Metha-Ramanathan for ε and k-semistable Decorated Sheaves

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

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