Two quotients are formed from the same pairing $\int _ X\theta \wedge \gamma ^{n-1}$. The volume quotient divides it by the determinant integral $D_\mu (\gamma )$, whereas the linear threshold $\kappa _ h(\theta )$ of corollary 11.6.2 divides it by the trace $\int _ Xh\wedge \gamma ^{n-1}$. This threshold is positive exactly when $\theta $ is big, and we call it the bigness threshold; for the pseudoeffective but nonbig background $\theta $ of theorem 12.14.6 it vanishes. The same metrics can approach both infima, but need not do so. In the radial model, the volume-minimizing metrics approach the bigness threshold only in dimension three, while a second family of smooth balanced metrics approaches it in every dimension by concentrating on the fiber over $p$.
Keep the data of theorem 12.14.6, and put
\[ T_\epsilon =\gamma _\epsilon ^ m,\qquad I_\epsilon =\int _ B(a+\epsilon )^{-2}\omega ^ m, \qquad M_\epsilon =m+I_\epsilon /H_\epsilon , \qquad g_\epsilon =M_\epsilon ^{-1/m}\gamma _\epsilon . \]
Write $Z=\{ p\} \times \mathbb P^1$. Its integration current $[Z]$ satisfies $\int _ Zh=1$ and annihilates $\theta $, since $\theta |_ Z=a(p)w=0$.
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
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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.
$\square$
For $n\geq 4$ the metrics $g_\epsilon $ still minimize the perturbed volumes, and their quotients converge to $\lambda _\mu (\theta )$, but their normalized pairings with $\theta $ stay bounded away from zero. This does not contradict nonbigness: the bigness threshold is approached by a different family.
Proposition 12.14.13.F8EA88FD522E For every $n\geq 3$ there are smooth balanced metrics $b_\rho $, $0<\rho <1$, with
\[ \int _ Xh\wedge b_\rho ^ m=1,\qquad b_\rho ^ m\rightharpoonup [Z],\qquad \int _ X\theta \wedge b_\rho ^ m\longrightarrow 0, \qquad D_\mu (b_\rho )\longrightarrow 0 \]
as $\rho \downarrow 0$.
Proof.
Choose a nonzero smooth function $f_\rho \geq 0$ on $B$, supported in $\{ x<\rho \} $ and normalized by $\int _ Bf_\rho \omega ^ m=1$. The form
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12.14.14
\begin{equation} \Gamma _\rho =\frac{f_\rho \omega ^ m+\rho h^ m}{1+n\rho } \end{equation}
is smooth, strictly positive and closed: the first summand has top degree on the base, and $h$ is Kähler. The positive-root construction in the proof of theorem 11.6.1 gives a unique smooth metric $b_\rho $ with $b_\rho ^ m=\Gamma _\rho $, and $b_\rho $ is balanced because $\Gamma _\rho $ is closed. Since $\int _ Xh^{m+1}=n$, its $h$-trace is one.
Put $C_\theta =\int _ X\theta \wedge h^ m$. Since $a<\rho /t$ on the support of $f_\rho $,
\[ 0\leq \int _ X\theta \wedge \Gamma _\rho \leq \frac{\rho /t+\rho C_\theta }{1+n\rho }\longrightarrow 0. \]
The lower bound in the proof of theorem 12.14.6 gives $D_\mu (b_\rho )\leq n^{-1/n}\int _ X\theta \wedge \Gamma _\rho $, so the determinant integrals also tend to zero. The probability measures $f_\rho \omega ^ m$ converge to the Dirac mass at $p$ because their supports shrink to that point. Fiber integration, as in the preceding proof, gives the current limit $[Z]$, while the term $\rho h^ m$ tends to zero.
$\square$
Thus the normalized current $[Z]$ is a weak limit of smooth balanced metrics normalized by their $h$-traces. It annihilates $\theta $, and its determinant-weighted measure vanishes by proposition 12.34.2. The two families give explicit instances of the separation between the volume and threshold denominators described in proposition 24.3.2, without any general smoothing result for Gauduchon currents.
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