16.29.25 Exact pullback and smooth changes of backgroundE5CE4F64BF58

Let $\mu :Y\to X$ be a modification and let $\boldsymbol \Gamma $ satisfy theorem 16.29.22. By proposition 18.4.3, its recovery identity pulls back to

This concerns the exact pulled backgrounds. The form $\mu ^*\eta $ may be degenerate, so these identities do not define a full system indexed by every Hermitian form on $Y$.

For a modification $\nu :\widetilde B\to B$, proposition 16.28.59 gives

so the transverse positivity test survives exact pullback. The same proposition also preserves ambient singularity volumes in an exact pulled background; it makes no comparison with an independent positive background upstairs.

Finally, the smooth transport $\mathsf G_ f$ of proposition 16.29.13 commutes with both $\mathsf E$ and $\mathsf R$. Therefore recovery survives the change from $h$ to $h+\mathrm{dd}^{\mathrm{c}}f$ for every smooth real $f$. The transported scalar slices need not remain invariant under $E$, and no invariance of the raw masses of independently normalized nonclosed-background representatives is inferred. The general recovery problem without the scalar symmetry or transverse positive-volume condition remains open.