Let $k$ be a field equipped with a complete non-Archimedean absolute value $\lvert\mathord{\cdot}\rvert$, which is not trivial. Let $X$ be an irreducible scheme of finite type over $\spec k$. One denotes by $X^{\mathrm{an}}$ the Berkovich analytic space associated with $X$ and by $j_{X}:X^{\mathrm{an}}\rightarrow X$ the map sending any $x\in X^{\mathrm{an}}$ to its associated scheme point.

1. 1.

   Let $V=\bigoplus_{n\in\mathbb{N}}V_{n}$ be a graded $k$-algebra. Let $\vvvert\mathord{\cdot}\vvvert$ be an algebra seminorm on $V$. For every $n\in\mathbb{N}$, this algebra seminorm induces by restriction a seminorm on the $k$-vector space $V_{n}$, denoted by $\lVert\mathord{\cdot}\rVert_{n}$. As $\vvvert\mathord{\cdot}\vvvert$ is sub-multiplicative, these seminorms satisfy the property

   $$\forall(m,n)\in\mathbb{N}^{2},s_{m}\in V_{m},s_{n}\in V_{n},\quad\lVert s_{m}\cdot s_{n}\rVert_{m+n}\leqslant\lVert s_{m}\rVert_{m}\cdot\lVert s_{n}\rVert_{n};\quad\lVert 1\rVert_{0}=1$$

   Conversly, given a familly of ultrametric seminorms $\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}}$ on $k$-vector spaces $V_{n}$ of $V$ satisfying these properties, the seminorm on the graded $k$-algebra $\bigoplus_{n\in\mathbb{N}}V_{n}$ defined by

   $$\forall\underline{s}=(s_{n})_{n\in\mathbb{N}},\quad\vvvert\underline{s}\vvvert:=\sup_{n\in\mathbb{N}}\lVert s_{n}\rVert_{n}$$

   is submultiplicative, hence is an algebra seminorm on $V$. In fact, let $\underline{s}=(s_{n})_{n\in\mathbb{N}}$ and $\underline{t}=(t_{n})_{n\in\mathbb{N}}$ be two elements of $\bigoplus_{n\in\mathbb{N}}V_{n}$ and $\underline{u}=(u_{n})_{n\in\mathbb{N}}=\underline{s}\cdot\underline{t}$, then one has

   $$u_{n}=\sum_{\begin{subarray}{c}(p,q)\in\mathbb{N}^{2}\\ p+q=n\end{subarray}}s_{p}\cdot t_{q}.$$

   By using the fact that the seminorm $\lVert\mathord{\cdot}\rVert_{n}$ is ultrametric, one obtains that

   $$\lVert u_{n}\rVert_{n}\leqslant\max_{\begin{subarray}{c}(p,q)\in\mathbb{N}^{2}\\ p+q=n\end{subarray}}\lVert s_{p}\cdot t_{q}\rVert_{n}\leqslant\max_{\begin{subarray}{c}(p,q)\in\mathbb{N}^{2}\\ p+q=n\end{subarray}}\lVert s_{p}\rVert_{p}\cdot\lVert t_{q}\rVert_{q},$$

   so $\vvvert\underline{u}\vvvert$ is bounded from above by $\vvvert\underline{s}\vvvert\cdot\vvvert\underline{t}\vvvert$. Denote by $\widehat{V}(\vvvert\mathord{\cdot}\vvvert)$ the separated completion of the seminormed algebra $(\bigoplus_{n\in\mathbb{N}}V_{n},\vvvert\mathord{\cdot}\vvvert)$.

   One denotes by $\Upsilon(V_{{\scriptscriptstyle\bullet}}(L))$ the set of all power-multiplicative ultrametric algebra norms $\vvvert\mathord{\cdot}\vvvert$ which satisfies

   $$\forall(s_{n})\in V_{{\scriptscriptstyle\bullet}}(L),\ \vvvert(s_{n})\vvvert=\sup_{n\in\mathbb{N}}\vvvert s_{n}\vvvert.$$

   This last condition is equivalent to the orthogonality of $\{V_{n}\}_{n\in\mathbb{N}}$ as $k$-linear subspaces.

