3.4. The functions σ and μ on X an [027E]

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3.4. The functions $\sigma$ and $\mu$ on $X^{\operatorname{an}}$

Throughout this subsection, we assume that $X$ is projective. Let $\widehat{\operatorname{Pic}}_{C^{0}}(X)$ denote the group of isomorphism classes of pairs $(L,h)$ consisting of an invertible sheaf $L$ on $X$ and a continuous metric $h$ of $L^{\operatorname{an}}$ . Fix $\overline{L}=(L,h)\in\widehat{\operatorname{Pic}}_{C^{0}}(X)$ . We assume that $L$ is generated by global sections. We define $\sigma_{\overline{L}}(x)$ to be

\sigma_{\overline{L}}(x):=\log\left(\frac{|\raisebox{1.72218pt}{.}|_{h}^{\operatorname{quot}}(x)}{|\raisebox{1.72218pt}{.}|_{h}(x)}\right).

Lemma 3.13.

For $\overline{L}$ and $\overline{L}^{\prime}\in\widehat{\operatorname{Pic}}_{C^{0}}(X)$ such that both $L$ and $L^{\prime}$ are generated by global sections, we have the following:

(1)

$\sigma_{\overline{L}}\geq 0$ on $X^{\operatorname{an}}$ .
(2)

$\sigma_{\overline{L}\otimes\overline{L}^{\prime}}(x)\leq\sigma_{\overline{L}}(x)+\sigma_{\overline{L}^{\prime}}(x)$ for $x\in X^{\operatorname{an}}$ .
(3)

If $\overline{L}\simeq\overline{L}^{\prime}$ , then $\sigma_{\overline{L}}=\sigma_{\overline{L}^{\prime}}$ on $X^{\operatorname{an}}$ .

Proof.

(1) and (3) are obvious. (2) follows from (3) in Lemma 3.5. ∎

We assume that $L$ is semiample. We set

{\mathbb{N}}(L):=\left\{n\in{\mathbb{Z}}_{\geq 1}\mid\text{$L^{\otimes n}$ is generated by global sections}\right\}.

Note that ${\mathbb{N}}(L)\not=\emptyset$ and ${\mathbb{N}}(L)$ forms a subsemigroup of ${\mathbb{Z}}_{\geq 1}$ with respect to the addition of ${\mathbb{Z}}_{\geq 1}$ . For $x\in X^{\operatorname{an}}$ , we define $\mu_{\overline{L}}(x)$ to be

\mu_{\overline{L}}(x):=\inf\left\{\left.\frac{\sigma_{\overline{L}^{\otimes n}}(x)}{n}\ \right|\ n\in{\mathbb{N}}(L)\right\}.

Note that $\mu_{\overline{L}}$ is upper-semicontinuous on $X^{\operatorname{an}}$ because $\sigma_{\overline{L}^{\otimes n}}$ is continuous for all $n\in{\mathbb{N}}(L)$ . We set

\widehat{\operatorname{Pic}}_{C^{0}}^{+}(X):=\{(L,h)\in\widehat{\operatorname{Pic}}_{C^{0}}(X)\mid\text{$L$ is semiample}\}.

Note that $\widehat{\operatorname{Pic}}_{C^{0}}^{+}(X)$ forms a semigroup with respect to $\otimes$ .

Lemma 3.14.

Let $\overline{L}=(L,h)$ and $\overline{L}^{\prime}=(L^{\prime},h^{\prime})$ be elements of $\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)$ . Then we have the following:

(1)

$\mu_{\overline{L}}\geq 0$ on $X^{\operatorname{an}}$ .
(2)

${\displaystyle\mu_{\overline{L}}(x)=\lim\limits_{\begin{subarray}{c}n\to\infty\\ n\in{\mathbb{N}}(L)\end{subarray}}\frac{\sigma_{\overline{L}^{\otimes n}}(x)}{n}}$ for $x\in X^{\operatorname{an}}$ .
(3)

$\mu_{\overline{L}\otimes\overline{L}^{\prime}}(x)\leq\mu_{\overline{L}}(x)+\mu_{\overline{L}^{\prime}}(x)$ for $x\in X^{\operatorname{an}}$ .
(4)

If $\overline{L}\simeq\overline{L}^{\prime}$ , then $\mu_{\overline{L}}=\mu_{\overline{L}^{\prime}}$ on $X^{\operatorname{an}}$ .
(5)

For $n\geq 0$ , $\mu_{\overline{L}^{\otimes n}}=n\mu_{\overline{L}}$ on $X^{\operatorname{an}}$ .

