PomeLemma 16.5.18.B42B448077CF The function $E^{\phi }_{\theta }\colon \mathcal{E}^{\infty }(X,\theta ;\phi )\rightarrow \mathbb {R}$ is concave.
Lemma 16.5.18B42B448077CF
Proof. Let $\varphi _0,\varphi _1\in \mathcal{E}^{\infty }(X,\theta ;\phi )$. We write $\varphi _ t=t\varphi _1+(1-t)\varphi _0$ for each $t\in [0,1]$.
We claim that for each $t\in [0,1]$, we have
806CE06D337B
16.5.19
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}E^{\phi }_{\theta }(\varphi _ t) =\int _ X \left(\varphi _1-\varphi _0\right)\, \theta _{\varphi _ t}^ n. \end{equation}
At the boundary $t=0,1$, the derivative is understood as the one-sided derivative.
Since $\varphi _0,\varphi _1\in \mathcal{E}^{\infty }(X,\theta ;\phi )$, the difference $\varphi _1-\varphi _0$ is quasi-everywhere bounded. Hence all mixed integrals below are finite. We compute the right derivative first. Fix $t\in [0,1)$ and take $s>0$ small enough so that $t+s\leq 1$. By the cocycle formula (16.5.15),
\[ \begin{aligned} \frac{E^{\phi }_{\theta }(\varphi _{t+s})-E^{\phi }_{\theta }(\varphi _ t)}{s} =& \frac{1}{(n+1)s}\sum _{j=0}^ n \int _ X(\varphi _{t+s}-\varphi _ t)\, \theta _{\varphi _{t+s}}^ j\wedge \theta _{\varphi _ t}^{n-j}\\ =& \frac{1}{n+1}\sum _{j=0}^ n \int _ X(\varphi _1-\varphi _0)\, \theta _{\varphi _{t+s}}^ j\wedge \theta _{\varphi _ t}^{n-j}. \end{aligned} \]
For each fixed $j$, the last integral is a polynomial in $s$ by the multi-linearity of the non-pluripolar product proposition 13.11.4 6. Letting $s\to 0+$, it converges to
\[ \int _ X(\varphi _1-\varphi _0)\, \theta _{\varphi _ t}^ n. \]
Thus the right derivative is given by the right-hand side of (16.5.19). The proof for the left derivative is identical, using $s<0$ with $t+s\in [0,1]$. Hence the left and right derivatives agree for $t\in (0,1)$, and the endpoint derivatives are the corresponding one-sided derivatives.
Now it suffices to show that the derivative (16.5.19) is decreasing in $t$. Take $0\leq s<t \leq 1$. We wish to show that
\[ \int _ X (\varphi _1-\varphi _0)\, \theta _{\varphi _ t}^ n\leq \int _ X (\varphi _1-\varphi _0)\, \theta _{\varphi _ s}^ n. \]
This can be equivalently written as
\[ \int _ X (\varphi _ t-\varphi _ s)\, \theta _{\varphi _ t}^ n\leq \int _ X (\varphi _ t-\varphi _ s)\, \theta _{\varphi _ s}^ n. \]
In particular, replacing $\varphi _0$ by $\varphi _ s$ and $\varphi _1$ by $\varphi _ t$, we are reduced to proving the following
\[ \int _ X (\varphi _1-\varphi _0)\, \theta _{\varphi _1}^ n\leq \int _ X (\varphi _1-\varphi _0)\, \theta _{\varphi _0}^ n. \]
But this is just a consequence of 16.5.16. $\square$
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