Corollary 18.15.1013B4DE048053

Proof. On $Y=\mathbb P^1\times \mathbb P^1$, put $\beta _ f=\pi _1^*\sigma +(f\circ \pi _1)\pi _2^*\sigma $. This form is semipositive, and the argument with horizontal sections in corollary 18.15.9 still gives

Indeed, the pullback of $q$ is $\beta _ f$-plurisubharmonic, and every horizontal section has background $\sigma $, also where $f$ vanishes.

Write $q_{\mathcal I}=P_\sigma [q]_{\mathcal I}$, let $E$ be the atomic part of $\mu _ q$, and set $g=w_ E\circ \pi _1$, with the Green potential of the preceding proof. Then $g$ is $\beta _ f$-plurisubharmonic, $q\circ \pi _1\sim _{\mathcal I}g$, and $g\sim q_{\mathcal I}\circ \pi _1$. Its complex Hessian is the same countable divisor current of finite mass minus $E(\mathbb P^1)\pi _1^*\sigma $. For every ideal competitor $v\leq 0$ for $q\circ \pi _1$, lemma 5.38.5 gives $v\preceq g$. Therefore

The last equality uses the ordinary identity just proved, since $q_{\mathcal I}$ is an ordinary curve model. Conversely, $q_{\mathcal I}\circ \pi _1$ is a normalized $\beta _ f$-plurisubharmonic ideal competitor, because ideal projection preserves the ideal type on the curve and multiplier ideals pull back. This proves the ideal identity on the base, including the cases $E=0$ and $E$ with countably many atoms.

Proposition 18.14.1 allows this semipositive background on the base and gives both identities upstairs. Normalization follows because the factors are independent. The metric space is the same isometric copy of the whole space of curve models, and ideal projection, analytic closure, geodesics and $L^1$ convergence upstairs follow as in the preceding proof. Since the relevant currents are still pulled back from the surface $Y$, the same proposition gives zero upper volumes, so the upper-volume comparison vanishes. Both $\Phi (q)$ and its polar measure are independent of $f$, so the stated identifications and the compatibility with ideal projection hold exactly, including at the limiting background. Analytic singularities depend only on the functions, so their closure is also unchanged.

To see that $\Theta _0$ is not big, let $n=k+2$, and take the smooth closed positive $(n-1,n-1)$-form and the Hermitian form

One has $\int _ M\Theta _0\wedge T=0$ and $\int _ M\Omega \wedge T=1$. If $\Theta _0+\mathrm{dd}^{\mathrm{c}}u\geq \varepsilon \Omega $ for some $u\in \mathrm{QPSH}(M)$ and $\varepsilon >0$, pairing with $T$ and using Stokes' theorem would give $0\geq \varepsilon $. Thus $\Theta _0$ is not big, although the model locus keeps its complete metric. $\square$