2. 2.

   For any invertible $\mathscr{O}_{X}$-module $L$, one denotes by $V_{\scriptscriptstyle\bullet}(L)$ the graded $k$-algebra $\bigoplus_{n\in\mathbb{N}}V_{n}(L)$ where $V_{n}(L):=H^{0}(X,L^{\otimes n})$. As $X$ is irreducible, $V_{0}(L)=k$.

   Let $f:Y\rightarrow X$ be a morphism of $k$-schemes. The morphism of $\mathscr{O}_{X}$-modules $L^{\otimes n}\to f_{*}f^{*}(L^{\otimes n})$ induces linear maps of $k$-vector spaces $V_{n}(L)\rightarrow V_{n}(f^{*}L)$ and graded homomorphism of degree $0$ of graded-$k$-algebras $V_{\scriptscriptstyle\bullet}(L)\rightarrow V_{\scriptscriptstyle\bullet}(f^{*}L)$. Denote by $V_{n}(L_{X|Y})$ and $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ the image vector space and image graded algebra.

   One denotes by $Tot(L^{\vee})$ the scheme $\spec(\sym_{\mathscr{O}_{X}}L)$ over $\spec k$, by $\pi_{X}$ the canonical morphism of schemes $Tot(L^{\vee})\rightarrow X$, and by $\mathbb{O}$ the reduced closed subscheme of zero section. If the graded $k$-algebra $V_{{\scriptscriptstyle\bullet}}(L)$ is of finite type, then one denotes by $\boldsymbol{0}$ the closed point of $\spec V_{{\scriptscriptstyle\bullet}}(L)$ given by the maximal ideal $V_{\geq 1}(L)$, and by $p_{X}(\mathbf{0})$ the canonical blow-up morphism $Tot(L^{\vee})\rightarrow\spec V_{{\scriptscriptstyle\bullet}}(L)$ along the sub-scheme $\mathbf{0}$.

   Denote the integer $\dim_{k}V_{n}(L)-1$ by $d_{n}$. If $L^{\otimes n}$ is globally generated, there is a morphism induced by $V_{n}(L)$

   $$\iota_{n}:\ X\rightarrow\mathbb{P}(V_{n}(L))\simeq\mathbb{P}^{d_{n}}.$$

3. 3.

   Let $L$ be an invertible $\mathscr{O}_{X}$-module. Let $\mathcal{F}_{X}$ be the sheaf of real-valued functions on $X^{\mathrm{an}}$. By pseudometric on $L$ one refers to a morphism of sheaves of sets $\phi:L\rightarrow j_{X,*}(\mathcal{F}_{X})$ such that, for any $x\in X^{\mathrm{an}}$, the map $\lvert\mathord{\cdot}\rvert_{\phi}(x):L(x)\rightarrow\mathbb{R}$ induced by $\phi$ is a seminorm on the one-dimensional vector space $L(x)$ over $\widehat{\kappa}(x)$. If, for any $x\in X^{\mathrm{an}}$, the map $\lvert\mathord{\cdot}\rvert_{\phi}(x)$ is a norm, one says that $\phi$ is a metric.

   We say that a pseudometric $\phi$ is (upper semi-)continuous if, for any Zariski open subset $U$ of $X$ and any section $s\in\Gamma(X,L)$, the function $(x\in U^{\mathrm{an}})\rightarrow|s|_{\phi}(x)$ is (upper semi-)continuous.

4. 4.

