Lemma 3.25 (00LC) · ScalingStacks

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Lemma 3.25. Let $\phi$ be an asymptotic Fubini-Study metric on $L$ . Then $\phi|_{Y}$ is an asymptotic Fubini-Study metric on $L|_{Y}$ .

00LD

Proof. Suppose that $\phi$ is the pointwise limit on $X^{\mathrm{an}}$ of $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})\}$ , where $\{\lVert\mathord{\cdot}\rVert_{n}\}$ are norms on $V_{n}(L)$ . Then $L|_{Y}$ is the pointwise limit of $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n,X|Y})\}$ on $Y^{\mathrm{an}}$ . ∎