18.18 Arbitrary curve backgrounds16EA76D7DAAB

Proof. On $B=Y\times C_0$, let $\pi ,s_0$ be the projections and put

The case of the zero obstacle in proposition 18.16.2 gives $P_{\gamma _\sigma }[w]=w$. Replace an ordinary competitor for $w\circ \pi $ by its maximum with $w\circ \pi $. Its restriction to every section $Y\times \{ z\} $ is then a competitor for $P_{\gamma _\sigma }[w]$ that is not identically $-\infty $, so

Use the atomic Green representatives $g_ i$ constructed in the proof of proposition 18.16.2, which are smooth in the coordinates of zero area. Their sum $g=\sum _ i g_ i\circ \pi _ i\circ \pi $ is $\gamma _{\sigma ,f}$-plurisubharmonic and multiplier-ideal equivalent to $w\circ \pi $, and its complex Hessian is a closed countable divisor current of finite mass minus a smooth form. If $w_{\mathcal I}=\sum _ iq_{i,\mathcal I}\circ \pi _ i$, then $g\sim w_{\mathcal I}\circ \pi $. By lemma 5.38.5, every ideal competitor is ordinarily dominated by $g$, hence bounded by

Here the last equality holds because each ideal projection $q_{i,\mathcal I}$ on a curve is an ordinary model, so the argument with sections just given applies to $w_{\mathcal I}$. By pullback and addition, $w_{\mathcal I}\circ \pi $ is itself an ideal competitor, which proves the ideal identity on $B$. Now apply proposition 18.14.1 to $M\to B$ and the divisor $B\times D$ with weight one, whose induced zero obstacle is zero. The two identities upstairs and the independence of the factors give the assertion and normalization.

For the statements on bigness, put $n=m+1+k$ and $\Omega =p^*\beta +s^*\tau +r^*\kappa >0$, with the positive product form $\beta $ fixed earlier. For $f=0$, the closed positive test form

satisfies $\int _ M\Theta _{\sigma ,0}\wedge T_0=0$ and $\int _ M\Omega \wedge T_0>0$. If $B_ i=0$, use instead

Then $\int _ M\Theta _{\sigma ,f}\wedge T_ i=0$ for every $f$, since the only nonzero term contains the factor $\int _{C_ i}\sigma _ i=B_ i$, while $\int _ M\Omega \wedge T_ i>0$. In either case, Stokes' theorem against the closed test form shows that no current $\Theta _{\sigma ,f}+\mathrm{dd}^{\mathrm{c}}u$ with $u$ quasi-plurisubharmonic is bounded below by a positive multiple of $\Omega $, so the background is not big.

Finally, write $\sigma _ i=B_ i\eta _ i/A_ i+\mathrm{dd}^{\mathrm{c}}\chi _ i$ by lemma 5.29.22. The smooth potential $-\sum _ i\chi _ i\circ \pi _ i\circ p$ has strictly positive current when every $B_ i>0$ and $f>0$, which proves bigness in that case. The identity $\mathrm d\Theta _{\sigma ,f}=p^*(\mathrm df)\wedge s^*\tau $ shows that the background is not closed for nonconstant $f$. $\square$

Proof. Restriction to coordinate sections on which the other potentials are finite determines each $q_ i$ up to a constant, and normalization fixes that constant. The preceding lemma and proposition 13.29.1 therefore identify the locus with the finite product of the spaces of curve models, with the sum of their polar metrics; factors of zero area are single points. Completeness, geodesics and the diameter formula follow coordinatewise. Under this identification, corollary 13.28.7, proposition 13.28.11 give closedness, geodesic convexity and the unique nearest-point retraction onto the ideal subspace, and the distances add. Fiber integration transfers $L^1$ convergence on the curves to $M$.

For analytic density, approximate each atomic polar measure by finite atomic measures with rational coefficients and mass at most $B_ i$. By the atomic Green construction and transport under smooth gauge changes, their models in background $\sigma _ i$ have the corresponding finitely many poles with rational coefficients, up to bounded remainders. Coordinates of zero area are smooth. Together with the unit coefficient of $D$, the poles in the coordinates of positive area allow the denominators to be cleared in local product monomials, which gives analytic models in the locus. They converge for the metric (18.18.2). Conversely, by proposition 5.34.1, a product model with analytic singularities restricts, on every coordinate curve section on which the other potentials are finite and whose $Z$-coordinate lies outside $D$, to a function with analytic singularities that is not identically $-\infty $. Each $q_ i$ therefore has polar curvature with finitely many atoms, and it is an ideal model by the classification of curve models. Closedness proves the analytic-closure assertion.

By proposition 18.14.1, applied to the divisor $B\times D$ with weight one, every model and every pairwise maximum in this locus has zero upper volume; for a maximum, use $\Psi $ plus the pullback of the maximum of the two potentials on $B$. Thus all three terms of the upper-volume comparison vanish, even in the backgrounds that are not big by the lemma.

To compare different backgrounds, put $a_ i=B_ i/A_ i$, and for a positive polar measure $\mu $ on $C_ i$ of mass at most $B_ i$ write $p_{\sigma _ i}(\mu )$ for the ordinary model in background $\sigma _ i$ with polar curvature $\mu $, as in proposition 13.29.1. Sending a curve model for $\eta _ i$ with polar measure $\mu _ i$, of mass at most $A_ i$, to $p_{\sigma _ i}(a_ i\mu _ i)$, which for $a_ i=0$ is the unique normalized potential, defines an isometry from the weighted locus $\mathcal S_ a$ of proposition 18.17.3 onto $\mathcal S_\sigma $ that maps the ideal subspaces onto each other. Applying proposition 18.17.5 with $a_ i=B_ i/A_ i$ and $b_ i=B'_ i/A_ i$ gives (18.18.4), the exact value for ordered areas, the contractions and the diameters of their fibers. In curve coordinates the contraction is

In the second case $B'_ i=0$, because $B'_ i\leq B_ i$. Atomic projection commutes with both operations. As in theorem 13.29.3, irrational area ratios need not preserve individual analytic models, but the ideal subspaces are preserved for every ratio.

If $B_ i>0$, the models with polar measure $B_ i\delta _ x$ for infinitely many distinct points $x\in C_ i$, with all other coordinates fixed at their models with zero polar curvature, form an infinite set of ideal product models with mutual distances $2B_ i$. This excludes total boundedness of both loci. If all areas are zero, every curve coordinate is fixed at its unique normalized smooth potential, which proves the last assertion. $\square$

Proof. The areas $B_{i,j}$ tend to zero, so the diameter formula applies. By theorem 5.29.23, all normalized $\sigma _{i,j}$-plurisubharmonic potentials converge in $L^1(C_ i)$ to $u_{\sigma _{i,0}}$, uniformly over this family. Integration along the fixed compact fibers bounds each pulled-back norm by a constant times the norm on the curve, and the finite sum gives the asserted uniform limit on $M$. The auxiliary coefficients do not enter these potentials or their metrics. $\square$