PomeFD3DFA6C79CFFD3DFA6C79CF
A smooth change of representative $(\theta ,u)\mapsto (\theta +\mathrm{dd}^{\mathrm{c}}f,u-f)$ preserves the current $\theta +\mathrm{dd}^{\mathrm{c}}u$ but moves the zero obstacle. The resulting envelopes remain ordinarily equivalent after transport. In a nonclosed background, however, their masses can differ.
Proposition 18.5.1.07B0D783C3F0 Let $u\in \mathrm{PSH}(X,\theta )$ and $f\in C^\infty (X,\mathbb R)$. Set $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$, $u'=u-f$, $m=\min _ Xf$ and $M=\max _ Xf$. Then
The same inequalities hold for the ordinary envelopes.
Proof. If $v$ is a competitor for $P_\theta [u]_{\mathcal I}$, then $v-f+m$ is $\theta '$-plurisubharmonic, nonpositive and multiplier-ideal equivalent to $u'$. Therefore $v\leq P_{\theta '}[u']_{\mathcal I}+f-m$, and taking the envelope gives the first inequality. Conversely, a competitor $v'$ in the new background gives the competitor $v'+f-M$ in the old one, which proves the second inequality. For the ordinary envelopes, the same calculation applies with the ordinary singularity order in place of multiplier-ideal equivalence. $\square$
For $u\in \mathrm{PSH}(X,\theta )$, put
This quantity depends on the decomposition of the current, even for smooth potentials.
Proposition 18.5.2.6E8985733CBC There are two smooth decompositions of the same positive current for which the corresponding values of $W$ are different.
Proof. Use the Hermitian form $\omega $ and the smooth function $g=u_ t$ of example 11.1.12, with $t\ne 0$ sufficiently small. Both $\omega $ and $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}g$ are Hermitian, and
For a smooth potential in a Hermitian background, all multiplier ideals are trivial and zero is a competitor for its multiplier-ideal envelope, so that envelope is zero. Consequently
By the explicit formula (11.1.13), the two integrals differ. $\square$
The underlying torus is Kähler; the dependence comes from the chosen nonclosed background. In the closed theory, the invariance of mass under bounded changes of the potential gives the missing step after proposition 18.5.1.
Singularity volumes, on the other hand, range over the entire ordinary type, so the bijection of competitors in the gauge change preserves them. Under bounded mass, proposition 13.25.1 accordingly transports ordinary models of positive volume and preserves their upper-volume comparison.
1A21A12C70DEFor each $u\in \mathrm{PSH}(X,\theta )$, the equality
is meaningful. We call it envelope goodness in the fixed background. Its relation to mass differs from the Kähler characterization: it can hold when the potential has infinite mass while its envelope has finite mass.
Proposition 18.5.5.A2602BC06784 If a Hermitian metric $\omega $ fails the bounded mass property of definition 13.14.14, there exists $U\in \mathrm{PSH}(X,\omega )$ such that
where $\mathcal E_{\mathrm{env}}(X,\omega ,0)$ is the relative envelope class of definition 16.28.61.
Proof. Take the potential $U$ of proposition 16.28.72, which has the stated ordinary envelope, class membership and infinite mass. Since $P_\omega [U]=V_\omega =0$, proposition 18.4.4 gives $P_\omega [U]_{\mathcal I}=0$. $\square$
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