Proposition 18.22.3.AD848996B76C The metric $\rho _\theta $ is independent of the Gauduchon form $\gamma $. Pullback is an isometry from the complete geodesic curve model space onto $(\mathcal M(\theta ),\rho _\theta )$, which has diameter $2A$. The ideal models form a closed geodesically convex subspace equal to the metric closure of the models with analytic singularities. Ideal projection is its unique nearest-point retraction and is $1$-Lipschitz. If $U=v\circ p$ is a model, its distance to that subspace is the nonatomic polar mass of $v$.
Metric convergence implies $L^1(X)$ convergence for every smooth positive volume form. The upper-volume comparison on $\mathcal M(\theta )$ vanishes identically. At $A=0$ this is the one-point metric space of the unique smooth model.
Proof.
The definition of pullback measure gives $p^*T_ v=\theta +\mathrm{dd}^{\mathrm{c}}(v\circ p)$: in the smooth decomposition $\beta =A\eta +\mathrm{dd}^{\mathrm{c}}h$, both sides are $A p^*\eta +\mathrm{dd}^{\mathrm{c}}((v+h)\circ p)$. Apply the Borel restriction and total-variation identities of 18.22.2 to get
\[ \lambda _{v\circ p}^\gamma =L_\gamma (\mu _ v),\qquad \rho _\theta (v\circ p,w\circ p) =\| \mu _ v-\mu _ w\| _{\mathrm{TV}}=d_\beta (v,w). \]
The envelope bijection and proposition 13.29.1 give the isometry, completeness and geodesics. Polar measures have mass at most $A$, so their mutual distance is at most $2A$. For $A>0$, the models of $A\delta _ s$ and $A\delta _ t$, $s\ne t$, attain this bound. For $A=0$, 18.22.1 gives the singleton.
Ideal projection corresponds to the atomic part of the polar measure by the same curve classification. If $\mu =\mu _{\mathrm{at}}+\mu _{\mathrm{diff}}$ and $\nu $ is atomic, the measures $\mu _{\mathrm{diff}}$ and $\mu _{\mathrm{at}}-\nu $ are mutually singular. Therefore
\[ \| \mu -\nu \| _{\mathrm{TV}} =\mu _{\mathrm{diff}}(C) +\| \mu _{\mathrm{at}}-\nu \| _{\mathrm{TV}}. \]
The minimum is attained exactly at $\nu =\mu _{\mathrm{at}}$. Taking the atomic part does not increase total variation, giving the Lipschitz estimate. Atomic polar measures of mass at most $A$ form a closed convex subset of the polar-measure space.
For $A>0$, the smooth-gauge model bijection in the proof of proposition 13.29.1 preserves analytic singularities in both directions: its envelope differs by a bounded function from the smoothly transported analytic model, by proposition 18.5.1. Thus the analytic density in corollary 13.28.7 carries from $A\eta $ to $\beta $. The analytic descent equivalence of 5.43.11 and the envelope bijection then give the stated closure upstairs. At zero area the unique smooth model is already analytic and ideal.
Curve metric convergence implies $L^1(C)$ convergence by proposition 13.29.1, and 5.43.10 gives convergence upstairs. Finally all top-degree masses, including those of pairwise maxima, vanish by 18.22.1. The upper-volume comparison therefore vanishes.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.