Section 1. (00MF) · ScalingStacks

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1. Introduction

Let $k$ be a field, $X$ be a projective scheme over $\spec k$ and $L$ be an ample invertible $\mathscr{O}_{X}$ -module. Let $Y$ be a closed subscheme of $X$ and $\mathscr{I}_{Y}$ be the corresponding ideal sheaf. Recall that Serre’s vanishing theorem (see for example [EGA, Théorème III.2.2.1]) asserts that there exists an integer $n_{Y}$ such that

H^{1}(X,\mathscr{I}_{Y}\otimes L^{\otimes n})=\{0\}

for any integer $n\geqslant n_{Y}$ . Therefore the exact sequence of coherent $\mathscr{O}_{X}$ -modules

induces a surjective $k$ -linear map from $H^{0}(X,L^{\otimes n})$ to $H^{0}(X,(\mathscr{O}_{X}/\mathscr{I}_{Y})\otimes L^{\otimes n})$ , which coincides with the restriction map $H^{0}(X,L^{\otimes n})\rightarrow H^{0}(Y,L|_{Y}^{\otimes n})$ if we identify $H^{0}(Y,L|_{Y}^{\otimes n})$ with $H^{0}(X,(\mathscr{O}_{X}/\mathscr{I}_{Y})\otimes L^{\otimes n})$ . In other words, for any integer $n\geqslant n_{Y}$ , any global section of $L|_{Y}^{\otimes n}$ extends to a global section of $L^{\otimes n}$ . We denote by $H^{0}(Y,L_{X|Y}^{\otimes n})$ the image vector space of restriction map

H^{0}(Y,L_{X|Y}^{\otimes n}):=\mathrm{Image}(H^{0}(X,L^{\otimes n})\longrightarrow H^{0}(Y,L|_{Y}^{\otimes n})).

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\}.

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.

Note that, to obtain an estimate of the form (3) for any closed point $Y$ , the semipositivity of the metric $\phi$ is actually necessary. In fact, for sufficiently positive integer $n$ , the ample invertible $\mathscr{O}_{X}$ -module determines a closed embedding $\iota_{n}:X\rightarrow\mathbb{P}(H^{0}(X,L^{\otimes n}))$ . The norm $\lVert\mathord{\cdot}\rVert_{n\phi}$ on $H^{0}(X,L^{\otimes n})$ defines a Fubini-Study metric on the universal invertible sheaf of the projective space $H^{0}(X,L^{\otimes n})$ . Denote by $\phi_{n}$ the continuous metric on $L$ the $n$ -th tensor power of which identifies with the restriction of the Fubini-Study metric. Then the estimate (3) in the case where $Y$ is a single closed point $\{x\}$ implies that the sequence of norms $\lvert\mathord{\cdot}\rvert_{\phi_{n}}(x)$ converges to $\lvert\mathord{\cdot}\rvert_{\phi}(x)$ . Moreover, it can be shown the subsequence $(|\mathord{\cdot}|_{\phi_{2^{m}}}(x))_{m\in\mathbb{N}}$ is decreasing. Therefore, the semipositivity of Fubini-Study metrics implies that of the metric $\phi$ .

The problem of metric extension has various applications, not only in complex analytic geometry, but also in Arakelov geometry. It is a key ingredient in the proof of the arithmetic Hilbert-Samuel theorem, see [AB95]. It has also been applied in the proof of Nakai-Moishezon criterion of arithmetic ampleness, cf. [Zha95], see also [Mor11].

From the adelic point of view of Arakelov geometry, one can replace the integral models of arithmetic objects by a family of Berkovich analytic objects (possibly equipped with metrics) parametrised by the set of finite places of a number field. The advantage of the adelic approach consists in treating all places of a number field in a uniform way. This motivates the research of the non-Archimedean analogue of notions and results in complex analytic geometry. It turns out that many usual analytic tools (such as $L^{2}$ estimates) do not work well in the non-Archimedean setting, and often new ideas are needed to develop the non-Archimedean analogue of complex analytic geometry and to unify the arguments in both settings.

In this article we undertake a study of the metric extension problem in (1) in the non-Archimedean analytic setting. Various notions of semipositivity of a metric have been proposed (see §3.1 12.), we adopt the one as being a uniform limit of Fubini-Study metrics. We establish the following result (see Theorem 4.5, Theorem 5.11), which improves considerably the metric extension theorem of [CMor18].