Proof.

(1) follows from (1) in Lemma 3.13.

(2) Since $\sigma_{\overline{L}^{\otimes(n+n^{\prime})}}(x)\leq\sigma_{\overline{L}^{\otimes n}}(x)+\sigma_{\overline{L}^{\otimes n^{\prime}}}(x)$ for $n,n^{\prime}\in{\mathbb{N}}(L)$ by (2) in Lemma 3.13, the assertion follows from Fekete’s lemma.

(3) and (4) follow from (2) and (3) in Lemma 3.13 together with (2), respectively.

(5) If $n=0$ , then the assertion is obvious, so that we may assume that $n\geq 1$ . We fix $n_{0}\in{\mathbb{N}}(L)$ . Then $n_{0}\in{\mathbb{N}}(L^{\otimes n})$ . Thus, by (2),

\mu_{\overline{L}^{\otimes n}}(x)=\lim_{m\to\infty}\frac{\sigma_{L^{\otimes mn_{0}n}}(x)}{mn_{0}}=n\lim_{m\to\infty}\frac{\sigma_{L^{\otimes mn_{0}n}}(x)}{mn_{0}n}=n\mu_{\overline{L}}(x).

∎

We set $\widehat{\operatorname{Pic}}_{C^{0}}(X)_{{\mathbb{Q}}}:=\widehat{\operatorname{Pic}}_{C^{0}}(X)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ and

\widehat{\operatorname{Pic}}_{C^{0}}^{+}(X)_{{\mathbb{Q}}}:=\{(L,h)\in\widehat{\operatorname{Pic}}_{C^{0}}(X)_{{\mathbb{Q}}}\mid\text{$L$ is semiample}\}.

Let $\iota:\widehat{\operatorname{Pic}}_{C^{0}}(X)\to\widehat{\operatorname{Pic}}_{C^{0}}(X)_{{\mathbb{Q}}}$ be the canonical homomorphism. For $\overline{L}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}$ , we choose a positive integer $n$ and $\overline{L}_{n}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)$ with $\iota(\overline{L}_{n})=\overline{L}^{\otimes n}$ . Then $\mu_{\overline{L}_{n}}(x)/n$ does not depend on the choice of $n$ and $\overline{L}_{n}$ . Indeed, let us choose another $n^{\prime}\in{\mathbb{Z}}_{\geq 1}$ and $\overline{L}_{n^{\prime}}\in\widehat{\operatorname{Pic}}_{C^{0}}^{+}(X)$ with $\iota(\overline{L}_{n^{\prime}})=\overline{L}^{\otimes n^{\prime}}$ . As $\iota(\overline{L}_{n}^{\otimes n^{\prime}})=\iota(\overline{L}_{n^{\prime}}^{\otimes n})=\overline{L}^{\otimes nn^{\prime}}$ , there is a positive integer $m$ such that $\overline{L}_{n}^{\otimes mn^{\prime}}=\overline{L}_{n^{\prime}}^{\otimes mn}$ . By (5) in Lemma 3.14,

mn^{\prime}\mu_{\overline{L}_{n}}(x)=\mu_{\overline{L}_{n}^{\otimes mn^{\prime}}}(x)=\mu_{\overline{L}_{n^{\prime}}^{\otimes mn}}(x)=mn\mu_{\overline{L}_{n^{\prime}}}(x),

that is, $\mu_{\overline{L}_{n}}(x)/n=\mu_{\overline{L}_{n^{\prime}}}(x)/n^{\prime}$ , as required. By abuse of notation, it is also denoted by $\mu_{\overline{L}}(x)$ .

Lemma 3.15.

For $\overline{L},\overline{L}^{\prime}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}$ , we have the following:

(1)

$\mu_{\overline{L}\otimes\overline{L}^{\prime}}(x)\leq\mu_{\overline{L}}(x)+\mu_{\overline{L}^{\prime}}(x)$ for $x\in X^{\operatorname{an}}$ .
(2)

For $a\in{\mathbb{Q}}_{\geq 0}$ , $\mu_{\overline{L}^{\otimes a}}=a\mu_{\overline{L}}$ on $X^{\operatorname{an}}$ .

