Definition 19.4.2.0F5DB984097B We will write
\[ \begin{aligned} \Gamma (\theta ,\varphi )\coloneqq & \left\{ (\nu (s),k) : k\in \mathbb {N}, s\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}(k\varphi ))^{\times } \right\} ,\\ \Delta _ k(\theta ,\varphi )\coloneqq & \mathrm{Conv}\left\{ k^{-1}\nu (s): s\in \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)^{\times } \right\} \subseteq \mathbb {R}^ n, \end{aligned} \]
where the second set is defined only for those $k$ such that $\mathrm{H}^0(X,L^ k\otimes \mathcal{I}(k\varphi ))\neq 0$.
When $\varphi =V_{\theta }$, we simply write $\Gamma (L)$ and $\Delta _ k(L)$ instead.
Here and in the sequel, the cross notation means excluding $0$. Here $\mathrm{Conv}$ denotes the convex hull. For large enough $k$, $\Delta _ k(\theta ,\varphi )$ is non-empty thanks to theorem
Definition 19.4.3.DCCC835656D0 Assume that $\varphi $ has analytic singularities. We define
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19.4.4
\begin{equation} \Gamma ^{\infty }(\theta ,\varphi )\coloneqq \left\{ (\nu (s),k):k\in \mathbb {N}, s\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}_{\infty }(k\varphi ))^{\times } \right\} . \end{equation}
For later use, we introduce a twisted version as well.
Definition 19.4.5.51F245B40203 If $T$ is a holomorphic line bundle on $X$, we introduce
\[ \begin{aligned} \Delta _{k,T}(\theta ,\varphi )\coloneqq & \mathrm{Conv}\left\{ k^{-1}\nu (s): s\in \mathrm{H}^0(X,T\otimes L^ k\otimes \mathcal{I}(k\varphi ))^{\times } \right\} \subseteq \mathbb {R}^ n,\\ \Delta _{k,T}(L)\coloneqq & \mathrm{Conv}\left\{ k^{-1}\nu (s): s\in \mathrm{H}^0(X,T\otimes L^ k)^{\times } \right\} \subseteq \mathbb {R}^ n \end{aligned} \]
for all $k\in \mathbb {Z}_{>0}$ such that the relevant spaces of sections are non-zero.
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