Corollary 18.15.9.DFCE5415EAB5 Let $M=\mathbb P^1\times \mathbb P^1\times \mathbb P^ k$, $k\geq 1$, with projections $h,s,r$. Let $\sigma $ and $\kappa $ be the probability Fubini–Study forms on $\mathbb P^1$ and $\mathbb P^ k$, respectively, and let $f$ be a positive nonconstant smooth real function on the first factor. Set
\[ \Theta =h^*\sigma +(f\circ h)s^*\sigma +r^*\kappa , \qquad \Psi =\psi \circ r, \]
where $\psi $ is the normalized hyperplane potential with $\kappa +\mathrm{dd}^{\mathrm{c}}\psi =[D]$. For every normalized ordinary model $q$ on $(\mathbb P^1,\sigma )$, put $\Phi _ f(q)=\Psi +q\circ h$. Then
\[ P_\Theta [\Phi _ f(q)]=\Phi _ f(q),\qquad P_\Theta [\Phi _ f(q)]_{\mathcal I} =\Phi _ f(P_\sigma [q]_{\mathcal I}). \]
On $\mathcal T_\Psi =\{ \Phi _ f(q):P_\sigma [q]=q\} $, define
\[ \begin{gathered} \rho _{\Psi ,f}(\Phi _ f(q_1),\Phi _ f(q_2)) =\| \mu _{q_1}-\mu _{q_2}\| _{\mathrm{TV}},\\ \mu _ q=\mathbf1_{\{ q=-\infty \} }(\sigma +\mathrm{dd}^{\mathrm{c}}q). \end{gathered} \]
All the conclusions of proposition 18.15.8 hold on this locus: completeness and geodesics, vanishing of the upper-volume comparison, a closed ideal subspace in which analytic models are dense, ideal projection as the nearest-point retraction, and $L^1$ convergence upstairs.
Proof.
On $Y=\mathbb P^1\times \mathbb P^1$, write $\pi _1,\pi _2$ for the projections and put $\beta =\pi _1^*\sigma +(f\circ \pi _1)\pi _2^*\sigma $. The restriction argument of example 18.15.5 applies to every normalized ordinary curve model and gives
\[ P_\beta [q\circ \pi _1]=q\circ \pi _1. \]
Indeed, the maximum of an ordinary competitor with $q\circ \pi _1$ restricts on every horizontal section to a competitor for $P_\sigma [q]=q$ that is $\sigma $-plurisubharmonic and not identically $-\infty $.
Put $q_{\mathcal I}=P_\sigma [q]_{\mathcal I}$ and let $E$ be the atomic part of $\mu _ q$. The Green potential $w_ E$ of lemma 20.3.20 satisfies $q_{\mathcal I}\sim w_ E$ and $q\sim _{\mathcal I}w_ E$. Its pullback satisfies
\[ \mathrm{dd}^{\mathrm{c}}(w_ E\circ \pi _1) =\sum _ xE(\{ x\} )[\{ x\} \times \mathbb P^1] -E(\mathbb P^1)\pi _1^*\sigma . \]
The divisor current is closed and has finite mass, and $w_ E\circ \pi _1$ is $\beta $-plurisubharmonic because $E(\mathbb P^1)\leq 1$. By corollary 18.4.9, its ordinary and ideal envelopes are equal. Multiplier ideals pull back by lemma 5.37.7, so ordinary equivalence and the ordinary identity just proved give
\[ P_\beta [q\circ \pi _1]_{\mathcal I} =P_\beta [w_ E\circ \pi _1]_{\mathcal I} =P_\beta [w_ E\circ \pi _1] =q_{\mathcal I}\circ \pi _1. \]
This includes the cases $E=0$ and $E$ with countably many atoms.
Apply proposition 18.14.1 to $M\to Y$ and the divisor $Y\times D$, whose induced zero obstacle is zero. The two identities on the base give the asserted identities upstairs. Thus $\Phi _ f$ identifies the locus isometrically with the whole space of curve models, and ideal projection again takes the atomic part. The metric, nearest-point and analytic-closure arguments of proposition 18.15.8 apply unchanged; rational atomic coefficients and the unit coefficient of the hyperplane give local analytic monomials. Each pairwise maximum still has the form $\Psi +\max (q_1,q_2)\circ h$, so the fibration proposition makes all three upper volumes zero. Finally, integration of a smooth volume form along the compact fibers of $h$ transfers $L^1$ convergence on the curve to $M$, as in that proposition. The background is positive and not closed, and, as in example 18.15.5, a dominating closed product form gives the bounded mass property.
$\square$
Pome
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