ScalingStacks

Lemma 3.7. Assume that $L$ is globally generated. Let $\lVert\mathord{\cdot}\rVert_{1}$ be a norm on $V_{1}(L)$ and let $\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})$ be the associated Fubini-Study metric. Then for any $x\in X^{\mathrm{an}}$ and $e(x)\in L(x)\setminus 0$,

$$\lvert e(x)\rvert_{\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})}=\inf_{\begin{subarray}{c}\lambda\in\widehat{\kappa}(x),\ s_{1}\in V_{1}(L)\\ s_{1}(x)=\lambda\cdot e(x)\end{subarray}}\lvert\lambda\rvert^{-1}\cdot\lVert s_{1}\rVert.$$

(with the convention that $0^{-1}=+\infty$)