ScalingStacks

Assume that $k$ is equipped with a complete absolute value $\lvert\mathord{\cdot}\rvert$ and that $L$ is equipped with a continuous metric $\phi=(\lvert\mathord{\cdot}\rvert_{\phi}(x))_{x\in X^{\mathrm{an}}}$, which induces by tensor power a continuous metric $n\phi$ on each $L^{\otimes n}$, $n\in\mathbb{N}$. Suppose in addition that the schemes $X$ and $Y$ are integral. Then the space of global sections $H^{0}(X,L^{\otimes n})$ is naturally equipped with a supremum norm $\lVert\mathord{\cdot}\rVert_{n\phi}$ associated with $n\phi$, defined as follows

$$\forall\,s\in H^{0}(X,L^{\otimes n}),\quad\lVert s\rVert_{n\phi}:=\sup_{x\in X^{\mathrm{an}}}|s(x)|_{n\phi}(x).$$

We denote by $\phi|_{Y}$ the restriction of the metric $\phi$ on $L|_{Y}$. A supremum norm $\lVert\mathord{\cdot}\rVert_{n\phi|_{Y}}$ on $H^{0}(Y,L|_{Y}^{\otimes n})$ is defined in a similar way. The metric extension problem compares the norm $\lVert\mathord{\cdot}\rVert_{n\phi|_{Y}}$ to the quotient norm (denoted by $\lVert\mathord{\cdot}\rVert_{n\phi,X|Y}$) of $\lVert\mathord{\cdot}\rVert_{n\phi}$ induced by the restriction map $H^{0}(X,L^{\otimes n})\rightarrow H^{0}(X,L|_{Y}^{\otimes n})$ with $n\in\mathbb{N}$, $n\geqslant n_{Y}$. Note that by definition we always have $\lVert\mathord{\cdot}\rVert_{n\phi,X|Y}\geqslant\lVert\mathord{\cdot}\rVert_{n\phi|_{Y}}$ on $H^{0}(Y,L_{X|Y}^{\otimes n})$. Therefore the metric extension problem can be interpreted as finding a uniform upper bound for

(1)

$$\inf_{\begin{subarray}{c}s\in H^{0}(X,L^{\otimes n})\\ s|_{Y}=t\end{subarray}}\frac{\lVert s\rVert_{n\phi}}{\lVert t\rVert_{n\phi|_{Y}}},\quad t\in H^{0}(Y,L|_{Y}^{\otimes n})\setminus\{0\}.$$