Proposition 17.14.6.BF40FB71C62C Let $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$. Then
\[ P_{\theta }[\Gamma ]_{\mathcal{I}}\in \mathcal{E}^{\mathrm{NA}}(X,\theta ;P_{\theta }[\phi ]_{\mathcal{I}}). \]
More generally, for any closed real smooth positive $(1,1)$-form $\omega $ on $X$, we have
\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\in \mathcal{E}^{\mathrm{NA}}(X,\theta +\omega ;P_{\theta +\omega }[\phi ]_{\mathcal{I}}). \]
Proof.
We prove the first assertion; the second is identical, replacing $\theta $ by $\theta +\omega $.
Set
\[ \widetilde\Gamma _\tau \coloneqq P_\theta [\Gamma _\tau ]_{\mathcal I}. \]
The family is decreasing by 16.6.28. If $\tau =(1-s)\tau _0+s\tau _1$, then the concavity of $\Gamma $ gives
\[ \Gamma _\tau \geq (1-s)\Gamma _{\tau _0}+s\Gamma _{\tau _1}. \]
Hence, by 16.6.28 and proposition 16.6.27,
\[ \widetilde\Gamma _\tau \geq (1-s)\widetilde\Gamma _{\tau _0}+s\widetilde\Gamma _{\tau _1}. \]
Thus the family is concave. Finally,
\[ \operatorname *{sup}_{\tau <\Gamma _{\max }}{}\! {}^{*}\widetilde\Gamma _\tau = P_\theta [\phi ]_{\mathcal I} \]
follows from proposition 16.6.30. Since $P_\theta [\phi ]_{\mathcal I}\geq \phi $, its mass is positive by the monotonicity theorem theorem 13.13.11. This proves the assertion.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.