Theorem 5.39.1230882F1D833

Proof. Choose a nonnegative cutoff function $\chi $ equal to one on $B(x,r)$ and supported in $B(x,2r)$, with second derivatives bounded by $Cr^{-2}$. Positivity and integration by parts, after subtracting the constant $v(x)$ from $v$, give

Here we used that the constant $v(x)$ integrates to zero against $\mathrm{dd}^{\mathrm{c}}\chi \wedge \omega _{\mathrm E}^{n-1}$. The matrix $(a_{ij})$ of Radon–Nikodym derivatives of the coefficient measures with respect to $\mu $ is positive semidefinite almost everywhere, so $|a_{ij}|\leq (a_{ii}+a_{jj})/2$. A ball cover proves the assertion about Hausdorff null sets. After a smooth local correction making $v$ plurisubharmonic, dividing (5.39.2) by $r^{2n-2}$ shows that all Lelong numbers vanish.

For the distance estimate, let $m=2n$ and let $\tau $ be the Euclidean trace of $\eta $. Then $\tau \, \mathrm{d}\lambda \leq C\mu $ on the regular locus and $|\xi |_\eta \leq C\sqrt\tau |\xi |_{\mathrm E}$ for tangent vectors $\xi $. Polar integration bounds the average $\eta $-length of the segments from $x$ to the points of $B(x,R)$ by a constant times $\int _{B(x,R)}\sqrt\tau (\zeta )|\zeta -x|^{1-m}\, \mathrm{d}\lambda (\zeta )$. On a dyadic annulus of radius $s$, the Cauchy–Schwarz inequality and (5.39.2) bound the contribution to this integral by $Cs^{1-m}s^{m/2}s^{(m-2+\alpha )/2}=Cs^{\alpha /2}$. Summing over the annuli gives the bound $CR^{\alpha /2}$.

Now let $x,y\in V\setminus A$, and average the lengths of the broken segments $[x,z]\cup [z,y]$ over $z$ in the ball of radius $|x-y|/2$ about the midpoint of $x$ and $y$. By the preceding bound, some such path has length at most $C|x-y|^{\alpha /2}$. For almost every $z$, both segments avoid $A$: proper analytic sets are locally countable unions of smooth strata of real codimension at least two, and the cones from a fixed regular point over these strata have zero volume. If $V$ is small enough, all these paths stay inside $U$.

By this estimate, all regular sequences converging to a point of $A$ have a common limit in the completion, and the identity of the regular locus extends continuously across $A$. By compactness, the image of this extension is the entire completion; different points of $A$ may be identified. A sequence converging to a point of $A$ cannot converge in the metric to a regular point, since $\eta $ is bounded below by a positive multiple of the Euclidean metric on a small closed neighborhood of that point. Hence the metric $\eta $ is incomplete when $A$ is nonempty. Finally, the local model $v=|z|^\alpha +|w|^2$, in coordinates $(z,w)\in \mathbb C\times \mathbb C^{n-1}$, has trace mass of exact order $r^{2n-2+\alpha }$ on balls centered on $\{ z=0\} $, and the distance to $\{ z=0\} $ is of exact order $|z|^{\alpha /2}$; this proves both sharpness claims. In later applications to potentials with logarithmic poles, the theorem is applied to the positive residual current that remains after the divisorial part has been removed. $\square$