00H4

Theorem 1.1. Let $\phi$ be an asymptotic Fubini-Study metric on $L$ . Then for any $\epsilon>0$ , there exists $n_{Y}\in\mathbb{N}$ such that, for any $n\geq n_{Y}$ and any $t_{n}\in H^{0}(Y,L|_{Y}^{\otimes n})$ , there exits $s_{n}\in V_{n}(L)$ such that $s_{n}|_{Y}=t_{n}$ and

\lVert s_{n}\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}

We mimic Grauert’s argument as in [Bos01] and in [Ran06], though some parts of the argument require adaptations. On the restricted section algebra, namely

V_{{\scriptscriptstyle\bullet}}(L_{X|Y}):=\bigoplus_{n\in\mathbb{N}}H^{0}(Y,L_{X|Y}^{\otimes n})=\mathrm{Image}(V_{{\scriptscriptstyle\bullet}}(L)\longrightarrow V_{{\scriptscriptstyle\bullet}}(L|_{Y})),

we gather all norms on graded pieces together, to have two algebra norms defined as follows

\forall\underline{t}=(t_{n})_{n\in\mathbb{N}}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\ \vvvert\underline{t}\vvvert_{\phi|_{Y}}:=\sup_{n\in\mathbb{N}}\lVert t_{n}\rVert_{n\phi|_{Y}},\ \vvvert\underline{t}\vvvert_{\phi,X|Y}:=\sup_{n\in\mathbb{N}}\lVert t_{n}\rVert_{n\phi,X|Y},

respectively. After completion with respect to these algebra norms, we get two commutative Banach algebras $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$ . The geometry of Berkovich spectra of these Banach algebras are studied in §3. They are intimately related to the dual unit disc bundle of $L$ with respect to the Fubini-Study envelop metric $\mathcal{P}(\phi)$ . It turns out that when $\phi$ is semipositive (more precisely, when $\phi$ is asymptotic Fubini-Study), the spectral seminorm of $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$ is $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ , and the two spectra coincide. This already permits us to reprove a statement of metric extension of Chen-Moriwaki in the non-trivial valuation case ([CMor18, Theorem 0.1], see Theorem 3.28). Note that since we equip $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ with Banach algebra norms instead of Fréchet algebra seminorms as used in [Bos01][Ran06], the spectrum gives rise to closed dual unit disc bundle $\overline{\mathbb{D}}^{\vee}(L|_{Y},\mathcal{P}(\phi)|_{Y})$ rather than the open dual disc bundle.

There are two independent methods to compare these two algebra norms. The geometric method in §4 exploits the holomorphic convexity of the aforementioned spectrum, and uses holomorphic functional calculus in non-Archimedean commutative Banach algebras, to construct a (continuous) Banach algebra homomorphism from $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$ to some perturbed version of $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y})$ . This construction is analogous to the use of Grauert’s vanishing theorem along with pseuodo-convexity to get continuous map between Fréchet seminormed coherent analytic sheaves in the $\mathbb{C}$ -analytic setting. In §5 we present an algebraic method with an extra assumption that $(k,\lvert\mathord{\cdot}\rvert)$ is discretely valued. The analytic convexity enters only in the semi-positivity of the continuous metric, equivalently the uniform approximation by Fubini-Study metrics. For $\phi$ being a Fubini-Study metric, we show directly by a delicate calculation that the two Banach algebra norms are affinoid algebra norms, which possesses strong finiteness property to give a uniform upper bound. The exact calculation depends heavily on the existence of a non-Archimedean orthogonal basis. We are unaware of any analogue of this affinoid algebra technique in the $\mathbb{C}$ -analytic setting, but it seems plausible to compare this with the use of Ohsawa-Takegoshi $L^{2}$ extension theorem (see Remark 5.10).

Note that our proof gives a sub-exponential upper bound as in (3). Whether the polynomial bound of (2) is obtainable in the non-Archimedean setting remains unexplored. It seems to us that the commutative Banach algebra techniques that we used here are insufficient to ameliorate the bound. We would like also to metion the recent work [MP17] which gives the hope of carrying out directly Grauert’s argument as in [Bos01] for the metric extension problem in the non-Archimedean setting.