Corollary 13.29.202F73F083FCE

Proof. Suppose first that $A=A_\sigma >0$, and write $\sigma =A\eta +\mathrm{dd}^{\mathrm{c}}h$. Put $v=u+h$ and $q_+=P_{A\eta }[v]_{\mathcal I}$. The gauge change preserves the curvature current and its pole set, hence both $\mu _ u$ and $D_ u$. By proposition 18.5.1, $q+h\sim q_+$. The gauge invariance of the relative envelope class and its invariance under equivalent references (proposition 16.28.62) give

By corollary 13.28.23, this is equivalent to $\mu _ u=D_ u$.

The gauge bounds for the ordinary and ideal envelopes, corollary 5.27.15 and the classifications in the positive background $A\eta $ identify the polar curvatures of $P_\sigma [u]$ and $P_\sigma [u]_{\mathcal I}$ with $\mu _ u$ and $D_ u$, respectively. Both envelopes are ordinary models, the ideal one by corollary 18.4.4. The classification of models in proposition 13.29.1 therefore shows that they are equal if and only if $\mu _ u=D_ u$.

For a smooth real $F$, the bijection $a\mapsto a+h$ between competitors gives

and the same identity for the ideal envelopes. Indeed, this bijection preserves the obstacle bound, the ordinary order and multiplier-ideal equivalence, and addition of $h$ commutes with upper regularization. Since $F\mapsto F+h$ is a bijection of the smooth obstacles, corollary 13.28.23 gives the remaining equivalence.

If $A_\sigma =0$, lemma 5.29.22 gives $u=-h+c$ for a constant $c$, and the curvature current of $u$ vanishes. Thus $\mu _ u=D_ u=0$. All potentials in this background are smooth and ordinarily equivalent, so their ordinary and ideal competitor families below every obstacle coincide, and they are nonempty after a constant translation. The reference $q$ is ordinarily equivalent to $u$, and proposition 16.28.62 gives $u\in \mathcal E_{\mathrm{env}}(Y,\sigma ,q)$. $\square$