Lemma 16.9.25.7F9EDAFF39F9 There is a constant $C>0$ depending only on $X$ and $\theta $ such that for any $\varphi \in \mathrm{PSH}(X,\theta )$ satisfying that $\theta _{\varphi }$ is a Kähler current, we have
\[ d_{S,\theta }\Bigl ((1-\epsilon )\varphi ,\varphi \Bigr )\leq C \epsilon \]
for small enough $\epsilon >0$ such that $(1-\epsilon )\varphi \in \mathrm{PSH}(X,\theta )$.
Proof.
After adding a constant, we may assume that $\varphi \leq 0$. Since $\theta _{\varphi }$ is a Kähler current, $(1-\epsilon )\varphi \in \mathrm{PSH}(X,\theta )$ for all small enough $\epsilon >0$. Fix such an $\epsilon $. Take any Kähler form $\omega $ on $X$ so that $\omega -\theta $ is still a Kähler form. Note that the choice of $\omega $ depends only on $X$ and $\theta $.
Since $\varphi \preceq (1-\epsilon )\varphi $, by 16.9.7 and the multi-linearity of the non-pluripolar product proposition 13.11.4 3, we can compute
\[ \begin{aligned} d_{S,\theta }\Bigl ((1-\epsilon )\varphi ,\varphi \Bigr )=& \frac{1}{n+1}\sum _{j=0}^ n \left(\int _ X \theta _{(1-\epsilon )\varphi }^ j \wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j \wedge \theta _{V_{\theta }}^{n-j}\right)\\ \leq & \frac{1}{n+1}\sum _{j=0}^ n \left(\int _ X \left((1-\epsilon )\theta _{\varphi }+\epsilon \omega \right)^ j \wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j \wedge \theta _{V_{\theta }}^{n-j}\right)\\ =& \sum _{m=1}^ n c_ m(\varphi )\epsilon ^ m, \end{aligned} \]
where, for $1\leq m\leq n$,
\[ c_ m(\varphi )=\frac{1}{n+1}\sum _{j=m}^ n \binom {j}{m} \sum _{\ell =0}^ m (-1)^{m-\ell }\binom {m}{\ell } \int _ X \omega ^\ell \wedge \theta _{\varphi }^{j-\ell }\wedge \theta _{V_{\theta }}^{n-j}. \]
Note that
\[ \begin{aligned} \left|c_ m(\varphi )\right| \leq & \frac{1}{n+1}\sum _{j=m}^ n \binom {j}{m} \sum _{\ell =0}^ m \binom {m}{\ell } \int _ X \omega ^\ell \wedge \theta _{\varphi }^{j-\ell }\wedge \theta _{V_{\theta }}^{n-j}\\ \leq & \frac{1}{n+1}\sum _{j=m}^ n \binom {j}{m} \sum _{\ell =0}^ m \binom {m}{\ell } \int _ X \omega ^\ell \wedge \theta _{V_{\theta }}^{n-\ell } \end{aligned} \]
where we used the monotonicity theorem theorem 13.13.11. Hence the whole expression is of the order of $\mathcal{O}(\epsilon )$, and our assertion follows.
$\square$
Pome
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