PomeProposition 19.4.26.97A03B9F1F87 The partial Okounkov body $\Delta _{\nu }(L,h)$ depends only on $\mathrm{dd}^{\mathrm{c}}h$, not on the explicit choices of $L,h_0,h$.
Proposition 19.4.2697A03B9F1F87
Proof. There are two different claims to prove, as detailed in the two steps below.
Step 1. Let $h_0'$ be another Hermitian metric on $L$. Set $\theta '=c_1(L,h_0')$. Write $\mathrm{dd}^{\mathrm{c}}f=\theta -\theta '$. Let $\varphi '=\varphi +f\in \mathrm{PSH}(X,\theta ')$. Then
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19.4.27
\begin{equation} \Delta _{\nu }(\theta ,\varphi )=\Delta _{\nu }(\theta ',\varphi '). \end{equation}
This is obvious since $\Gamma (\theta ,\varphi )=\Gamma (\theta ',\varphi ')$.
Step 2. Let $L'$ be another big line bundle on $X$. By Step 1, we may assume that the reference Hermitian metric $h'_0$ on $L'$ is such that $c_1(L',h'_0)=\theta $.
Let $h'$ be a plurisubharmonic metric on $L'$ with $c_1(L,h)=c_1(L',h')$. Then
\[ \Delta _{\nu }(L,h)=\Delta _{\nu }(L',h'). \]
By the approximation step in the construction of $\Delta _\nu (L,h)$ and the continuity of the extension in theorem 19.2.10, it is enough to prove the assertion when $c_1(L,h)$ has analytic singularities. After taking a birational resolution, it suffices to deal with the case where $c_1(L,h)$ has analytic singularities along an effective $\mathbb {Q}$-divisor $D$. By rescaling, we may also assume that $D$ is a divisor. By 19.4.23, we further reduce to the case where $c_1(L,h)$ is not singular.
In this case, the assertion is proved in proposition 19.4.9. $\square$
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