PomeProposition 19.4.12.1ED4AF0D9A23 For any holomorphic line bundle $T$ on $X$, as $k\to \infty $
\[ \Delta _{k,T}(L)\xrightarrow {d_{\mathrm{Haus}}} \Delta _{\nu }(L). \]
Proposition 19.4.121ED4AF0D9A23
Proof. As $L$ is big, we can take $k_0\in \mathbb {Z}_{>0}$ so that
6BF24D61A02C$T^{-1}\otimes L^{k_0}$ admits a non-zero global holomorphic section $s_0$, and
0D44384F7F7B$T\otimes L^{k_0}$ admits a non-zero global holomorphic section $s_1$.
Then for $k\in \mathbb {Z}_{>k_0}$, we have injective linear maps
\[ \mathrm{H}^0(X,L^{k-k_0})\xrightarrow {\times s_1}\mathrm{H}^0(X,T\otimes L^{k})\xrightarrow {\times s_0}\mathrm{H}^0(X, L^{k+k_0}). \]
It follows that
\[ (k-k_0)\Delta _{k-k_0}(L)+\nu (s_1)\subseteq k\Delta _{k,T}(L)\subseteq (k+k_0)\Delta _{k+k_0}(L)-\nu (s_0). \]
Using theorem 19.2.5, we conclude. $\square$
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