Bogomolov unstability on arithmetic surfaces

In this paper, we will consider an arithmetic analogue of Bogomolov unstability theorem, i.e. if $(E, h)$ is a torsion free Hermitian sheaf on an arithmetic surface $X$ and $\widehat{\operatorname{deg}}\left((\rank E - 1) \widehat{c}_{1}{E, h}^2 - (2 \rank E) \widehat{c}_{2}{E, h}\right) > 0$, then there is a non-zero saturated subsheaf $F$ of $E$ such that the point $\widehat{c}_{1}{F, \submet{h}{F}}/\!\rank F - \widehat{c}_{{1}{E, h}/\!\rank E$ lies in the positive cone of $X$.

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Published 1 January 1994