ScalingStacks

Theorem 4.1 . [027R]

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Theorem 4.1.

We assume that LL is very ample. Let ‖.‖\|\raisebox{1.72218pt}{.}\| be a norm of H0​(X,L)H^{0}(X,L) and hh a continuous metric of LanL^{\mathrm{an}} given by {|.|(H0​(X,L),‖.‖)quot​(x)}x∈Xan\big\{|\raisebox{1.72218pt}{.}|_{(H^{0}(X,L),\|\raisebox{1.20552pt}{.}\|)}^{\mathrm{quot}}(x)\big\}_{x\in X^{\mathrm{an}}}. Let YY be a closed subschme of XX and l∈H0​(Y,L|Y)l\in H^{0}(Y,\left.{L}\right|_{{Y}}). Then, for any ϵ>0\epsilon>0, there are a positive integer nn and s∈H0​(X,L⊗n)s\in H^{0}(X,{L}^{\otimes n}) such that s|Y=l⊗n\left.{s}\right|_{{Y}}={l}^{\otimes n} and ‖s‖h⊗n≤en​ϵ​(‖l‖Y,h)n\|s\|_{{h}^{\otimes n}}\leq e^{n\epsilon}(\|l\|_{Y,h})^{n}.

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