Problem 25.23.77FAD0FAF17B7

Problem 25.23.7.7FAD0FAF17B7 Consider a connected compact Kähler manifold $X$ of dimension $n$ and a smooth flag $Y_{\bullet }$ on $X$. Consider a closed smooth real $(1,1)$-form $\theta $ on $X$ representing a big cohomology class and $\varphi \in \mathrm{PSH}(X,\theta )$ with $\int _ X \theta _{\varphi }^ n>0$.

Can one define a refined notion of partial Okounkov bodies $\Delta '_{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ contained in $\Delta _{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ with volume given by $\frac{1}{n!}\int _ X \theta _{\varphi }^ n$?