Proposition 12.6.49.94A159D3DBDB Let $r>1$ be the ratio $r_0$ or $r_ M$ of the end. After discarding finitely many terms, write $L_ j=s_ j(f+\epsilon _ j)$ with $s_ j>0$, $s_ j\to s_0>0$ and $\epsilon _ j\to 0^+$. Here $f=a$, $s_ j=c_ j$ at the first face, and $f=M-a$, $s_ j=-c_ j$ at the second. Set $J_ j=\int _ B\sigma /L_ j^2$. Then
F61793B0A157
12.6.50
\begin{equation} G_ j=\frac{\sigma }{nL_ j^2}+mA_ j^{m-1}\wedge w, \qquad t_ j:=\int _ Xh\wedge G_ j=\frac{J_ j}{n}+mb_ j^{m-1}. \end{equation}
For $1<r<2$, $t_ j\asymp \epsilon _ j^{r-2}$; for $r=2$, $t_ j\asymp \log (1/\epsilon _ j)$. For $r>2$, the entire sequence converges weakly to the finite positive closed current
\[ G_0=\frac{\sigma }{nL_0^2}+mA_0^{m-1}\wedge w, \qquad A_0=b\kappa +\mathrm{dd}^{\mathrm{c}}\phi _0, \]
and $t_ j\to t_0=\int _ Xh\wedge G_0>0$.
Proof.
On $\mathscr S$, $H(c_ j,e_ j)=nb_ j^ m$, so $A_ j^ m=\sigma /(nL_ j)$. Substituting this in (12.6.34) gives the formula for $G_ j$. Stokes' theorem on $B$, for the closed form $\kappa $, gives $\int _ B\kappa \wedge A_ j^{m-1}=b_ j^{m-1}$, which proves the trace identity. The factorization of $L_ j$ uses $\epsilon _ j=e_ j/c_ j$ at the first end and $\epsilon _ j=e_ j/(-c_ j)-M$ at the second, and the coefficient limits give $s_ j\to s_0>0$.
On each tube around an extreme component, $f\asymp \operatorname {dist}(Z_ i)^{k_ i}$, so the integral of $f^{-2}$ is comparable to
\[ \int _0^{\varrho _0}\frac{\varrho ^{\operatorname {codim}_{\mathbb R}Z_ i-1}}{(\varrho ^{k_ i}+\epsilon _ j)^2}\, \mathrm{d}\varrho , \]
where $\varrho $ is normal radius, $\varrho _0>0$ is small and $r_ i=\operatorname {codim}_{\mathbb R}Z_ i/k_ i$. Splitting the integral at $\varrho =\epsilon _ j^{1/k_ i}$ gives order $\epsilon _ j^{r_ i-2}$ for $r_ i<2$, order $\log (1/\epsilon _ j)$ for $r_ i=2$, and a finite limit for $r_ i>2$. The finite sum over components is governed by $r=\min _ i r_ i$. Since $b_ j$ and $s_ j$ have positive finite limits, these estimates prove the three alternatives for $t_ j$.
If $r>2$, then $f^{-2}$ is integrable and dominates $L_ j^{-2}$ up to a fixed constant factor, so $\sigma /L_ j^2\to \sigma /L_0^2$ in total variation. Since $w\wedge w=0$, we have $A_ j^{m-1}\wedge w=(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)^{m-1}\wedge w$, which converges weakly by proposition 12.6.47. This proves $G_ j\rightharpoonup G_0$, and $G_0$ is positive and closed as a weak limit of positive closed forms. Pairing with $h$ gives the limit of the traces, which is positive because of the term $\sigma /(nL_0^2)$.
$\square$
Pome
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