(3)

Let $\overline{L}_{1},\ldots,\overline{L}_{r}$ be elements of $\widehat{\operatorname{Pic}}_{C_{0}}(X)_{{\mathbb{Q}}}$ . We assume that there are open intervals $I_{1},\ldots,I_{r}$ of ${\mathbb{R}}$ such that

\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}

for all $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$ . Then, for a fixed $x\in X^{\operatorname{an}}$ , there is a continuous function $f:I_{1}\times\cdots\times I_{r}\to{\mathbb{R}}$ such that

f(t_{1},\ldots,t_{r})=\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)

for all $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$ .

Proof.

(1) and (2) are consequences of (3) and (5) in Lemma 3.14, respectively.

(3) We set

f_{0}(t_{1},\ldots,t_{r}):=\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)

for $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$ . By (1) and (2), for $\lambda\in[0,1]\cap{\mathbb{Q}}$ and $(t_{1},\ldots,t_{r}),(t^{\prime}_{1},\ldots,t^{\prime}_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$ , we have

f_{0}(\lambda(t_{1},\ldots,t_{r})+(1-\lambda)(t^{\prime}_{1},\ldots,t^{\prime}_{r}))\\ \hskip-50.00008pt=\mu^{\operatorname{a}}_{(\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}})^{\otimes\lambda}\otimes(\overline{L}\otimes\overline{L}_{1}^{\otimes t^{\prime}_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t^{\prime}_{r}})^{\otimes(1-\lambda)}}(x)\\ \leq\lambda\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)+(1-\lambda)\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t^{\prime}_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t^{\prime}_{r}}}(x)\\ =\lambda f_{0}(t_{1},\ldots,t_{r})+(1-\lambda)f_{0}(t^{\prime}_{1},\ldots,t^{\prime}_{r}),

that is, $f_{0}$ is concave on $(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$ . Therefore, the assertion (3) follows from [7, Corollary 1.3.2]. ∎

Let $(L,h)$ be an element of $\widehat{\operatorname{Pic}}_{C^{0}}^{+}(X)_{{\mathbb{Q}}}$ . We say that $h$ is semipositive if there is a positive integer $n$ such that $L^{\otimes n}\in\operatorname{Pic}(X)$ and $h^{n}$ is semipositive. The following characterization of the semipositivity of $h$ is a consequence of Proposition 3.10.

Proposition 3.16.

For $\overline{L}=(L,h)\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}$ , $h$ is semipositive if and only if $\mu_{\overline{L}}=0$ on $X^{\operatorname{an}}$ .

We assume that $|\raisebox{1.72218pt}{.}|$ is non-trivial. Let ${\mathscr{X}}$ be a model of $X$ over $\operatorname{Spec}(\mathfrak{o}_{k})$ . Let $L\in\operatorname{Pic}(X)\otimes{\mathbb{Q}}$ and ${\mathscr{L}}\in\operatorname{Pic}({\mathscr{X}})\otimes{\mathbb{Q}}$ with $\left.{{\mathscr{L}}}\right|_{{X}}=L$ . Let $m$ be a positive integer such that $L^{\otimes m}\in\operatorname{Pic}(X)$ . Then we define $\overline{L}=(L,h)$ to be

(L,h):=\left(L^{\otimes m},\left\{|\raisebox{1.72218pt}{.}|_{{\mathscr{L}}^{\otimes m}}(x)\right\}_{x\in X^{\operatorname{an}}}\right)^{\otimes 1/m}.

Proposition 3.17.

If $L$ is ample and ${\mathscr{L}}$ is nef, then $h$ is semipositive.

Proof.

