Proposition 13.29.1.C9C98321BEFC Polar curvature identifies $\mathcal M(\sigma )$ with the set of polar measures of mass at most $A_\sigma $. Write $p_\sigma (\mu )$ for the model with polar measure $\mu $. Then
\[ U_\sigma (p_\sigma (\mu ))=A_\sigma -\mu (Y),\qquad d_\sigma (p_\sigma (\mu ),p_\sigma (\nu ))=\| \mu -\nu \| _{\mathrm{TV}}, \]
and ideal projection corresponds to taking the atomic part. The space $(\mathcal M(\sigma ),d_\sigma )$ is complete and geodesic, and convergence for $d_\sigma $ implies convergence in $L^1$ and in capacity. If $A_\sigma =0$, its only member is the smooth potential $u_\sigma $ of lemma 5.29.22.
Proof.
Suppose first that $A=A_\sigma >0$, and write $\sigma =A\eta +\mathrm{dd}^{\mathrm{c}}h$. For $v\in \mathrm{PSH}(Y,A\eta )$, the inequalities for ordinary envelopes in proposition 18.5.1 give
\[ P_{A\eta }[v]+\min _ Yh \leq P_\sigma [v-h]+h \leq P_{A\eta }[v]+\max _ Yh. \]
For a model $p$ in background $A\eta $, the potential $q=P_\sigma [p-h]$ therefore satisfies $q+h\sim p$, and since the envelope depends only on the ordinary type, $P_\sigma [q]=P_\sigma [p-h]=q$. Conversely, for $q\in \mathcal M(\sigma )$, the same bounds with $v=q+h$ show that $P_{A\eta }[q+h]\sim q+h$, and taking the envelope back recovers $q$. Thus $p\mapsto P_\sigma [p-h]$ is a bijection from $\mathcal M(A\eta )$ onto $\mathcal M(\sigma )$.
Smooth gauge changes preserve curvature currents, and ordinary equivalence preserves their polar parts by corollary 5.27.15. Consequently the bijection preserves polar curvature. It also preserves singularity volumes, and subtracting the same smooth function commutes with maxima. The classification and the distance formula of lemma 13.28.1, theorem 13.28.4, applied to the positive form $A\eta $, now give the first assertions. The same gauge bounds for ideal envelopes show that the transported ideal projections are ordinarily equivalent. Both are ordinary models by corollary 18.4.4, and ordinary models of the same type are equal. Thus the bijection commutes with ideal projection, which still takes the atomic part. Completeness and the geodesics follow from the identification with polar measures, as in corollary 13.28.7.
For convergence to a model of positive mass, note that $\sigma $ is big because $\sigma +\mathrm{dd}^{\mathrm{c}}(-h)=A\eta >0$, and that the convergence of the polar masses gives a common positive lower-volume bound; then apply proposition 13.22.4, theorem 13.23.3. For a limit of zero mass, the current estimate (13.28.9) follows from the same polar decomposition as before. Normalized compactness and the uniqueness of harmonic functions give $L^1$ convergence, and lemma 5.27.11 on a finite cover gives convergence in capacity.
If $A_\sigma =0$, lemma 5.29.22 leaves only $u_\sigma $ after normalization. Its curvature current vanishes, and both envelope operations fix it. All assertions reduce to statements about the zero measure and the one-point metric space.
$\square$
Pome
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