Metha-Ramanathan for {\epsilon} and k-semistable Decorated Sheaves

Abstract

This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of {\epsilon}-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the {\epsilon} and k-(semi)stablility. We first prove the existence of a (unique) {\epsilon}-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of {\epsilon}-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope {\epsilon}-semistable decorated sheaves. In Section 4 we reach the same results in the k-semistability case that we did in the {\epsilon}-semistability, but only for decorated sheaves of rank less or equal than 3.