Theorem 17.5.41.0A392CD9486E Let $X$ be a connected compact Kähler manifold of dimension $n\geq 1$, and let $T_1,\ldots ,T_ n$ be closed positive $(1,1)$-currents on $X$. Then
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17.5.42
\begin{equation} \int _ X\langle T_1\wedge \cdots \wedge T_ n\rangle \geq \prod _{i=1}^ n\left(\int _ X\langle T_ i^ n\rangle \right)^{1/n}. \end{equation}
Consequently the functions
\[ T\longmapsto \left(\int _ X\langle T^ n\rangle \right)^{1/n}, \qquad T\longmapsto \log \left(\int _ X\langle T^ n\rangle \right) \]
are concave on the convex cone of closed positive $(1,1)$-currents on $X$, with the convention $\log 0=-\infty $ for the second function.
Proof.
All the masses involved are finite by theorem 13.5.4. If one of the masses $\int _ X\langle T_ i^ n\rangle $ vanishes, the right side of (17.5.42) is zero and there is nothing to prove. Otherwise put $m_ i=\int _ X\langle T_ i^ n\rangle >0$. Both sides of (17.5.42) are positively homogeneous of degree one in each $T_ i$, by the multilinearity of non-pluripolar products (proposition 13.2.6), so after replacing $T_ i$ by $m_ i^{-1/n}T_ i$ we may assume $m_ i=1$ for all $i$.
By theorem 17.5.37, the class of each $T_ i$ is big. Let $\theta _ i$ be a smooth closed real $(1,1)$-form in the class of $T_ i$ (definition 10.2.17). By corollary 10.2.19 we can write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}u_ i$ with $u_ i\in \mathrm{PSH}(X,\theta _ i)$. Set $\phi _ i=P_{\theta _ i}[u_ i]$. By the mass preservation in proposition 16.21.2(3) and by theorem 16.22.11, $\phi _ i$ is a model potential of mass one.
Let $\nu =\omega ^ n/\int _ X\omega ^ n$. This smooth probability measure is non-pluripolar, as noted after definition 17.5.2. In each big class $\{ \theta _ i\} $, theorem 17.5.15 gives
\[ w_ i\in \mathcal E(X,\theta _ i,\phi _ i),\qquad S_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}w_ i\geq 0,\qquad \langle S_ i^ n\rangle =\nu . \]
By proposition 16.22.14, $P_{\theta _ i}[w_ i]=P_{\theta _ i}[\phi _ i]=\phi _ i$. Applying the mixed mass preservation of proposition 16.21.2(3) first to the tuple $(w_1,\ldots ,w_ n)$ and then to the tuple $(u_1,\ldots ,u_ n)$, we obtain
\[ \int _ X\langle S_1\wedge \cdots \wedge S_ n\rangle =\int _ X\left\langle \bigwedge _{i=1}^ n (\theta _ i+\mathrm{dd}^{\mathrm{c}}\phi _ i)\right\rangle =\int _ X\langle T_1\wedge \cdots \wedge T_ n\rangle . \]
On the other hand, proposition 13.5.8, with all densities equal to one, gives $\langle S_1\wedge \cdots \wedge S_ n\rangle \geq \nu $. Hence the common value of these mixed masses is at least $\nu (X)=1$, which is the normalized inequality. Undoing the scalings proves (17.5.42).
For the concavity, let $T,S$ be closed positive $(1,1)$-currents and $0<t<1$; for $t=0,1$ there is nothing to prove. Put $a=\int _ X\langle T^ n\rangle $ and $b=\int _ X\langle S^ n\rangle $. By multilinearity,
\[ \int _ X\langle ((1-t)T+tS)^ n\rangle =\sum _{k=0}^ n\binom nk(1-t)^ kt^{n-k} \int _ X\langle T^ k\wedge S^{n-k}\rangle . \]
For $1\leq k\leq n-1$, (17.5.42) with $k$ factors $T$ and $n-k$ factors $S$ gives $\int _ X\langle T^ k\wedge S^{n-k}\rangle \geq a^{k/n}b^{(n-k)/n}$; the terms with $k=n$ and $k=0$ are $a$ and $b$, also when one of them vanishes. The binomial formula therefore yields
\[ \int _ X\langle ((1-t)T+tS)^ n\rangle \geq \big ((1-t)a^{1/n}+tb^{1/n}\big )^ n. \]
Taking $n$th roots proves that the function $T\mapsto (\int _ X\langle T^ n\rangle )^{1/n}$ is concave. If $a$ and $b$ are both positive, then, since the logarithm is increasing and concave,
\[ \begin{aligned} \log \Big(\int _ X\langle ((1-t)T+tS)^ n\rangle \Big) & \geq n\log \big ((1-t)a^{1/n}+tb^{1/n}\big )\\ & \geq (1-t)\log a+t\log b. \end{aligned} \]
If one of them is zero, the right side is $-\infty $ for $0<t<1$, so there is nothing to prove.
$\square$
Pome
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