Proposition 12.6.51.2AC49E96484C Suppose $1<r\leq 2$, and put $T_ j=G_ j/t_ j$. Every weak cluster limit of $(T_ j)$ has the form $p_ B^*\nu $, where $\nu $ is a probability measure on $B$ supported on the extreme components with ratio $r$. Such a limit is positive and closed, has trace one relative to $h$, and satisfies $\theta _0\wedge p_ B^*\nu =0$. If those components reduce to a single point $p$, the whole sequence converges to the current of integration $[\{ p\} \times C]$.
The trace-normalized smooth metrics $\eta _ j=t_ j^{-1/m}\gamma _ j$ satisfy
\[ \int _ X\theta _ j\wedge \eta _ j^ m=D_\mu (\eta _ j) =\frac{nb_ j^ m}{t_ j}\longrightarrow 0. \]
Both quantities have order $\epsilon _ j^{2-r}$ when $1<r<2$, and order $1/\log (1/\epsilon _ j)$ when $r=2$. Their quotient remains one.
Proof.
Let $\nu _ j=\sigma /(J_ jL_ j^2)$. These are probability measures on $B$, and
\[ T_ j=\frac{J_ j}{nt_ j}p_ B^*\nu _ j +\frac m{t_ j}A_ j^{m-1}\wedge w. \]
The first coefficient tends to one. The second current is positive with trace $mb_ j^{m-1}/t_ j\to 0$, hence converges weakly to zero. Compactness of probability measures on $B$ and weak continuity of pullback under the submersion $p_ B$ therefore show that every cluster limit is $p_ B^*\nu $ for a cluster limit $\nu $ of $(\nu _ j)$. Away from small tubes around the extreme set, the density $1/L_ j^2$ is uniformly bounded, while $J_ j\to \infty $; that region receives no mass in the limit. The tube integrals for components with ratio greater than $r$ are $o(J_ j)$ by the preceding estimates, so only components with ratio $r$ keep mass.
For a limit $T=p_ B^*\nu $, $\kappa \wedge T=0$ for degree reasons on $B$, while $L_0w\wedge T=0$ because $L_0$ vanishes on the support of $\nu $. Its trace is one because $\int _ Cw=1$. A top-degree measure on $B$ is closed, so its pullback is closed as well. If the support is the single point $p$, every cluster limit of $(\nu _ j)$ is $\delta _ p$; compactness then gives convergence of the whole sequence, and $p_ B^*\delta _ p$ is the current of integration over the fiber $\{ p\} \times C$.
The minimizing quotient is one on $\mathscr S$, and before trace normalization both its numerator and its denominator equal $nb_ j^ m$. Each is multiplied by $t_ j^{-1}$ under the normalization, and the trace estimates give their limits and rates.
$\square$
Pome
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