Proposition 3.4.1 . [01KG]

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Proposition 3.4.1.

Let $F$ be a number field, let $X$ be a projective smooth curve over $F$ . Let $\overline{L}$ and $\overline{M}$ be two admissible metrized line bundles over $X$ . Assume that $\deg(L)=\ell$ , $\deg(M)=m$ are positive. and $({\widehat{c}}_{1}(\overline{L})^{2}|X)=({\widehat{c}}_{1}(\overline{M})^{2}|X)=0$ . Then, the essential minimum of $\overline{L}\otimes\overline{M}$ satisfies the following inequality :

e(\overline{L}\otimes\overline{M})\geq-\frac{1}{2(\ell+m)\ell m}({\widehat{c}}_{1}(m\overline{L}-\ell\overline{M})^{2}|X).

Moreover, the right hand side of this inequality is always nonnegative and vanishes if and only if some power of $\overline{L}^{m}\otimes\overline{M}^{-\ell}$ is constant.