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# Mathematical Research Letters

## Volume 12 (2005)

### Number 6

### A quantitative sharpening of Moriwaki’s arithmetic Bogomolov inequality

Pages: 877 – 884

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n6.a8

#### Author

#### Abstract

A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soulé’s proof of a vanishing theorem on arithmetic surfaces.