   The pair $(L,\phi)$ is called a pseudometrized invertible $\mathscr{O}_{X}$-module. For any $\epsilon\in\mathbb{R}$, the following subset of $Tot(L^{\vee})^{\mathrm{an}}$, equipped with induced topology

   $$\{(x,e^{\vee}(x))\in Tot(L^{\vee})^{\mathrm{an}}:\lvert e^{\vee}(s)\rvert(x)\leq\lvert s\rvert_{\phi}(x)\cdot\mathrm{e}^{\epsilon}\ (resp.<\mathrm{e}^{\epsilon})\}$$

   is called the dual closed (resp. open) disc bundle of radius $\mathrm{e}^{\epsilon}$ of the pseudometrized pair $(L,\phi)$, where $s$ is a local section of $L$. We denote it by $\overline{\mathbb{D}}^{\vee}(L,\phi,\epsilon)$ (resp. $\mathbb{D}^{\vee}(L,\phi,\epsilon))$.

5. 5.

   Let $f:Y\rightarrow X$ be a morphism of separated $k$-schemes of finite type. Let $L$ be an invertible $\mathscr{O}_{X}$-module, equipped with a pseudometric $\phi$. We define a pseudometric $f^{*}\phi$ on $f^{*}(L)$ such that, for any section $s$ of $L$ on a Zariski open subset $U$ of $X$, one has

   $$\forall\,y\in f^{-1}(U)^{\mathrm{an}},\quad|f^{*}(s)|_{f^{*}\phi}(y)=|s|_{\phi}(f^{\mathrm{an}}(y)).$$

   Since $f^{\mathrm{an}}:Y^{\mathrm{an}}\rightarrow X^{\mathrm{an}}$ is continuous (Proposition 2.94), if the metric $\phi$ is continuous, so is $f^{*}\phi$. If $Y$ is a subscheme of $X$ and if $f:Y\rightarrow X$ is the canonical immersion, the restricted metric $f^{*}\phi$ is also denoted by $\phi|_{Y}$.

6. 6.

   Any map $f:X^{\mathrm{an}}\rightarrow\mathbb{R}\cup\{+\infty\}$ determines a pseudometric $\tau_{f}$ on $\mathscr{O}_{X}$ such that, for any regular function $a$ of $X$ on a Zariski open subset $U$, one has (with the convention $\mathrm{e}^{-\infty}=0$)

   $$\forall\,x\in U^{\mathrm{an}},\quad\lvert a\rvert_{\phi_{f}}(x)=|a|(x)\cdot\mathrm{e}^{-f(x)}.$$

   Note that $f\mapsto\tau_{f}$ defines a bijection between the set of maps $X^{\mathrm{an}}\rightarrow\mathbb{R}\cup\{+\infty\}$ and that of pseudometrics on $\mathscr{O}_{X}$, which maps the set of real-valued functions bijectively to that of pseudometrics on $\mathscr{O}_{X}$. Moreover, a pseudometric $\phi_{f}$ is continuous if and only if $f$ is continuous on $X^{\mathrm{an}}$. The trivial invertible sheaf $\mathscr{O}_{X}$ equipped with the pseudometric $\tau_{f}$ is denoted by $\mathscr{O}_{X}(f)$. The metric corresponding to the identically vanishing function is called the trivial metric on $\mathscr{O}_{X}$.

7. 7.

   Let $\phi_{1}$ and $\phi_{2}$ be two metrics on $L$. The distance of these two pseudometrics is a generalized positive real number (in $\mathbb{R}_{+}\cup\{+\infty\}$) defined by

   $$\dist(\phi_{1},\phi_{2})=\sup_{x\in X^{\mathrm{an}}}\Big|\log\Big|\frac{\phi_{1}(x)}{\phi_{2}(x)}\Big|_{\widehat{\kappa}(x)}\Big|.$$

   If $X$ is proper and $\phi_{1}$, $\phi_{2}$ are continuous metrics, then $\dist(\phi_{1},\phi_{2})\in\mathbb{R}_{+}$.

8. 8.

