The Gauduchon metrics that minimize the quotient defining the volume in (10.19.9) can be identified explicitly: they are the equation metrics themselves. Their normalized squares converge smoothly at the nef boundary, although the normalized metrics may diverge. Recall from definition 12.2.1 that the root volume $\lambda _{\mu _0}$, with $\lambda _{\mu _0}^3=\mathcal V_{\mu _0}$, is the infimum over Gauduchon metrics $\gamma $ of $\int _ X\theta \wedge \gamma ^2$ divided by the denominator $D_{\mu _0}(\gamma )=\int _ X(\mu _0/\gamma ^3)^{1/3}\gamma ^3$. For positive functions of $\epsilon $, we write $f\asymp g$ if $f/g$ is bounded above and below by positive constants as $\epsilon \downarrow 0$.
Proposition 10.19.11.983D2EEDBE68 Let $a,b\geq 0$, $\theta =a\alpha +b\beta $ and $\Omega _\epsilon =\theta +\epsilon \omega $. For every $\epsilon >0$, the metric $\Omega _\epsilon $ is Gauduchon, and it minimizes the quotient defining $\lambda _{\mu _0}(\Omega _\epsilon )$. Put
\[ I_\epsilon =8\pi \bigl (ab+2(a+b)\epsilon +3\epsilon ^2\bigr ), \qquad T_\epsilon =\frac{\Omega _\epsilon ^2}{I_\epsilon }, \qquad \gamma _\epsilon =\frac{\Omega _\epsilon }{\sqrt{I_\epsilon }}. \]
Then $\int _ X\omega \wedge T_\epsilon =1$, $\mathrm{dd}^{\mathrm{c}}T_\epsilon =0$ and $T_\epsilon =\gamma _\epsilon ^2$. As $\epsilon \downarrow 0$, the forms $T_\epsilon $ converge smoothly to
\[ T_0=\begin{cases} (4\pi )^{-1}\alpha \wedge \beta ,& a,b>0,\\ (8\pi )^{-1}\alpha \wedge (\beta +D),& a>0,\ b=0,\\ (8\pi )^{-1}\beta \wedge (\alpha +D),& a=0,\ b>0,\\ (24\pi )^{-1}\omega ^2,& a=b=0. \end{cases} \]
In every case $\theta \wedge T_0=0$. If $q$ is the number of positive coefficients among $a,b$, the denominator of the normalized minimizing metric satisfies
\[ D_{\mu _0}(\gamma _\epsilon ) =\frac{\bigl (24\pi \epsilon (a+\epsilon )(b+\epsilon )\bigr )^{2/3}}{I_\epsilon } \asymp \epsilon ^{q/3}, \]
where the two comparison constants may depend on the fixed pair $(a,b)$.
Proof.
Write $x=a+\epsilon $, $y=b+\epsilon $ and $z=\epsilon $. Since $\alpha ^2=\beta ^2=D^2=0$,
\[ \Omega _\epsilon ^2 =2xy\alpha \wedge \beta +2xz\alpha \wedge D+2yz\beta \wedge D. \]
The forms $\alpha ,\beta $ are closed and $\mathrm{dd}^{\mathrm{c}}D=2\alpha \wedge \beta $. Hence the first term is closed, and $\mathrm{dd}^{\mathrm{c}}$ of each of the last two terms contains $\alpha ^2$ or $\beta ^2$ and therefore vanishes. Thus $\mathrm{dd}^{\mathrm{c}}(\Omega _\epsilon ^2)=0$. Since the zero potential solves the normalized equation for $\Omega _\epsilon $ and $\mu _0$, the characterization of minimizing metrics in theorem 12.2.3 shows that $\Omega _\epsilon $ is a minimizer.
Integration along the fibers gives $\int _ X\alpha \wedge \beta \wedge D=4\pi $. Hence
\[ \int _ X\omega \wedge \Omega _\epsilon ^2 =8\pi (xy+xz+yz)=I_\epsilon , \]
which is the trace normalization. Dividing the three coefficients in the wedge expansion of $\Omega _\epsilon ^2$ by $I_\epsilon $ gives each of the stated limits; the convergence holds in every smooth norm, because the forms $\alpha \wedge \beta $, $\alpha \wedge D$ and $\beta \wedge D$ are fixed. The identities $\alpha ^2=\beta ^2=0$ then give $\theta \wedge T_0=0$ in all four cases.
Set $c_\epsilon =24\pi \epsilon (a+\epsilon )(b+\epsilon )$. The equation for $\Omega _\epsilon $ gives
\[ \gamma _\epsilon ^3 =\frac{c_\epsilon }{I_\epsilon ^{3/2}}\mu _0. \]
Substituting this in the definition of $D_{\mu _0}$ and using $\int _ X\mu _0=1$ yields $D_{\mu _0}(\gamma _\epsilon )=c_\epsilon ^{2/3}/I_\epsilon $. For fixed $(a,b)$ one has $c_\epsilon \asymp \epsilon ^{3-q}$ and $I_\epsilon \asymp \epsilon ^{2-q}$, so the quotient is of order $\epsilon ^{q/3}$.
$\square$
In a coframe adapted to the three rank-one forms $\alpha $, $\beta $ and $D$, the three complementary products in the proof show that the limiting positive $(2,2)$-forms have coefficient matrices of rank $1$, $2$ and $3$ when $q=2$, $1$ and $0$, respectively. If $a,b>0$, the normalized metrics themselves converge smoothly to
\[ \frac{a\alpha +b\beta }{\sqrt{8\pi ab}}. \]
If $a>0$ and $b=0$, however, the coefficient of $\alpha $ in $\gamma _\epsilon $ grows like a positive constant times $\epsilon ^{-1/2}$, while its other two coefficients tend to zero. Thus the metrics $\gamma _\epsilon $ are unbounded relative to the fixed metric $\omega $, although their squares converge smoothly; the same holds with $\alpha $ and $\beta $ interchanged. This divergence is unavoidable. In dimension three, a positive $(2,2)$-form of rank two is not the square of a semipositive $(1,1)$-form: the coefficients of such a square are the pairwise products of the eigenvalues of the $(1,1)$-form, and the number of nonzero products is three, one or zero. At the vertex $a=b=0$, the normalized metric instead stays equal to $\omega /\sqrt{24\pi }$.
For a nonzero boundary class, the smooth limit $T_0$ annihilates $\theta $, and the denominators $D_{\mu _0}(\gamma _\epsilon )$ tend to zero; no concentration of the coefficient measures occurs in this family. This degeneration of the minimizing tests for a fixed density differs from the sequence of Gauduchon forms used in lemma 12.19.13 to prove that the lower mass vanishes. At the vertex, the denominator stays positive, and the order three comes entirely from the scaling $\Omega _\epsilon =\epsilon \omega $ of the background.
Pome
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