PomeLemma 19.4.43.FD338F03BE48 Assume that $\varphi $ has analytic singularities and $\theta _{\varphi }$ is a Kähler current. Then (19.4.42) holds.
Lemma 19.4.43FD338F03BE48
Proof. Up to replacing $X$ by a modification and twisting $T$ accordingly, we may assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$, see proposition 19.4.32 and theorem 5.18.8.
Take $\epsilon \in \mathbb {Q}\cap (0,1)$. In this case, for large enough $k\in \mathbb {Z}_{>0}$ we have
\[ \begin{split} \mathrm{H}^0\left(X,T\otimes L^ k\otimes \mathcal{I}_{\infty }(k\varphi )\right) \subseteq & \mathrm{H}^0\left(X,T\otimes L^ k\otimes \mathcal{I}(k\varphi )\right) \\ \subseteq & \mathrm{H}^0\left(X,T\otimes L^ k\otimes \mathcal{I}_{\infty }(k(1-\epsilon )\varphi )\right). \end{split} \]
Take an integer $N\in \mathbb {Z}_{>0}$ so that $ND$ is a divisor and $N\epsilon $ is an integer.
Let $\Delta '$ be the limit of a subsequence of $(\Delta _{k,T}(\theta ,\varphi ))_ k$, say the sequence defined by the indices $k_1,k_2,\ldots $. Thanks to theorem 19.7.4, it suffices to show that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$.
There exists $t\in \{ 0,1,\ldots ,N-1\} $ such that $k_ i\equiv t$ modulo $N$ for infinitely many $i$, up to replacing $(k_ i)_ i$ by a subsequence, we may assume that $k_ i\equiv t$ modulo $N$ for all $i$. Write $k_ i=N g_ i+t$. Then for large enough $i$, we have
\[ \begin{split} \mathrm{H}^0\left(X,T\otimes L^{-N+t} \otimes L^{N(g_ i+1)}\otimes \mathcal{I}_{\infty }\left(N(g_ i+1)\varphi \right)\right) \subseteq \mathrm{H}^0\left(X,T\otimes L^{k_ i}\otimes \mathcal{I}(k_ i\varphi )\right) \\ \subseteq \mathrm{H}^0\left(X,T\otimes L^ t \otimes L^{Ng_ i}\otimes \mathcal{I}_{\infty }\left(g_ i N(1-\epsilon )\varphi \right)\right). \end{split} \]
So
\[ \begin{split} (g_ i+1) \Delta _{g_ i+1,T\otimes L^{-N+t}}(NL-ND)+N(g_ i+1)\nu (D)\subseteq (Ng_ i+t)\Delta _{k_ i,T}(\theta ,\varphi )\\ \subseteq g_ i\Delta _{g_ i,T\otimes L^ t}\left(NL-N(1-\epsilon )D\right)+Ng_ i(1-\epsilon )\nu (D). \end{split} \]
Letting $i\to \infty $, by proposition 19.4.12,
\[ \Delta _{\nu }(L-D)+\nu (D)\subseteq \Delta '\subseteq \Delta _{\nu }(L-(1-\epsilon )D)+(1-\epsilon )\nu (D). \]
Letting $\epsilon \to 0+$, using the continuity of classical Okounkov bodies in the big cone recalled in theorem 19.5.4, we find that
\[ \Delta '=\Delta _{\nu }(L-D)+\nu (D)=\Delta _{\nu }(\theta ,\varphi ), \]
where we applied 19.4.23 as well. Our assertion follows. $\square$
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