19.5.5 Definitions of partial Okounkov bodies232CC36315A0

Let $\theta $ be a closed real smooth $(1,1)$-form on $X$ representing a big cohomology class $\alpha $.

Let $T=\theta _{\varphi }\in \mathcal{Z}_+(X,\alpha )$. We shall define a convex body $\Delta _{Y_{\bullet }}(T)\subseteq \mathbb {R}^ n$, which is also written as $\Delta _{Y_{\bullet }}(\theta ,\varphi )$. This convex body is called the partial Okounkov body of $T$ with respect to the flag $Y_{\bullet }$.