Corollary 10.4.5D52AB2ADBD8C

Proof. Assume first that all $\alpha _ i$ are Kähler. Then all the intersection numbers $\int _ X\prod _ i\alpha _ i^{k_ i}$ are positive, and we put $b(k)=\log \int _ X\prod _ i\alpha _ i^{k_ i}$ for $k=(k_1,\ldots ,k_ r)$ with $\sum _ ik_ i=n$. If $n=1$, only one $k_ i$ is positive and there is nothing to prove; let $n\geq 2$. Let $i\neq j$ with $k_ i,k_ j\geq 1$, let $\Omega $ be the wedge product of Kähler representatives of the remaining $n-2$ factors, counted with multiplicity, and let $e_ i$ be the $i$th coordinate vector. Then (10.4.4) for $\alpha _ i,\alpha _ j$ reads

Fix the coordinates other than $i,j$ and put $s=k_ i+k_ j$. The inequality says that the finite sequence $l\mapsto b(\ldots ,l,\ldots ,s-l,\ldots )$, with $l$ in the $i$th and $s-l$ in the $j$th place, $0\leq l\leq s$, is concave: its consecutive differences decrease. A concave sequence lies above the linear interpolation between its endpoints, since the average of its first $l$ differences is at least the average of all $s$ differences. Consequently

where $k^{(i)}$ is obtained from $k$ by putting all $s$ factors in the $i$th place and $k^{(j)}$ by putting them in the $j$th place.

We now prove $b(k)\geq \sum _ i(k_ i/n)\log \int _ X\alpha _ i^ n$ by induction on the number of positive coordinates of $k$. If only one coordinate is positive, this is an equality. Otherwise choose $i\ne j$ with $k_ i,k_ j\geq 1$. The vectors $k^{(i)}$ and $k^{(j)}$ have one positive coordinate fewer, and inserting the induction hypothesis for them into the last display gives the claim: for $m\ne i,j$ the coefficient of $\log \int _ X\alpha _ m^ n$ is $(k_ i/s+k_ j/s)\, k_ m/n=k_ m/n$, and the coefficients of $\log \int _ X\alpha _ i^ n$ and $\log \int _ X\alpha _ j^ n$ are $(k_ i/s)(s/n)=k_ i/n$ and $k_ j/n$. Exponentiating proves the corollary for Kähler classes.

For nef classes, $\alpha _ i+\varepsilon \{ \omega \} $ is Kähler for every $\varepsilon >0$ by proposition 10.4.2, so the inequality holds for these classes. Each intersection number involved is a polynomial in $\varepsilon $ and tends to the corresponding number for the $\alpha _ i$ as $\varepsilon \downarrow 0$; in particular the limits $\int _ X\alpha _ i^ n$ are nonnegative. Since $t\mapsto t^{k_ i/n}$ is continuous on $[0,\infty )$, the inequality passes to the limit. $\square$