Theorem 12.6.41.72BDB634BD0E Under the hypotheses of lemma 12.6.39 on the extreme sets, the metric completion of $\mathscr S$ is a closed Euclidean strip if $r_0,r_ M>1$, a closed Euclidean half-plane if exactly one of them exceeds one, and the whole Euclidean plane otherwise. Each boundary line of the completion is parametrized by $b>0$, and its points are represented by the forms
\begin{align*} \theta _+(b)& =b\kappa +\frac{H_ a}{nb^ m}a w, & H_ a& =\int _ B\frac{\sigma }{a}\quad (r_0>1),\\ \theta _-(b)& =b\kappa +\frac{H_-}{nb^ m}(1-a/M)w, & H_-& =\int _ B\frac{\sigma }{1-a/M}\quad (r_ M>1). \end{align*}
These forms are smooth, semipositive, nef, nonclosed and not big, and their variational volume is one. The completion has no other boundary points.
Proof.
The completion of the convex product $\mathbb R\times I$ of proposition 12.6.36 is its Euclidean closure. Along a Cauchy sequence, $Y$ converges, so $b$ has a finite positive limit, and $R$ converges to a point of $I$ or to a finite endpoint of $I$. Lemma 12.6.39 decides which endpoints are finite and gives the three alternatives.
On $\mathscr S$ we have $e=Q(v)/(nb^ m)$ and $c=ve$. At the end $v\to \infty $, $vQ(v)\to H_ a$ by monotone convergence, which gives the coefficient limit $\theta _+(b)$. At the end $v\to -1/M$, write $1+va=1-a/M+(v+1/M)a$; dominated convergence gives $Q(v)\to H_-$, and hence the limit $\theta _-(b)$. These are the coefficient limits at all finite ends. The forms are semipositive, hence nef, and they are nonclosed. Increasing their fiber coefficient by $\epsilon >0$ makes them positive, and (12.6.33) shows that the resulting volumes tend to one as $\epsilon \downarrow 0$. Monotonicity bounds the root volume of the limit form by this limit. Conversely, each fixed Gauduchon quotient is continuous in $\epsilon $, so the limit of the root volumes is at most every quotient of the limit form, hence at most its root volume. Thus the variational volume of $\theta _\pm (b)$ is one.
Suppose that some quasi-plurisubharmonic $u$ satisfied
\[ \theta _\pm +\mathrm{dd}^{\mathrm{c}}u\geq \delta (\kappa +w),\qquad \delta >0. \]
For every smooth nonnegative top-degree form $\eta $ on $B$, its pullback is $\mathrm{dd}^{\mathrm{c}}$-closed for degree reasons, and $\kappa \wedge \eta =0$. Stokes' theorem for currents therefore gives
\[ \int _ B(L_\pm -\delta )\eta \geq 0. \]
where $L_\pm $ is the fiber coefficient of $\theta _\pm $. But $L_\pm $ vanishes somewhere, and a nonzero $\eta $ supported where $L_\pm <\delta $ contradicts this inequality. Thus $\theta _\pm $ is not big.
$\square$
Pome
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