Pome

Proposition 18.11.1760C8726BAE2

Proposition 18.11.1.760C8726BAE2 Let $\chi \in C^\infty (X,\mathbb R)$ and $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}\chi >0$. The map $[u]\mapsto [u-\chi ]$ is a uniform isomorphism

\[ (\mathfrak S(X,\omega ),\tau _{\omega ,F}) \longrightarrow (\mathfrak S(X,\omega '),\tau _{\omega ',F-\chi }). \]

For $s>0$, $[u]\mapsto [su]$ is a uniform isomorphism from $\tau _{\omega ,F}$ to $\tau _{s\omega ,sF}$. Both maps transport the ordinary and the ideal envelopes exactly.

Proof. For either envelope operator $P$, the correspondence of competitors gives

\[ P_{\omega ',F-\chi }[u-\chi ]=P_{\omega ,F}[u]-\chi , \qquad P_{s\omega ,sF}[su]=sP_{\omega ,F}[u]. \]

Smooth additions leave all multiplier ideals and divisorial Lelong numbers unchanged. Scaling multiplies the divisorial and envelope distances by $s$ and changes the ideal weights by $t\mapsto st$. Each map and its inverse therefore transport the generating entourages to generating entourages. $\square$

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