The twisted cotangent bundle of a hyperkahler manifold

Let X be a complex projective Hyperk ̈ahler manifold. By a recent result of H ̈oring and Peternell the cotangent bundle of X is not pseudoeffective. One way to measure this negativity more precisely is to give sufficient conditions on an ample line bundle A such that the twist ΩX ⊗ A is pseudoeffective. I will give a sufficient condition that depends only on the Segre classes and the Beauville–Fujiki form of X. Then I will discuss when this sufficent condition is also necessary. This is a joint work with Andreas H ̈oring.