ScalingStacks

Lemma 3.3 . [026S]

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Lemma 3.3.

Let hh be a continuous metric of LanL^{\mathrm{an}} (cf. Lemma 3.1). Let (e0,…,er)(e_{0},\ldots,e_{r}) be an orthogonal basis of VV with respect to ‖.‖\|\raisebox{1.72218pt}{.}\|. Then, for s∈H0​(X,L)s\in H^{0}(X,L),

|s|V¯quot​(x)=|s|h​(x)maxi=0,…,r⁡{|e~i|h​(x)‖ei‖}|s|_{\overline{V}}^{\mathrm{quot}}(x)=\frac{|s|_{h}(x)}{{\displaystyle\max_{i=0,\ldots,r}\left\{\frac{|\tilde{e}_{i}|_{h}(x)}{\|e_{i}\|}\right\}}}

on XanX^{\mathrm{an}}.

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