Remarks on the positivity of the cotangent bundle of a K3 surface

TL;DR

This paper investigates the positivity properties of the cotangent bundle of K3 surfaces, computing cones and providing counterexamples to existing conjectures about nefness and semistability.

Contribution

It computes the pseudoeffective and nef cones of the projectivized cotangent bundle of K3 surfaces and constructs explicit counterexamples to a question on nefness.

Findings

Computed pseudoeffective and nef cones for many K3 surfaces

Constructed explicit smooth curves with non-nef cotangent restrictions

Provided counterexamples to a question of Campana-Peternell

Abstract

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell.