Let $L$ be a line bundle on $\mathrm{X}$. Let $0$ be the neutral element of $\mathrm{X}$ and let us fix a trivialization $L_{0}$ of $L$ at $0$.

The line bundle $L^{\otimes 2}$ is canonically decomposed as the tensor product of an even and an odd line bundle :

$$L^{\otimes 2}=(L\otimes[-1]^{*}L)\otimes(L\otimes[-1]^{*}L^{-1}).$$

By the theorem of the cube, an even line bundle $L$ satisfies $[m]^{*}L\simeq L^{\otimes m^{2}}$, while for an odd line bundle $L$, one has $[m]^{*}L\simeq L^{\otimes m}$ ; moreover, there are in each case a unique isomorphism compatible with the trivialization at the origin. By Lemma 2.4.1, an even (resp. an odd) line bundle possesses a canonical continuous metric making this isomorphism an isometry. This furnishes a canonical metric on $L^{\otimes 2}$, hence on $L$. According to this lemma, this metric is semi-positive if $L$ is ample and even. Using a Lemma of Künnemann, ([17], Lemme 2.3), one proves that this also holds if $L$ is algebraically equivalent to $0$. In any case, the canonical metrics are admissible.