Lemma 5.5 (00M2) · ScalingStacks

00M2

Lemma 5.5. Assume that $\phi$ is a Fubini-Study metric. For any $\epsilon>0$ , there exists $\boldsymbol{\delta}\in\mathbb{R}^{d+1}$ with $|\boldsymbol{\delta}|\leq\epsilon$ such that

\forall n\in\mathbb{N},\quad\dist(\lVert\mathord{\cdot}\rVert_{n\phi},\lVert\mathord{\cdot}\rVert_{n\phi(\boldsymbol{\delta})})\leq n\epsilon.

00M3

Proof. Choose an arbitrary $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$ . Then

\dist(\lVert\mathord{\cdot}\rVert_{\phi},\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})})\leq\epsilon.

By Proposition 3.12, we have

\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})}))=\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\phi(\boldsymbol{\delta}))\leq\epsilon,

by the assumption and Proposition 3.11,

\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi})=\phi,

so the conlusion holds. ∎