We remark that ${\mathfrak{V}}_{s}\otimes_{\tilde{K}}\tilde{L}\simeq({\mathfrak{V}}\hat{\otimes}_{{K^{\circ}}}L^{\circ})_{s}$. Hence the result follows from the fact that a nef line bundle $\mathcal{L}$ on a proper variety over $\tilde{K}$ remains nef after pull back to $\tilde{F}$. This is proven in the projective case in [EFM, Remark 1.3.25] and the proper case follows from Chow’s lemma and the projection formula. ∎

Let ${\mathfrak{V}}_{\rm red}$ be the induced reduced structure on ${\mathfrak{V}}$. Since ${\mathfrak{V}}_{\rm red}\to{\mathfrak{V}}$ is finite (in fact an immersion), we deduce that the induced map $({\mathfrak{V}}_{\rm red})_{s}\to{\mathfrak{V}}_{s}$ between the special fibers is finite. By projection formula, we conclude that ${\mathfrak{L}}$ is nef if and only if ${\mathfrak{L}}_{\rm red}$ is nef. ∎

By definition if ${\mathfrak{L}}$ is nef, then $\|\ \|_{\mathfrak{L}}$ is semipositive, so we only have to prove the reverse implication. Hence we assume that ${\|\hskip 4.30554pt\|}_{\mathfrak{L}}$ is a semipositive formal metric and we have to show that ${\mathfrak{L}}$ is nef. Using Lemma 3.3, we can replace $K$ by ${\mathbb{C}}_{K}$ and hence assume that $K$ is algebraically closed.

By definition of semipositivity, there is a nef ${K^{\circ}}$-model ${\mathfrak{M}}$ of $L$ on some model ${\mathfrak{W}}$ of $V$ with ${\|\hskip 4.30554pt\|}_{\mathfrak{L}}={\|\hskip 4.30554pt\|}_{\mathfrak{M}}$. There exists a model ${\mathfrak{X}}$ of $V$ which dominates both ${\mathfrak{V}}$ and ${\mathfrak{W}}$. Let $\pi:{\mathfrak{X}}\to{\mathfrak{V}}$ be the induced morphism. Since the induced morphism on the special fibers $\pi_{s}\colon{\mathfrak{X}}_{s}\to{\mathfrak{V}}_{s}$ is proper and surjective, by the projection formula, ${\mathfrak{L}}$ is nef if and only $\pi^{*}{\mathfrak{L}}$ is nef. Hence replacing $({\mathfrak{V}},{\mathfrak{L}})$ by $({\mathfrak{X}},\pi^{*}{\mathfrak{L}})$, we can assume that ${\mathfrak{V}}$ dominates ${\mathfrak{W}}$.

Let ${\mathfrak{V}}_{\rm red}$ be the reduced structure on ${\mathfrak{V}}$. Hence ${\mathfrak{V}}_{\rm red}\to{\mathfrak{V}}$ is finite. Locally, ${\mathfrak{V}}_{\rm red}$ is given by ${\rm Spf}(A)$ for some reduced admissible ${K^{\circ}}$-algebra. Let ${\mathscr{A}}\coloneqq A\otimes_{{K^{\circ}}}K$. It is a strictly $K$-affinoid algebra, and by [BGR84, 6.4.3] $A^{\prime}\coloneqq{\mathscr{A}}^{\circ}$ is an admissible $K^{\circ}$-algebra, and moreover $A\to A^{\prime}$ is finite and induces an isomorphism on the generic fibers. By [BGR84, 7.2.6 Proposition 3], we can glue the morphisms ${\rm Spf}(A^{\prime})\to{\rm Spf}(A)$ to get a model ${\mathfrak{V}}^{\prime}$ of $V_{\rm red}$ such that ${\mathfrak{V}}^{\prime}\to{\mathfrak{V}}_{\rm red}$ is finite. In particular, we deduce that the induced morphisms ${\mathfrak{V}}^{\prime}_{s}\to({\mathfrak{V}}_{\rm red})_{s}\to{\mathfrak{V}}_{s}$ are proper and surjective, and we conclude from the projection formula that ${\mathfrak{L}}$ is nef if and only if its pull back ${\mathfrak{L}}^{\prime}$ to ${\mathfrak{V}}^{\prime}$ is nef.

