Lemma 17.10.53.B3BB9434F2D9 There is $A,B>0$ so that for any $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$, we have
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17.10.54
\begin{equation} d_1(\varphi ,\phi )\geq -\int _ X \theta _{\phi }^ n\cdot \sup _ X \varphi \geq -Ad_1(\varphi ,\phi )-B. \end{equation}
Proof.
When $\sup _ X \varphi \leq 0$, the right-hand part of (17.10.54) is trivial. Since $\varphi \leq \phi $, we find
\[ d_1(\phi ,\varphi )=-E_{\theta }^{\phi }(\varphi )\geq -\int _ X \theta _{\phi }^ n\cdot \sup _{\{ \phi \neq -\infty \} } (\varphi -\phi )=-\int _ X \theta _{\phi }^ n\cdot \sup _ X \varphi , \]
where the last equality follows from proposition 16.3.21.
We can therefore assume that $\sup _ X \varphi > 0$. In this case, the left-hand part of (17.10.54) is trivial. Take a Kähler form $\omega \geq \theta $ on $X$. Then thanks to theorem 16.3.12, we can find a constant $C>0$ so that
\[ \theta _{\phi }^ n\leq C\omega ^ n. \]
Thanks to proposition 5.17.3, there is a constant $C'>0$, independent of the choice of $\varphi $, so that
\[ \int _ X \left( \phi -\varphi +\sup _ X \varphi \right)\, \theta _{\phi }^ n\leq C'. \]
We estimate
\[ \begin{aligned} I_{\theta }(\phi ,\varphi )\geq & \int _ X |\varphi -\phi |\, \theta _{\phi }^ n\\ \geq & \left(\sup _ X \varphi \right)\int _ X \theta _{\phi }^ n-\int _ X \left( \phi -\varphi +\sup _ X \varphi \right)\, \theta _{\phi }^ n \\ \geq & \left(\sup _ X \varphi \right)\int _ X \theta _{\phi }^ n-C'. \end{aligned} \]
The right-hand part of (17.10.54) then follows from theorem 17.10.48.
$\square$
Pome
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