In this chapter we take up a problem already discussed in Section 3.1. We try to understand how µ-(semi)stable sheaves behave under restriction to hypersurfaces. At present, there are three quite different approaches to this question, and we will treat them in separate sections. None of these methods covers the results of the others completely.

The theorems of Mehta and Ramanathan 7.2.1 and 7.2.8 show that the restriction of a µ-stable or µ-semistable sheaf to a general hypersurface of sufficiently high degree is again µ-stable or µ-semistable, respectively. It has the disadvantage that it is not effective, i.e. there is no control of the degree of the hypersurface, which could, a priori, depend on the sheaf itself. However, such a bound, depending only on the rank of the sheaf and the degree of the variety, is provided by Flenner's Theorem 7.1.1. Since it is based on a careful exploitation of the Grauert-Mülich Theorem in the refined form 3.1.5, it works only in characteristic zero and for µ-semistable sheaves. In that respect, Bogomolov's Theorem 7.3.5 is the strongest, though one has to restrict to the case of smooth surfaces. It says that the restriction of a µ-stable vector bundle on a surface to any curve of sufficiently high degree is again µ-stable, whereas the theorems mentioned before provide information for general hypersurfaces only. Moreover, the bound in Bogomolov's theorem depends on the invariants of the bundle only.