By construction, ${\mathfrak{V}}^{\prime}$ is locally of the form ${\rm Spf}({\mathscr{A}}^{\circ})$, hence we deduce that ${\mathfrak{V}}^{\prime}_{s}$ is locally given by ${\rm Spec}(\tilde{{\mathscr{A}}})$ which is reduced. Now we use the fact that on an admissible formal scheme with reduced special fibre and with $K$ algebraically closed, the metric ${\|\hskip 4.30554pt\|}_{{\mathfrak{L}}^{\prime}}$ determines the model ${\mathfrak{L}}^{\prime}$ up to isomorphism (see [Gub98, Proposition 7.5]). Using that ${\|\hskip 4.30554pt\|}_{{\mathfrak{M}}^{\prime}}={\|\hskip 4.30554pt\|}_{\mathfrak{L}}={\|\hskip 4.30554pt\|}_{{\mathfrak{L}}^{\prime}}$ for the pull-back ${\mathfrak{M}}^{\prime}$ of ${\mathfrak{M}}$ to ${\mathfrak{V}}^{\prime}$, we deduce that ${\mathfrak{M}}^{\prime}\cong{\mathfrak{L}}^{\prime}$. As above, the pull-back ${\mathfrak{M}}^{\prime}$ of ${\mathfrak{M}}$ is nef and hence ${\mathfrak{L}}^{\prime}$ is nef. ∎

For each $i\in I$, let ${\mathscr{X}}_{i}$ be the closed subscheme of ${\mathscr{X}}$ defined as the topological closure of $X_{i}$ in ${\mathscr{X}}$ equipped with the reduced structure. We then get for each $i\in I$ a cartesian diagram

Since the morphism $\coprod_{i\in I}{\mathscr{X}}_{i}\to{\mathscr{X}}$ is finite surjective, the projection formula shows that ${\mathscr{L}}$ is nef on ${\mathscr{X}}$ if and only if ${\mathscr{L}}_{|{\mathscr{X}}_{i}}$ is nef on ${\mathscr{X}}_{i}$ for all $i$. ∎

Property (a) is obvious from the definitions and the other properties follow from [GK15, Proposition 6.4] by using base change to ${\mathbb{C}}_{K}$ and Lemma 3.3. ∎

The proof follows mainly the arguments in [GK15, Proposition 6.4]. By Lemma 3.3 and Lemma 3.4, we may assume that $K$ is algebraically closed and that $V$ is reduced. Let $({\mathfrak{V}},{\mathfrak{L}})$ be a formal ${K^{\circ}}$-model of $(V,L)$ with ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathfrak{L}}$. We assume that ${\|\hskip 4.30554pt\|}$ is semipositive in every $x\in V$. We choose a closed curve $C$ in ${\mathfrak{V}}_{s}$ which is proper over ${\tilde{K}}$. We have to show that $\deg_{\mathfrak{L}}(C)\geq 0$. By surjectivity of the reduction map $\pi:V\to{\mathfrak{V}}_{s}$, there is $x\in V$ such that $\pi(x)$ is the generic point of $C$. Since ${\|\hskip 4.30554pt\|}$ is semipositive in $x$, there is a compact strictly $K$-affinoid neighborhood $W$ of $x$ and a nef formal ${K^{\circ}}$-model $({\mathfrak{W}},{\mathfrak{M}})$ of $(W,L|_{W})$ such that ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathfrak{M}}$ over $W$. Using Proposition 3.5, we may always replace the models ${\mathfrak{W}}$ and ${\mathfrak{V}}$ by dominating formal ${K^{\circ}}$-models and the line bundles ${\mathfrak{M}}$ and ${\mathfrak{L}}$ by their pull-backs. By [BL93b, Corollary 5.4], we may therefore assume that ${\mathfrak{W}}$ is a formal open subset of ${\mathfrak{V}}$. Then ${\mathfrak{L}}|_{\mathfrak{W}}$ is also a formal ${K^{\circ}}$-model of $L|_{W}$ and hence Proposition 3.5 shows that ${\mathfrak{L}}|_{\mathfrak{W}}$ is nef.