First we assume that ${\mathscr{L}}$ is ample. We choose a positive integer $n$ such that ${\mathscr{L}}^{\otimes n}\in\operatorname{Pic}({\mathscr{X}})$ and ${\mathscr{L}}^{\otimes n}$ is very ample. Then we have an embedding $\iota:{\mathscr{X}}\to{\mathbb{P}}(H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n}))$ and ${\mathscr{L}}^{\otimes n}=\iota^{*}({\mathscr{O}}_{{\mathbb{P}}(H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n}))}(1))$ . Let $(e_{1},\ldots,e_{r})$ be a free basis of $H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n})$ . We define a norm $\|\raisebox{1.72218pt}{.}\|$ of $H^{0}(X,L^{\otimes n})$ to be

\|a_{1}e_{1}+\cdots+a_{r}e_{r}\|:=\max\{|a_{1}|,\ldots,|a_{r}|\}.

Note that $(H^{0}(X,L^{\otimes n}),\|\raisebox{1.72218pt}{.}\|)_{\leq 1}=H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n})$ , so that, by Proposition 3.8, we have $|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(H,\|\raisebox{1.20552pt}{.}\|)}(x)=|\raisebox{1.72218pt}{.}|_{\mathscr{L}^{\otimes n}}(x)$ for $x\in X^{\operatorname{an}}$ . Thus $h$ is semipositive.

In general, let ${\mathscr{A}}$ be an ample invertible sheaf on ${\mathscr{X}}$ and $A:=\left.{{\mathscr{A}}}\right|_{{X}}$ . We choose $\delta\in{\mathbb{Q}}_{>0}$ such that $L\otimes A^{\otimes a}$ is ample for all $a\in(-\delta,\delta)\cap{\mathbb{Q}}$ . Note that $\overline{L}\otimes\left(A,|\raisebox{1.72218pt}{.}|_{{\mathscr{A}}}\right)^{\otimes\epsilon}=\left(L\otimes A^{\otimes\epsilon},|\raisebox{1.72218pt}{.}|_{{\mathscr{L}}\otimes{\mathscr{A}}^{\otimes\epsilon}}\right)$ , so that $\mu_{\overline{L}\otimes\left(A,|\raisebox{1.20552pt}{.}|_{{\mathscr{A}}}\right)^{\otimes\epsilon}}=0$ for $\epsilon\in(0,\delta)\cap{\mathbb{Q}}$ by the previous observation together with Proposition 3.16. On the other hand, by (3) in Lemma 3.15,

\mu_{\overline{L}}(x)=\lim_{\begin{subarray}{c}\epsilon\downarrow 0\\ \epsilon\in{\mathbb{Q}}\end{subarray}}\mu_{\overline{L}\otimes(A,|\raisebox{1.20552pt}{.}|_{{\mathscr{A}}})^{\otimes\epsilon}}(x).

Therefore, $\mu_{\overline{L}}=0$ , and hence $h$ is semipositive by Proposition 3.16. ∎

Remark 3.18.

Assume that the absolute value $|\raisebox{1.72218pt}{.}|$ is non-trivial. Let $L$ be an ample invertible sheaf on $X$ , equipped with a semipositive continuous metric $h$ . Then there exists a sequence $\{(\mathscr{X}_{n},\mathscr{L}_{n})\}_{n\geqslant 1}$ , where $\mathscr{X}_{n}$ is a model of $X$ and $\mathscr{L}_{n}$ is a nef invertible sheaf on $\mathscr{X}_{n}$ such that $\mathscr{L}_{n}|_{X}=L^{\otimes n}$ and that $h_{n}=(|\raisebox{1.72218pt}{.}|_{\mathscr{L}_{n}}(x)^{1/n})_{x\in X^{\mathrm{an}}}$ converges uniformly to $h$ . This follows from Proposition 3.10 and the comparison between quotient metrics and model metrics (via the embedding into the projective spaces of lattices). Combining with Proposition 3.17 and Corollary 3.11, we obtain that, in the non-trivial valuation case, our semipositivity coincides with that of Zhang [12] and Moriwaki [8]. We refer the readers to [6, §6] and to [2, §6.8] for the descriptions of the semipositivity in terms of plurisubharmonic currents. Note that their semipositivity is also equivalent to our semipositivity.

3.4. The functions σ and μ on X an [027E]

3.4. The functions σ\sigma and μ\mu on XanX^{\operatorname{an}}

Lemma 3.13.

Proof.

Lemma 3.14.

Proof.

Lemma 3.15.

Proof.

Proposition 3.16.

Proposition 3.17.

Proof.

Remark 3.18.

3.4. The functions $\sigma$ and $\mu$ on $X^{\operatorname{an}}$