   Let $L_{1}$ and $L_{2}$ be invertible $\mathscr{O}_{X}$-modules, and $\phi_{1}$ and $\phi_{2}$ be pseudometrics on $L_{1}$ and $L_{2}$ respectively. The pseudometric $\phi_{1}$ and $\phi_{2}$ induce by passing to tensor product a metric on $L_{1}\otimes L_{2}$, denoted by $\phi_{1}+\phi_{2}$. For any Zariski open subset $U$ of $X$ and any $(s_{1},s_{2})\in\Gamma(U,L_{1})\times\Gamma(U,L_{2})$, one has

   $$\forall\,x\in U^{\mathrm{an}},\quad\lvert s_{1}\cdot s_{2}\rvert_{\phi_{1}+\phi_{2}}(x)=\lvert s_{1}\rvert_{\phi_{1}}(x)\cdot\lvert s_{2}\rvert_{\phi_{2}}(x).$$

   If $\phi_{1}$ and $\phi_{2}$ are continuous, then $\phi_{1}+\phi_{2}$ is also continuous.

   In particular, for any $\epsilon\in\mathbb{R}$, we denote by $\phi(\epsilon)$ the pseudometric $\phi+\tau_{\mathrm{e}^{\epsilon}}$ on $L\otimes\mathscr{O}_{X}=L$.

   Moreover, any metric $\phi$ on $L$ determines by passing to its dual a metric $-\phi$ on $L^{\vee}$ such that, for any Zariski open subset $U$ of $X$ and any $(s,\alpha)\in\Gamma(U,L)\times\Gamma(U,L^{\vee})$, one has

   $$\forall\,x\in U^{\mathrm{an}},\quad\lvert\alpha(s)\rvert(x)=\lvert\alpha\rvert_{-\phi}(x)\cdot\lvert s\rvert_{\phi}(x).$$

   If the metric $\phi$ is continuous, so is $-\phi$.

9. 9.

   Let $L$ be an invertible $\mathscr{O}_{X}$-module, and $n\in\mathbb{N}\setminus\{0\}$. A pseudometric $\phi$ on $L$ determines by tensor power a pseudometric on $L^{\otimes n}$ for any $n\in\mathbb{N}\setminus\{0\}$, denoted by $n\phi$. By convention, $0\phi$ denotes the trivial metric on $L^{\otimes 0}\cong\mathscr{O}_{X}$ (see 6. above).

   Similarly, assume given a pseudometric $\phi$ on $L^{\otimes n}$. We denote by $\frac{1}{n}\phi$ the pseudometric on $L$ such that, for any Zariski open subset $U$ of $X$ and any section $s\in\Gamma(U,L)$, one has

   $$\lVert s\rVert_{\frac{1}{n}\phi}=\lVert s^{n}\rVert_{\phi}^{1/n}.$$

   If the pseudometric $\phi$ is continuous, then also is $\frac{1}{n}\phi$.

10. 10.

    Let $L$ be an invertible $\mathscr{O}_{X}$-module. For any $n$ such that $L^{\otimes n}$ is globally generated, let $\lVert\mathord{\cdot}\rVert_{n}$ be a norm on $V_{n}(L)$. For any $x\in X^{\mathrm{an}}$, the evaluation map

    $$V_{n}(L)\otimes_{k}\widehat{\kappa}(x)\longrightarrow L^{\otimes n}(x)$$

    induces a quotient norm of $\lVert\mathord{\cdot}\rVert_{n,\widehat{\kappa}(x)}$ on the $\widehat{\kappa}(x)$-vector space $L^{\otimes n}(x)$, denoted by $\lVert\mathord{\cdot}\rVert_{n,X|x}$. This gives rise to a metric on $L^{\otimes n}$, which we call the Fubini-Study metric associated with $\lVert\mathord{\cdot}\rVert_{n}$ on $L^{\otimes n}$, denoted by $\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})(x)$. The metric $\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})$ on $L$ is called the $n$-th Fubini-Study metric associated with $\lVert\mathord{\cdot}\rVert_{n}$ on $L$.

