Lemma 18.17.19040CE491D09

Proof. For $a_ i>0$, scaling the ordinary competitors shows that $a_ iq_ i$ is an ordinary model for $a_ i\eta _ i$. Its polar curvature is $a_ i\mu _ i$, where $\mu _ i$ is the polar curvature of $q_ i$. Taking atomic parts and applying theorem 13.29.3, we see that its ideal model is $a_ iq_{i,\mathcal I}$.

Put $w_ a=\sum _{i\in J(a)}a_ iq_ i\circ \pi _ i$. If there are at least two active factors, lemma 18.16.1 makes the corresponding sum an ordinary model on the product of the active curves, with its positive product background. For one active factor, this is the identity on the curve. The identity is preserved under pullback to $Y$, by proposition 5.43.5: every competitor in the pulled-back background descends, and so do its obstacle bound and its ordinary order, since the projection is surjective. If there are no active factors, the normalized model is zero by the maximum principle. Hence $P_{\beta _ a}[w_ a]=w_ a$ in all cases.

On $B=Y\times C_0$, use the background $\beta _{a,f}=\pi ^*\beta _ a+(f\circ \pi )s_0^*\tau $. As in corollary 18.16.6, replace an ordinary competitor by its maximum with $w_ a\circ \pi $ and restrict it to each section $Y\times \{ z\} $. This gives $P_{\beta _{a,f}}[w_ a\circ \pi ]=w_ a\circ \pi $.

For the ideal identity, take $g_ a=\sum _{i\in J(a)}a_ i g_ i\circ \pi _ i\circ \pi $, with the atomic Green potentials $g_ i$ of lemma 20.3.20. Scaling, addition and pullback preserve multiplier-ideal equivalence by lemma 5.37.4, lemma 5.37.7. The complex Hessian of $g_ a$ is a closed countable divisor current of finite mass minus a smooth form, and $g_ a$ is $\beta _{a,f}$-plurisubharmonic. It is ordinarily equivalent to $w_{a,\mathcal I}\circ \pi $, where $w_{a,\mathcal I}=\sum _{i\in J(a)}a_ iq_{i,\mathcal I}\circ \pi _ i$. By lemma 5.38.5, every ideal competitor is an ordinary competitor for $g_ a$, so by the ordinary identity applied to the models $a_ iq_{i,\mathcal I}$ it is bounded by $w_{a,\mathcal I}\circ \pi $. The latter is itself an ideal competitor, so the ideal envelope equals it. When $J(a)$ is empty, both envelopes on $B$ are zero directly. Finally, proposition 18.14.1, applied to $M\to B$ and the divisor $B\times D$ with weight one, gives the identities upstairs, and independence of the factors gives normalization. $\square$