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 $L$.

We show the power-multiplicativity of the supremum algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$, and that the normed section algebras are reduced Banach algebras.

We embed the Berkovich spectrum of normed section algebra into the analytification of the spectrum of the section algebra.

We study distances between Fubini-Study metrics, and gives explicit expressions for Fubini-Study metrics admitting non-Archimedean orthogonal basis. Many of the results here are also obtained in [CMor18] or in [BE18, Section 6].