Since $W$ is a neighbourhood of $x$ and since $V$ is boundaryless, we conclude that the boundary of $W$ is the topological boundary of $W$ in $V$ (see [Ber90, Corollary 2.5.13(ii), Proposition 3.1.3(ii)]). In particular, $x$ is no boundary point of $W$ as $W$ is a neighborhood of $x$. Using [CD12, Lemma 6.5.1], such interior points are characterized by the property that the closure of the reduction in ${\mathfrak{W}}_{s}$ is proper over ${\tilde{K}}$. We conclude that the closure of $\pi(x)$ in ${\mathfrak{W}}_{s}$ is equal to $C$. Since ${\mathfrak{L}}|_{\mathfrak{W}}$ is nef, it follows that $\deg_{\mathfrak{L}}(C)\geq 0$. ∎

Since formal and algebraic metrics are the same as noted in Remark 2.5 and hence also the same as piecewise linear metrics, we deduce from Proposition 2.10 (d) that $\|\ \|$ is an algebraic metric. If the given metrics are semipositive in $x$, then it remains to prove that $\|\ \|$ is semipositive in $x$. By base change again, we may assume that $K$ is algebraically closed. By Lemma 3.6, we may assume that $X$ is a proper variety over $K$.

Let us pick models ${\mathscr{L}}_{1}$, ${\mathscr{L}}_{2}$ and ${\mathscr{L}}$ of $L$ defining the model metrics $\|\ \|$, $\|\ \|_{1}$ and $\|\ \|_{2}$. There is a ${K^{\circ}}$-model ${\mathscr{X}}$ of $X$ on which ${\mathscr{L}}_{1}$, ${\mathscr{L}}_{2}$ and ${\mathscr{L}}$ are determined. There is an open neighbourhood $W$ of $x$ in ${X^{\rm an}}$ such that ${\|\hskip 4.30554pt\|}_{1}$ and ${\|\hskip 4.30554pt\|}_{2}$ are semipositive in all points of $W$. We will show that ${\|\hskip 4.30554pt\|}$ is semipositive in every point of $W$. By [GK15, 6.5], it is equivalent to show that $\deg_{\mathscr{L}}(C)\geq 0$ for any closed curve $C$ of ${\mathscr{X}}_{s}$ contained in the reduction of $W$. Moreover, the same result yields that ${\mathscr{L}}_{1}$ and ${\mathscr{L}}_{2}$ restrict to nef line bundles on $C$. By [GK15, Theorem 4.1], there is a closed curve $Y$ in $X$ such that $C$ is an irreducible component of the special fibre of the closure $\overline{Y}$ in ${\mathscr{X}}$. By restriction, we may assume that $X=Y$ is a curve and hence $C$ is an irreducible component of ${\mathscr{X}}_{s}$. Let ${\mathfrak{X}}$ be the formal completion of ${\mathscr{X}}$ and let ${\mathfrak{L}},{\mathfrak{L}}_{1},{\mathfrak{L}}_{2}$ be the line bundles on ${\mathfrak{X}}$ induced by the pull-backs of ${\mathscr{L}},{\mathscr{L}}_{1},{\mathscr{L}}_{2}$.

