Let $k$ be a field and $X$ be a projective scheme over $\operatorname{Spec}k$, equipped with an ample invertible $\mathcal{O}_{X}$-module $L$. If $Y$ is a closed subscheme of $X$, then for sufficiently positive integer $n$, any section $\ell$ of $L|_{Y}^{\otimes n}$ on $Y$ extends to a global section of $L^{\otimes n}$ on $X$. In other words, the restriction map $H^{0}(X,L^{\otimes n})\rightarrow H^{0}(Y,L|_{Y}^{\otimes n})$ is surjective. A simple proof of this result relies on Serre’s vanishing theorem, which ensures that $H^{1}(X,\mathcal{I}_{Y}\otimes L^{\otimes n})=0$ for sufficiently positive integer $n$, where $\mathcal{I}_{Y}$ is the ideal sheaf of $Y$.

The metrized version (with $k=\mathbb{C}$) of this result has been widely studied in the literature and has divers applications in complex analytic geometry and in arithmetic geometry. We assume that the ample invertible sheaf $L$ is equipped with a continuous (with respect to the analytic topology) metric $|\raisebox{1.72218pt}{.}|_{h}$, which induces a continuous metric $|\raisebox{1.72218pt}{.}|_{h^{n}}$ on each tensor power sheaf $L^{\otimes n}$, where $n\in\mathbb{N}$, $n\geqslant 1$. The metric $|\raisebox{1.72218pt}{.}|_{h^{n}}$ leads to a supremum norm $\|\raisebox{1.72218pt}{.}\|_{h^{n}}$ on the global section space $H^{0}(X,L)$ such that

$$\forall\,s\in H^{0}(X,L),\;\|s\|_{h^{n}}=\sup_{x\in X(\mathbb{C})}|s|_{h^{n}}(x).$$

Similarly, it induces a supremum norm $\|\raisebox{1.72218pt}{.}\|_{Y,h^{n}}$ on the space $H^{0}(Y,L|_{Y}^{\otimes n})$ with $\|s\|_{Y,h^{n}}=\sup_{y\in Y(\mathbb{C})}|s|_{h^{n}}(y)$. Note that for any section $s\in H^{0}(X,L^{\otimes n})$ one has $\|s{|_{Y}}\|_{Y,h^{n}}\leqslant\|s\|_{h^{n}}$. The metric extension problem consists of studying the extension of global sections of $L|_{Y}$ to those of $L$ with an estimation on the supremum norms. Note that a positivity condition on the metric $h$ is in general necessary to obtain interesting upper bounds. This problem has been studied by using Hörmander’s $L^{2}$ estimates (see [3] for example), under smoothness conditions on the metric. More recently, it has proved (without any regularity condition) that, if the metric $|\raisebox{1.72218pt}{.}|_{h}$ is semi-positive, then for any $\epsilon>0$ and any section $l\in H^{0}(Y,L|_{Y})$ there exists an integer $n\geqslant 1$ and $s\in H^{0}(X,L^{\otimes n})$ such that $s{|_{Y}}=l^{\otimes n}$ and that $\|s\|_{h^{n}}\leqslant e^{\epsilon n}\|s{|_{Y}}\|_{Y,h^{n}}$. We refer the readers to [10, 9] for more details.

The purpose of this article is to study the non-archimedean counterpart of the above problem. We will establish the following result (see Theorem 4.2 and Corollary 1.2).

The semi-positivity condition of the metric means that the metric $|\raisebox{1.72218pt}{.}|_{h}$ can be written as a uniform limit of Fubini-Study metrics. We will show that, if the absolute value $|\raisebox{1.72218pt}{.}|$ is non-trivial, then this condition is equivalent to the classical semi-positivity condition (namely uniform limit of nef model metrics, see Proposition 3.17) of Zhang [12], see also [4, 8], and compare with the complex analytic case [11]. The advantage of the new definition is that it also works in the trivial valuation case, where the model metrics are too restrictive. We use an argument of extension of scalars to the ring of formal Laurent series to obtain the result of the above theorem in the trivial valuation case.

The article is organized as follows. In the first section we introduce the notation of the article and prove some preliminary results, most of which concern finite dimensional normed vector spaces over a non-archimedean field. In the second section, we discuss some property of model metrics. In the third section, we study various properties of continuous metrics on an invertible sheaf, where an emphasis is made on the positivity of such metrics. Finally, in the fourth section, we prove the extension theorem.