Proposition 12.14.10.0E9506DB4E70 The metrics $g_\epsilon $ are balanced, satisfy $\int _ Xh\wedge g_\epsilon ^ m=1$, and
662BE6BF9C61
12.14.11
\begin{equation} \int _ X\theta \wedge g_\epsilon ^ m =\frac{n-\epsilon I_\epsilon /H_\epsilon }{M_\epsilon }, \qquad D_\mu (g_\epsilon ) =\frac{(n/H_\epsilon )^{m/n}H_\epsilon }{M_\epsilon }. \end{equation}
Their volume quotients tend to $n^{1/n}$ as $\epsilon \downarrow 0$. If $n=3$, then $g_\epsilon ^2$ converges weakly to $[Z]$, and
F4D7155E150D
12.14.12
\begin{equation} \int _ X\theta \wedge g_\epsilon ^2 \sim \frac6{\log (1/\epsilon )}, \qquad D_\mu (g_\epsilon ) \sim \frac{2\, 3^{2/3}}{\log (1/\epsilon )}. \end{equation}
If $n\geq 4$, put $M_0=m+(m-1)^2/[m(m-2)]$. Then
\[ \int _ X\theta \wedge g_\epsilon ^ m\longrightarrow \frac n{M_0}>0, \qquad D_\mu (g_\epsilon )\longrightarrow \frac{n^{m/n}}{M_0}>0. \]
Thus this family approaches the bigness threshold, which is zero, exactly when $n=3$.
Proof.
By the power formula in the proof of theorem 12.14.6 and (12.14.3),
\[ T_\epsilon =\frac{\omega ^ m}{H_\epsilon (a+\epsilon )^2} +mA_\epsilon ^{m-1}\wedge w. \]
Since $A_\epsilon $ is cohomologous to $\omega $, closed Stokes gives $\int _ B\omega \wedge A_\epsilon ^{m-1}=1$. Consequently the two terms have $h$-traces $I_\epsilon /H_\epsilon $ and $m$, which proves the trace normalization of $g_\epsilon ^ m=T_\epsilon /M_\epsilon $. Subtracting $\epsilon w$ from $\theta _\epsilon $ subtracts $\epsilon I_\epsilon /H_\epsilon $ from the pairing $\int _ X\theta _\epsilon \wedge T_\epsilon =n$. Rescaling $\gamma _\epsilon $ by $M_\epsilon ^{-1/m}$ divides both the linear pairing and $D_\mu $ by $M_\epsilon $, proving (12.14.11).
Polar integration gives
\[ I_\epsilon =m t^2\int _0^1 \frac{x^{m-1}}{(x+t\epsilon )^2}\, \mathrm{d}x. \]
When $m=2$, put $q=\epsilon /2$. Direct integration yields
\[ I_\epsilon =\tfrac 12\left(\log \frac{1+q}{q} +\frac q{1+q}-1\right), \qquad H_\epsilon =1-q\log \frac{1+q}{q}. \]
Hence $I_\epsilon \sim \tfrac 12\log (1/\epsilon )$ and $\epsilon I_\epsilon \to 0$. Since $H_\epsilon \to 1$, the asymptotics in (12.14.12) follow from (12.14.11).
For the weak limit when $n=3$, note that the mixed positive term $2A_\epsilon \wedge w$ has $h$-trace two; divided by $M_\epsilon \to \infty $, it tends weakly to zero. The horizontal probability measure
\[ I_\epsilon ^{-1}(a+\epsilon )^{-2}\omega ^2 \]
on $B$ concentrates at $p$: outside any fixed neighborhood of $p$ its unnormalized density stays bounded, while $I_\epsilon \to \infty $. For a smooth $(1,1)$-test form on $X$, integration along the fibers $\{ b\} \times \mathbb P^1$ gives a smooth function on $B$. The convergence of these probability measures to the Dirac mass at $p$ therefore shows that the horizontal term of $g_\epsilon ^2$ converges to $[Z]$, because its coefficient $I_\epsilon /(H_\epsilon M_\epsilon )$ tends to one.
For $m\geq 3$, monotone convergence instead gives
\[ I_\epsilon \longrightarrow m t^2\int _0^1x^{m-3}\, \mathrm{d}x =\frac{(m-1)^2}{m(m-2)}. \]
In both cases $\epsilon I_\epsilon \to 0$. Using $H_\epsilon \to 1$ in (12.14.11) proves the remaining limits, including that of the volume quotients.
$\square$
Pome
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