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). ∎