Remarks on the positivity of the cotangent bundle of an Enriques surface

Abstract

Let $S$ be an Enriques surface. In this paper we study the semistability of the restriction $\Omega_{S}|_C$ for a general curve $C \in |H|$, where $H$ is a globally generated and ample line bundle on $S$. We show that $\Omega_{S}|_C$ is semistable when $H^2 \ge 6$, or when $H^2 \ge 2$ and $S$ is very general. Moreover, we give explicit constructions of families of smooth irreducible curves that destabilize $\Omega_S$.