Proposition 16.29.14.3B3109886F9F There are a compact complex surface $X$, Hermitian forms $\omega ,h$ on $X$, and a smooth real function $f$ with $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f>0$, with the following property. For every $a<c$ there is a system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\omega )$ such that the limits
\[ \lim _{\epsilon \downarrow 0} \int _ X\mathrm{MA}^{\mathrm{ext}}_{\omega +\epsilon h} (\Gamma ^{\epsilon h}_ a), \qquad \lim _{\epsilon \downarrow 0} \int _ X\mathrm{MA}^{\mathrm{ext}}_{\omega '+\epsilon h} ((\mathsf G_ f\boldsymbol \Gamma )^{\epsilon h}_ a) \]
both exist and are different.
Proof.
Take the torus $X$ and the smooth perturbation of proposition 18.5.2, so that $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f$ is Hermitian and $\int _ X(\omega ')^2\neq \int _ X\omega ^2$, and fix any Hermitian form $h$. In every background $\omega +\eta $, let every slice with $\tau <c$ be zero. This is an envelope system for every $a<c$, since zero is the multiplier-ideal envelope of the unit-ideal class in a positive background. Its first limit is
\[ \lim _{\epsilon \downarrow 0}\int _ X(\omega +\epsilon h)^2 =\int _ X\omega ^2. \]
The gauge bijection sends every slice to $P_{\omega '+\eta }[-f]_{\mathcal I}=0$, since $f$ is smooth and the new background is positive. The second limit is therefore $\int _ X(\omega ')^2$, which is different.
$\square$
Pome
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