ScalingStacks

In the complex analytic setting, namely when $(k,\lvert\mathord{\cdot}\rvert)$ is $\mathbb{C}$ equipped with the usual absolute value, the metric extension problem has been studied by different authors using various approaches. Assume that the metric $\phi$ is strictly positive (namely, for any local section $s$ of $L$ over an open subscheme $U$ of $X$, the function $(x\in U^{\mathrm{an}})\mapsto\log|s(x)|_{\phi}$ is strongly plurisubharmonic). In the case where $X$ is smooth, by Gromov’s theorem we can compare the sup norm $\lVert\mathord{\cdot}\rVert_{n\phi}$ to the $L^{2}$ norm $\lVert\mathord{\cdot}\rVert_{n\phi,L^{2}}$ defined as

$$\forall\,s\in H^{0}(X,L^{\otimes n}),\quad\lVert s\rVert_{n\phi,L^{2}}:=\bigg(\int_{X^{\mathrm{an}}}|s(x)|_{n\phi}(x)\,\mathrm{d}V\bigg)^{\frac{1}{2}},$$

where $\mathrm{d}V$ is a probability measure on $X^{\mathrm{an}}$ which is locally equivalent with Lebesgue measure with a smooth Radon-Nikodym density. Therefore, in the case where $X$ and $Y$ are both smooth, we can apply the Andreotti-Vesentini-Hömander’s $L^{2}$ technique or $L^{2}$-extension theorems of Ohsawa-Takegoshi type [OT87] and get an inequality (see for example [Tia90] and [Man93])

(2)

$$\lVert\mathord{\cdot}\rVert_{n\phi|_{Y}}\geqslant C^{\prime}(\phi,Y,X)n^{-d}\lVert\mathord{\cdot}\rVert_{n\phi,X|Y},\qquad n\geqslant n_{Y},$$

where $C^{\prime}(\phi,Y,X)$ is a positive constant. Alternatively, one can apply Grauert’s argument of pseudo-convexity of the (open) dual unit disc bundle of $(L,\phi)$ to produce, for any $\epsilon>0$, a slightly weaker inequality of the form

(3)

$$\lVert\mathord{\cdot}\rVert_{n\phi|_{Y}}\geqslant C_{\epsilon}(\phi,Y,X)\mathrm{e}^{-\epsilon n}\lVert\mathord{\cdot}\rVert_{n\phi,X|Y},\qquad n\geqslant n_{Y},$$

where $C_{\epsilon}(\phi,Y,X)$ are is a positive constant depending on $\epsilon$. We refer to [Bos01] and [Ran06] for more details.