We have seen in the proof of Proposition 3.5 that we can associate to ${\mathfrak{X}}$ a canonical formal model ${\mathfrak{X}}^{\prime}$ of ${X^{\rm an}}$ with reduced special fibre and a canonical finite surjective morphism $\iota:{\mathfrak{X}}^{\prime}\to{\mathfrak{X}}$. So there is a closed curve $C^{\prime}$ in ${\mathfrak{X}}^{\prime}_{s}$ which maps onto $C$ in ${\mathfrak{X}}_{s}={\mathscr{X}}_{s}$. Let ${\mathfrak{L}}^{\prime},{\mathfrak{L}}_{1}^{\prime},{\mathfrak{L}}_{2}^{\prime}$ be the line bundles on ${\mathfrak{X}}^{\prime}$ given by pull-back of ${\mathfrak{L}},{\mathfrak{L}}_{1},{\mathfrak{L}}_{2}$. Note that ${\mathfrak{L}}^{\prime},{\mathfrak{L}}_{1}^{\prime},{\mathfrak{L}}_{2}^{\prime}$ are formal models of the metrics ${\|\hskip 4.30554pt\|},{\|\hskip 4.30554pt\|}_{1},{\|\hskip 4.30554pt\|}_{2}$ on ${L^{\rm an}}$. By projection formula, the line bundles ${\mathfrak{L}}_{1}^{\prime},{\mathfrak{L}}_{2}^{\prime}$ restrict to nef line bundles on $C^{\prime}$ and it remains to show that

(3.11.1)

$$\deg_{{\mathfrak{L}}^{\prime}}(C^{\prime})\geq 0.$$

Let $\zeta$ be the generic point of $C^{\prime}$. Then there is a unique point $\xi$ in ${X^{\rm an}}$ with reduction $\zeta$. This follows from [Ber90, Proposition 2.4.4] since $\zeta$ has a formal affine open neighbourhood in ${\mathfrak{X}}^{\prime}$ of the form ${\rm Spf}({\mathscr{A}}^{\circ})$ for a strictly $K$-affinoid algebra ${\mathscr{A}}$. Using $\|\ \|=\min(\|\ \|_{1},\|\ \|_{2})$, we may assume ${\|\hskip 4.30554pt\|}(\xi)={\|\hskip 4.30554pt\|}_{1}(\xi)$. Since ${L^{\rm an}}$ is algebraic, there is a non-trivial meromorphic section $t$ of ${\mathfrak{L}}^{\prime}$. Note that the restriction of $t$ to the generic fibre ${L^{\rm an}}$ induces also a meromorphic section $t_{1}$ of ${\mathfrak{L}}_{1}^{\prime}$. The meromorphic section $t/t_{1}$ of ${\mathfrak{M}}:={\mathfrak{L}}^{\prime}\otimes({\mathfrak{L}}_{1}^{\prime})^{-1}$ restricts to the trivial section $1$ of ${\mathcal{O}}_{X^{\rm an}}$ and we have

$$\|t/t_{1}\|_{\mathfrak{M}}=\|t\|/\|t_{1}\|_{1}=\|t\|/\|t\|_{1}\leq 1.$$

By [Gub98, Proposition 7.5], we deduce that $t/t_{1}$ is a global section of $\mathfrak{M}$. The definition of formal metrics and $\|t/t_{1}\|_{\mathfrak{M}}(\xi)=\|t\|(\xi)/\|t\|_{1}(\xi)=1$ yield that $\{y\in{X^{\rm an}}\mid\|t/t_{1}\|_{\mathfrak{M}}(y)\geq 1\}$ is the generic fibre of a formal open neighbourhood ${\mathfrak{U}}$ of $\zeta$. Hence [Gub98, Proposition 7.5] again shows that $t/t_{1}$ is a nowhere vanishing regular section of $\mathfrak{M}$ on ${\mathfrak{U}}$. We conclude that the restriction of the global section $t/t_{1}$ to $C^{\prime}$ is not identically zero inducing an effective Cartier divisor $D$ on $C^{\prime}$. This shows

$$\deg_{\mathfrak{M}}(C^{\prime})=\deg_{D}(C^{\prime})\geq 0.$$

Using that ${\mathfrak{L}}_{1}^{\prime}$ is nef on $C^{\prime}$ and ${\mathfrak{L}}^{\prime}={\mathfrak{M}}\otimes{\mathfrak{L}}_{1}^{\prime}$, we get

$$\deg_{{\mathfrak{L}}^{\prime}}(C^{\prime})\geq\deg_{{\mathfrak{L}}_{1}^{\prime}}(C^{\prime})\geq 0$$

proving (3.11.1). ∎