A quantitative sharpening of Moriwaki's arithmetic Bogomolov inequality (Q812533)

\textit{A. Moriwaki} [Math. Res. Lett. 1, 601--611 (1994; Zbl 0858.14011)] proved that if a torsion-free coherent sheaf \(\overline E\) of rank \(r\geq2\) on an arithmetic surface \(X\) has negative discriminant \(\Delta\), then it has a nonzero saturated subsheaf \(\overline E'\) of rank \(r'\) such that \(\xi_{\overline E',\overline E}:=\widehat c_1(\overline E')/r' - \widehat c_1(\overline E)/r\) lies in the positive cone of \(X\). Thus \(\overline E'\) is destabilizing with respect to every polarization of \(X\). This was also shown in the projective case by F. Bogomolov. The present paper shows that \(\overline E'\) may be chosen so that the inequality \(\xi_{\overline E',\overline E}\geq-\Delta/r^2(r-1)\) also holds. This is the exact arithmetic analogue of a result known already in the geometric case; see for example Theorem 7.3.4 of [\textit{D. Huybrechts} and \textit{M. Lehn}, The geometry of moduli spaces of sheaves, Aspects Math. E 31 (1997; Zbl 0872.14002)]. As an application, the paper shows how one can slightly simplify a vanishing theorem of \textit{C. Soulé} [Invent. Math. 116, 577--599 (1994; Zbl 0834.14013)].