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

Let $S$ be an Enriques surface. In this paper we study the semistability of the restriction ${\mathrm{\Omega }}_S{\mathrm{\mid }}_C$ for a general curve $C\in \mathrm{\mid }H\mathrm{\mid }$, where $H$ is a globally generated and ample line bundle on $S$. We show that ${\mathrm{\Omega }}_S{\mathrm{\mid }}_C$ 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 ${\mathrm{\Omega }}_S$.