    In particular, let $\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}}$ be a basis of $V_{n}(L)$ and let $\lVert\mathord{\cdot}\rVert_{n}$ be a ultrametric norm on $V_{1}(L)$ with respect to which $\{s_{1,j}\}_{j\in\{0,\dots,d_{1}\}}$ is an orthogonal basis. Such a metric $\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})$ is studied and heavily used in [CMor18], and is said to be diagonalizable in [BE18].

11. 11.

    Similarly, let $\vvvert\mathord{\cdot}\vvvert$ be an algebra norm on $V_{{\scriptscriptstyle\bullet}}(L)$. For any $x\in X^{\mathrm{an}}$, the evaluation map induces a $\widehat{\kappa}(x)$-algebra homomorphism

    $$V_{{\scriptscriptstyle\bullet}}(L)\otimes\widehat{\kappa}(x)\rightarrow\bigoplus_{n\in\mathbb{N}}L^{\otimes n}(x)=:V_{{\scriptscriptstyle\bullet}}(L)(x)$$

    This algebra homomorphism induces a quotient algebra norm of the scalar extension $\vvvert\mathord{\cdot}\vvvert_{\widehat{\kappa}(x)}$ on $V_{{\scriptscriptstyle\bullet}}(L)(x)$, denoted by $\vvvert\mathord{\cdot}\vvvert_{X|x}$. Let $\widehat{V}(L,\vvvert\mathord{\cdot}\vvvert)(x)$ denote the separated completion of $(V_{{\scriptscriptstyle\bullet}}(L)(x),\vvvert\mathord{\cdot}\vvvert_{X|x})$. Once a non-zero element $e_{1}(x)\in L(x)$ is chosen, the second algebra can be identified with $\widehat{\kappa}(x)[T]$ by sending $e_{1}(x)$ to $T$.

12. 12.

    Let $L$ be an invertible $\mathscr{O}_{X}$ module. Let $\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}}$ be a familly of norms on $\{V_{n}(L)\}_{n\in\mathbb{N}}$. If the sequence of metrics $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})\}_{n\in\mathbb{N}}$ converges pointwisely to a limit metric, we denote it by $\mathcal{P}(\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}})$ and call it the Fubini-Study envelop metric associated with $\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}}$.

    Note that if the convergence is uniform for $x\in X^{\mathrm{an}}$, since Fubini-Study metrics $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})\}_{n\in\mathbb{N}}$ are continuous, the envelop metric will also be continuous. Conversely, if $X$ is proper over $\spec k$ and the envelop metric is continuous, then the convergence is uniform in $x\in X^{\mathrm{an}}$ as $X^{\mathrm{an}}$ is Hausdorff and compact by Theorem 2.95. A metric $\phi$ on $L$ is asymptotic Fubini-Study if it is a Fubini-Study envelop metric and the convergence is uniform for $x\in X^{\mathrm{an}}$ (see [BE18, Definition 6.1]). Asymptotic Fubini-Study metrics are thus continuous. Note that asymptotic Fubini-Study property in this sense is equivalent to the notion of semipositive metric by the terminology of [CMor18]. We refer to [BFJ16, §5.4] and [BE18, §6.1] for a clear discussion of other various notions of semipositivity that have been proposed and studied in [Zha95], [Gu98], [Mor11], [BFJ16], [CLD12], [BMPS], [CMor18], [GM16] and literature therein.

    In particular, let $\vvvert\mathord{\cdot}\vvvert$ be an algebra seminorm on $V_{{\scriptscriptstyle\bullet}}(L)$, and let $\{\lVert\mathord{\cdot}\rVert_{n}\}_{n\in\mathbb{N}}$ be the associated familly of seminorms on $\{V_{n}(L)\}_{n\in\mathbb{N}}$. The seminorms $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})(x)\}_{n\in\mathbb{N}}$ on $L(x)$ satisfy sub-multiplicative property, so they converges to a limit seminorm on $L(x)$. This gives rise to a pseuodometric on $L$, called the Fubini-Study envelop pseudometric associated with $\vvvert\mathord{\cdot}\vvvert$. We denote it by $\mathcal{P}(\vvvert\mathord{\cdot}\vvvert)$. It is not necessarily continuous.

