19.5.1 The traditional approach to the Okounkov body problemAB52CD73DBE6

Theorem 19.5.4.E839B4A35A2A For any big cohomology class $\alpha $ on $X$, the set $\Delta _{Y_{\bullet }}(\alpha )\subseteq \mathbb {R}^ n$ is a convex body satisfying the following properties:

1. D06584225E6D

   We have

   \[ \operatorname{vol}\Delta _{Y_{\bullet }}(\alpha )=\frac{1}{n!}\operatorname{vol}\alpha ; \]
2. 046A9D0BEEA7

   given another big cohomology class $\alpha '$ on $X$, we have

   \[ \Delta _{Y_{\bullet }}(\alpha )+\Delta _{Y_{\bullet }}(\alpha ')\subseteq \Delta _{Y_{\bullet }}(\alpha +\alpha '); \]
3. 7D58AF38EA3C

   let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism where $Y$ is a Kähler manifold. Assume that $(W_{\bullet },g)$ is the lifting of $Y_{\bullet }$ to $Y$. Then

   \[ \Delta _{W_{\bullet }}(\pi ^*\alpha )=\Delta _{Y_{\bullet }}(\alpha )g; \]
4. 35F74103F4DE

   the map $\alpha \mapsto \Delta _{Y_{\bullet }}(\alpha )$ is continuous in the big cone with respect to the Hausdorff metric;

5. 111D84FB6B1C

   for any small enough $t>0$, we have

   \[ \left\{ y\in \mathbb {R}^{n-1}: (t,y)\in \Delta _{Y_{\bullet }}(\alpha ) \right\} =\Delta _{Y_1\supseteq \cdots \supseteq Y_ n}\Bigl ((\alpha -t\{ Y_1\} )|_{Y_1}\Bigr ). \]