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

TL;DR

This paper investigates the semistability of the cotangent bundle restricted to curves on Enriques surfaces, providing conditions for semistability and explicit destabilizing examples.

Contribution

It establishes new criteria for the semistability of the cotangent bundle on Enriques surfaces and constructs explicit destabilizing curve families.

Findings

Semistability holds when H^2 ≥ 6

Semistability holds for very general S when H^2 ≥ 2

Explicit destabilizing families of curves are constructed

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 $\Omega_S$.