13. 13.

    Assume that $X$ is proper over $\spec k$. Note that $X^{\mathrm{an}}$ is then a compact Hausdorff space (see [Ber, Theorem 3.4.8]). Let $L$ be an invertible $\mathscr{O}_{X}$-module and $\phi$ be an upper semicontinuous metric on $L$ (see 3. above). As $X^{\mathrm{an}}$ is compact, any upper semicontinuous function on $X^{\mathrm{an}}$ is bounded from above and attains its maximal value. In particular, for any $s\in V_{1}(L)$, one has

    $$\lVert s\rVert_{\phi}:=\sup_{x\in X^{\mathrm{an}}}\lvert s\rvert_{\phi}(x)<+\infty.$$

    Moreover, $\lVert\mathord{\cdot}\rVert_{\phi}:V_{1}(L)\rightarrow\mathbb{R}_{\geq 0}$ is a norm on $V_{1}(L)$. This norm is ultrametric since the absolute value $\lvert\mathord{\cdot}\rvert$ on $k$ is non-Archimedean and $L$ is of rank $1$. We denote by $\vvvert\mathord{\cdot}\vvvert_{\phi}$ the norm on the $k$-vector space $V_{\scriptscriptstyle\bullet}(L)$ defined as

    $$\forall\,\underline{s}=(s_{n})_{n\in\mathbb{N}}\in V_{\scriptscriptstyle\bullet}(L),\quad\vvvert\underline{s}\vvvert_{\phi}:=\sup_{n\in\mathbb{N}}\lVert s_{n}\rVert_{n\phi}.$$

    Note that the $k$-algebra $V_{\scriptscriptstyle\bullet}(L)$ equipped with this norm forms a normed $k$-algebra. In fact, since $0\phi$ is the trivial metric on $\mathscr{O}_{X}$, one has $\vvvert\mathbf{1}\vvvert_{0\phi}=1$, where $\mathbf{1}$ denotes the unit section of $\mathscr{O}_{X}$. Moreover, for $s_{n}\in V_{n}(L)$ and $s_{m}\in V_{m}(L)$, we have

    $$\begin{split}\lVert s_{m}\cdot s_{n}\rVert_{n\phi}=&\sup_{x\in X^{\mathrm{an}}}\lvert s_{m}\rvert_{m\phi}(x)\cdot\lvert s_{m}\rvert_{m\phi}(x)\\ \leqslant&\sup_{x\in X^{\mathrm{an}}}\lvert s_{m}\rvert_{m\phi}(x)\cdot\sup_{x\in X^{\mathrm{an}}}\lvert s_{n}\rvert_{n\phi}(x)=\lVert s_{m}\rVert_{m\phi}\cdot\lVert s_{n}\rVert_{n\phi}\end{split}$$

    Then $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is an algebra norm by 1.

    In addition, the familly of norms $\{\lVert\mathord{\cdot}\rVert_{n\phi}\}_{n\in\mathbb{N}}$ satisfies the power-multiplicative property for homogeneous elements:

    $$\forall s_{n}\in V_{n}(L),\forall m\in\mathbb{N},\quad\lVert(s_{n})^{m}\rVert_{nm\phi}=(\lVert s_{n}\rVert_{n\phi})^{m}.$$

    In fact, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is power-multiplicative also for non-homogeneous elements (see Proposition 3.1).

    Moreover, for every $n\in\mathbb{N}$, there is a metric $\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n\phi})$ on $L$, namely the $n$-th Fubini-Study metrics on $L$ associated with $\lVert\mathord{\cdot}\rVert_{n\phi}$.

