23.1.60 Concentration at the critical weight4B61F80092C8

Proof. Use the correction chosen in the proof of theorem 23.1.56. Its plateau and its late supplement satisfy

Therefore $m_\eta \to m:=K\ell >0$. Shrink the fixed $c$ so that $m$ satisfies all the established positivity margins. On each fixed finite $x$ interval, $r_\eta $ and all its derivatives tend to zero; the late supplement escapes the interval. Thus $g_\eta $ converges smoothly there to

It is zero for $x\leq -2$ and equals $-m(x-\mu _\psi )$ for $x\geq -1$. The same margins give $\mu +g'>0$ and $g''+\mu (1-\mu )\geq 0$ at finite $x$. Across $H$, its extension has logarithmic singularity of coefficient $m$: a local defining function $\zeta $ gives $g=m\log |\zeta |^2+O(1)$. Its upper semicontinuous extension is $\omega $-plurisubharmonic, has supremum zero and belongs to $\mathcal R$.

The bound (23.1.58) and $F_\eta \leq C\eta $ imply that, on compact base sets away from $H$, each fixed-depth exceptional set lies in an exponentially shrinking tube about the local holomorphic zero section $w=0$. These tubes have capacity tending to zero. Indeed, on a relatively compact coordinate chart, $\log |w|^2$ is locally integrable and plurisubharmonic; the weighted Chern–Levine–Nirenberg estimate of section 5.27 bounds the relative capacity of $\{ \log |w|^2<-T\} $ by $C/T$. A finite cover and the local capacity comparison transfer this estimate to $\operatorname {Cap}_\omega $.

The corrections converge uniformly on compact base sets away from $H$, including $\xi =0$, and the supremum normalization is $O(\eta )$. Since $g$ has its logarithmic pole on $H$, the estimate (5.29.4) gives neighborhoods of $H$ with arbitrarily small capacity. Removing one such neighborhood proves $u_\eta \to g$ in capacity.

Assume $\sqrt t\, W(t)$ is bounded but does not tend to zero. Choose the preceding sequence of cores with weighted mass bounded below by $c_0>0$. The proved upper bound makes their full weighted measures uniformly finite, so a subsequence has a weak cluster measure $\nu $. For each fixed closed neighborhood $H_ r$ of $H$, all sufficiently late cores lie in $H_ r$. Portmanteau gives $\nu (H_ r)\geq c_0$; decreasing continuity of the finite measure gives $\nu (H)\geq c_0$. But $\nu _{W,g}(H)=0$ by its defining non-pluripolar truncation. $\square$

Proof. The sufficient conditions are proposition 23.1.47. The necessary uniform mass and tail conditions follow from corollary 23.1.55, whose smooth potentials are invariant. If $\sqrt t\, W(t)$ is unbounded, theorem 23.1.56 disproves universal individual finiteness and prevents a map into finite measures.

For uniform absolute continuity, the concentration regions of theorem 23.1.52 lie in $\{ x\geq c_1/\eta \} $ for a fixed $c_1>0$, hence in shrinking neighborhoods of $H$. The globally defined $h_ H=\log (|z_0|^2/|z|^2)$ has supremum zero and $\omega +\mathrm{dd}^{\mathrm{c}}h_ H=\beta +[H]\geq 0$. Its sublevel estimate (5.29.4) shows that these neighborhoods have capacity tending to zero. If $\limsup _{t\to +\infty }\sqrt t\, W(t)>0$, choose the scales so that $D\eta ^{-1}W(B\eta ^{-2})$ stays bounded below. The weighted masses on these regions then violate uniform absolute continuity. Finally, lemma 23.1.61 disproves capacity-to-weak continuity whenever $\sqrt t\, W(t)$ is bounded but does not tend to zero. $\square$

Proof. Apply theorem 23.1.62 to $\sqrt t\, W_{p,a}(t)\sim t^{1/2-p}(\log t)^{-a}$. A smooth decreasing modification on a compact interval changes none of these conditions. $\square$