set, its non-pluripolar measure is zero.

We claim that $P_\theta [q_\nu ]_{\mathcal I}=q_\nu $. Let $w\in \mathrm{PSH}(Y,\theta )$ satisfy $w\sim _{\mathcal I}q_\nu $. The Lelong-number criterion of theorem 5.37.3 gives $\nu (w,p_ j)=a_ j$, where $\nu (w,p_ j)$ is the Lelong number of $w$ at $p_ j$. On a curve, this Lelong number is exactly the mass of the atom at $p_ j$ of the positive measure $\theta +\mathrm{dd}^{\mathrm{c}}w$. By distributional Stokes, that measure has total mass one, so the prescribed atoms already exhaust its mass and $\theta +\mathrm{dd}^{\mathrm{c}}w=\nu $. Thus $\mathrm{dd}^{\mathrm{c}}(w-q_\nu )=0$, and the difference is a harmonic distribution, hence a smooth function that is constant on the compact connected curve. Among these translates, the greatest nonpositive one is $q_\nu $, which proves the claim. By proposition 18.4.4, every such $q_\nu $ is also an ordinary model.

Now let $u\in \mathcal K_\theta $. Its curvature $\nu =\theta +\mathrm{dd}^{\mathrm{c}}u$ is a positive probability measure. Choose finitely supported probability measures $\nu _ j$ converging weakly to $\nu $. For instance, partition the compact metric space $Y$ into finitely many Borel sets of diameter tending to zero, place the mass of each set at a chosen point of it, and omit the sets of mass zero; uniform continuity of each continuous test function gives weak convergence.

The weights can be made rational. For a partition with weights $a_1,\ldots ,a_ N$, choose a large integer $M$, replace $a_ i$ by $b_ i=\lfloor Ma_ i\rfloor /M$ for $i<N$, put $b_ N=1-\sum _{i<N}b_ i$, and discard zero weights. The total absolute change is at most $2(N-1)/M$, which can be made to tend to zero along the partitions, so weak convergence is preserved. Writing $b_ i=m_ i/M$, the corresponding Green potential has the analytic type

with smooth local remainders.

Let $q_ j=q_{\nu _ j}$. By normalized compactness, every subsequence has a further subsequence converging in $L^1$ to some $q\in \mathcal K_\theta $. Passing to the limit in the curvature identity in the sense of distributions gives

By the same harmonic argument, $q-u$ is constant, and the constant is zero because both suprema vanish. Every cluster value is therefore $u$, so the whole sequence converges to $u$ in $L^1$. This proves the density assertion and identifies the completion.

To see incompleteness concretely, choose a nonconstant smooth real function $f$. For sufficiently small $\varepsilon >0$, one has $\theta +\varepsilon \mathrm{dd}^{\mathrm{c}}f>0$, so $u=\varepsilon f-\sup _ Y(\varepsilon f)$ is a smooth, normalized and nonconstant element of $\mathcal K_\theta $. Its multiplier ideals are all trivial, so zero is a competitor and $P_\theta [u]_{\mathcal I}=0$. Thus $u\notin \mathfrak M_{\mathcal I}(Y,\theta )$, although $u$ lies in the completion. The set of ideal models is therefore not closed in its completion, and the metric is not complete. $\square$

Proof. Choose finitely supported probability measures $\nu _ j$ with rational weights converging weakly to $\theta $, and let $q_ j$ be their normalized Green potentials; the preceding proof gives $q_ j\to 0$ in $L^1$. Put $a=1-\tau $ and $r_ j=P_\theta [a q_ j]_{\mathcal I}$. The scaled potential lies in the same background, since

Since $aq_ j$ is a competitor, $aq_ j\leq r_ j\leq 0$, and hence $r_ j\to 0$ in $L^1$. By projection, $r_ j$ is an ideal model. By proposition 18.4.8, it has exactly the logarithmic coefficients of $aq_ j$, which are rational, with bounded local remainders, and it is locally bounded away from the finite pole set. On the curve, removing the polar part of the curvature therefore gives

By distributional Stokes, this measure has total mass $1-a=\tau $. The positive curvature measures $\theta +\mathrm{dd}^{\mathrm{c}}r_ j$ have mass one and converge as distributions, hence weakly, to $\theta $. Since $\nu _ j\rightharpoonup \theta $, the last identity gives $\mathrm{MA}_\theta (r_ j)\rightharpoonup \tau \theta $. Finally, $P_\theta [0]_{\mathcal I}=0$, so the distance in the statement is $\int _ Y|r_ j|\, \mathrm{d}V$, which tends to zero. The limiting potential $0$ has non-pluripolar measure $\theta $, of mass one. $\square$