    One denotes by $\widehat{V}_{\scriptscriptstyle\bullet}(L,\phi)$ the separated completion of the normed $k$-algebra $(V_{\scriptscriptstyle\bullet}(L),\vvvert\mathord{\cdot}\vvvert_{\phi})$. More generally, if $V_{\scriptscriptstyle\bullet}$ is a graded sub-$k$-algebra of $V_{\scriptscriptstyle\bullet}(L)$, by abuse of notation we still denote by $\vvvert\mathord{\cdot}\vvvert_{\phi}$ the restriction of the norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on $V_{\scriptscriptstyle\bullet}$ and denote by $\widehat{V}_{{\scriptscriptstyle\bullet}}(\phi)$ the separated completion of the normed algebra $(V_{\scriptscriptstyle\bullet},\vvvert\mathord{\cdot}\vvvert_{\phi})$. The restricted norm is also power-multiplicative.

    In particular, for any $N\in\mathbb{N}\setminus\{0\}$, if one takes $V_{{\scriptscriptstyle\bullet}}$ to be $\bigoplus_{n\in\mathbb{N}}V_{nN}(L)$, denoted by $V_{{\scriptscriptstyle\bullet}}^{(N)}(L)$, we denote by $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(N)}(L,\phi)$ the separated completion of $(V_{{\scriptscriptstyle\bullet}}^{(N)}(L),\vvvert\mathord{\cdot}\vvvert_{\phi})$.

14. 14.

    Assume that $X$ is proper over $\spec k$. Let $L$ be an invertible $\mathscr{O}_{X}$-module and $\phi$ be an upper semicontinuous metric on $L$. Let $f:Y\rightarrow X$ be a morphism of $k$-schemes of finite type. Let $\lVert\mathord{\cdot}\rVert_{n\phi,X|Y}$ be the quotient norm of $\lVert\mathord{\cdot}\rVert_{n\phi}$ on $V_{n}(L_{X|Y})$. It is ultrametric. Let $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$ be the quotient algebra norm of $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$. In fact,

    $$\forall\underline{t}=(t_{n})_{n\in\mathbb{N}}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\quad\vvvert\underline{t}\vvvert_{\phi,X|Y}=\sup_{\begin{subarray}{c}n\in\mathbb{N}\end{subarray}}\lVert t_{n}\rVert_{n\phi,X|Y}.$$

    One denotes by $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$ the separated completion of the normed $k$-algebra $(V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}})$. In particular, for any $N\in\mathbb{N}$, we denote by $V_{{\scriptscriptstyle\bullet}}^{(N)}(L_{X|Y})$ the graded $k$-algebra $\bigoplus_{n\in\mathbb{N}}V_{nN}(L_{X|Y})$. This is a sub-algebra of $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$. The restriction of $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$ on this sub-algebra is still denoted by $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$. One denotes by $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(N)}(L_{X|Y},\phi_{X|Y})$ the separated completion of $(V_{{\scriptscriptstyle\bullet}}^{(N)}(L_{X|Y}),\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y})$. In particular, if $f$ is the canonical immersion associated with a sub-scheme, we get a Banach $k$-algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(N)}(L_{X|Y},\phi_{X|Y})$.

In the rest of the article, we make the following assumptions. For algebro-geometric data: let $X$ be an integral projective scheme over $\spec k$ of pure dimension $d$, $Y$ be a reduced closed sub-scheme of $X$ with its canonical closed immersion $i_{Y}:Y\rightarrow X$, and $L$ be an ample invertible $\mathscr{O}_{X}$-module. One can find $M\in\mathbb{N}$ such that $L^{\otimes M}$ is very ample and for any $n\geq M$, the restriction map from $V_{n}(L)$ to $V_{n}(L|_{Y})$ is surjective, so $V_{n}(L|_{Y})=V_{n}(L_{X|Y})$. For the metric data, let $\phi$ be an upper-semicontinuous metric on $L$.