Lemma 17.15.9.8571F1340B16 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then for any closed smooth positive $(1,1)$-forms $\omega $ and $\omega '$ on $X$, we have
\[ P_{\theta +\theta '+\omega +\omega '}[\Gamma +\Gamma ']=P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma ']. \]
Proof.
Observe that
\[ \begin{aligned} P_{\theta +\theta '+\omega +\omega '}[\Gamma +\Gamma ']_{\max }=& \left(P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma '] \right)_{\max }\\ =& \Gamma _{\max }+\Gamma '_{\max }. \end{aligned} \]
Take $\tau \in \mathbb {R}$ less than this common value. We need to verify that
\[ (\Gamma +\Gamma ')_{\tau }\sim _ P \Bigl ( P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma ']\Bigr )_{\tau }. \]
By definition, this means that
\[ \sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(\Gamma _{\delta }+\Gamma '_{\tau -\delta }\right)\sim _ P \sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(P_{\theta +\omega }[\Gamma _{\delta }]+P_{\theta '+\omega '}[\Gamma '_{\tau -\delta }]\right). \]
This is a consequence of proposition 16.8.27 and proposition 16.8.28.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.