Corollary 18.16.6A5CCF14B62D6

Proof. On $B=Y\times C_0$, let $\pi ,s_0$ be the projections and put $\beta _ f=\pi ^*\beta +(f\circ \pi )s_0^*\tau $. For $w=\sum _ iq_ i\circ \pi _ i$, an ordinary competitor $v\leq 0$ for $w\circ \pi $ may be replaced by its maximum with $w\circ \pi $. Its restriction to every section $Y\times \{ z\} $ is then a $\beta $-plurisubharmonic competitor for $P_\beta [w]=w$, not identically $-\infty $, by lemma 18.16.1. Thus

Take the atomic Green potentials $g_ i$ from the proof of the lemma and put $g=\sum _ i g_ i\circ \pi _ i\circ \pi $. Its complex Hessian is a closed countable divisor current of finite mass minus a smooth form. It is $\beta _ f$-plurisubharmonic for every $f\geq 0$, and it is multiplier-ideal equivalent to $w\circ \pi $. Put $w_{\mathcal I}=\sum _ iq_{i,\mathcal I}\circ \pi _ i$. Since $g\sim w_{\mathcal I}\circ \pi $, every ideal competitor $v\leq 0$ is bounded above by

Here lemma 5.38.5 gives $v\preceq g$, and the ordinary identity applies to $w_{\mathcal I}$. Conversely, $w_{\mathcal I}\circ \pi $ is a normalized ideal competitor, because ideal projection preserves the ideal type on each curve and the ideal type is compatible with pullback and addition. This computes the ideal envelope on $B$ without requiring $\beta _ f$ to be positive.

Apply proposition 18.14.1 to $M\to B$ and the divisor $B\times D$. Its induced zero obstacle is zero, which gives the two identities upstairs and normalization. Restriction to a fixed section of the auxiliary curve identifies this locus isometrically with the finite product of the spaces of curve models. Completeness, geodesics and the nearest-point property of ideal projection therefore follow as in proposition 18.16.5, and the same rational atomic monomials give analytic density. For $f=1$, the background is positive and proposition 18.4.8 shows that every analytic model lies in the ideal subspace. Analytic singularities and the computed ideal projection are independent of $f$, so this analytic-closure assertion holds for every nonnegative coefficient. Fiber integration, now including the factor $C_0$, gives the same $L^1$ convergence upstairs. Each model and each pairwise maximum is $\Psi $ plus a pullback from $B$, whose dimension is smaller than that of $M$, so their upper volumes vanish by proposition 18.14.1; the upper-volume comparison of definition 13.22.1 therefore vanishes for every $f$.

Each polar measure $\mu _ i$ has mass at most $A_ i$, so the sum distance is at most $2\sum _ iA_ i$. By lemma 13.28.1, this bound is attained by placing mass $A_ i$ at two distinct points of each $C_ i$. Hence the diameter does not change, even at the limiting coefficient.

Finally, $\mathrm d\Theta _ f=p^*(\mathrm df)\wedge s^*\tau $ is nonzero for nonconstant $f$, and $\Theta _ f$ is strictly positive when $f>0$. To see that $\Theta _0$ is not big, put $n=m+1+k$ and use the smooth closed positive $(n-1,n-1)$-form $T$ and the Hermitian form $\Omega $ given by

Then $\int _ M\Theta _0\wedge T=0$, whereas $\int _ M\Omega \wedge T>0$. By Stokes' theorem, a lower bound $\Theta _0+\mathrm{dd}^{\mathrm{c}}u\geq \varepsilon \Omega $ with $\varepsilon >0$ would contradict these identities. Thus the background $\Theta _0$ is not big. $\square$