Proposition 18.20.2.85AC8B179E4B The map $\Phi _\alpha (q)\mapsto \Phi _\sigma (q)$ is an isometry from $\mathcal S_\alpha $ onto the locus $\mathcal S_\sigma $ of proposition 18.18.3, and it maps the ideal subspaces and the models with analytic singularities onto each other. Consequently $\mathcal S_\alpha $ is complete and geodesic, with diameter $2\sum _ iB_ i$. Its ideal subspace is closed, geodesically convex and equal to the metric closure of the models with analytic singularities. Ideal projection is the unique nearest-point retraction onto this subspace and is $1$-Lipschitz, and the distance to the subspace is the sum of the nonatomic polar masses.
Changing $T$ or $\alpha $ does not change this abstract metric space. All the area comparisons and the contractions commuting with ideal projection of proposition 18.18.3 hold, even for different auxiliary data. The upper-volume comparison vanishes on the locus, and metric convergence implies $L^1(M)$ convergence for every smooth positive volume form. If $T$ and $Z$ are fixed and the curve backgrounds converge uniformly to backgrounds of zero area, then, for any choice of the semipositive terms $\alpha $, the potentials converge in $L^1(M)$, uniformly over the model tuples, to the sum of $\Psi $ and the pullbacks of the normalized smooth potentials of the limiting curve backgrounds, as in corollary 18.18.5.
Proof.
Applied to model tuples, the lemma shows that every $\Phi _\alpha (q)$ is a normalized ordinary model and computes its ideal projection coordinatewise. Restrictions to curve sections on which the other potentials are finite determine the tuple up to constants, and the normalization of the coordinates determines these constants. Thus the map is well defined and bijective, and the formulas for the polar curvatures make it an isometry preserving ideal projection.
Each potential in either locus is pulled back from $Y\times Z$. Analytic presentations pull back under the projection that forgets the auxiliary factor. Conversely, by proposition 5.34.1, the restriction of a potential with analytic singularities to a section with fixed auxiliary coordinate has analytic singularities and is not identically $-\infty $. Hence a potential in either locus has analytic singularities exactly when the common potential on $Y\times Z$ does, and the isometry maps the analytic models onto each other. The statements on the complete metric, the ideal closure, geodesics, nearest points and area comparisons now follow from proposition 18.18.3.
A pairwise maximum is $\Psi $ plus the pullback of the maximum of two $\Gamma _\alpha $-plurisubharmonic functions on $B$. The reduction along the divisor of weight one gives zero upper volume for it and for both models, so the upper-volume comparison vanishes. For $L^1$ convergence, integration along the fixed compact fibers bounds the norm of each pulled-back curve potential by a constant times its norm on the curve. The same estimate and theorem 5.29.23 give the asserted uniform limit at zero area. Neither estimate involves $\alpha $.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.