Thomas Peternell, Semipositive Tangent Bundles and Canonical Extensions

Given a projective complex manifold M with an ample polarization there is canonically associated an affine bundle Z over M. The question I will discuss is under which circumstances Z is an affine variety, or at least Stein. This is related to the global structure of M, specifically to the semipositivity of the tangent bundle. I will explain the main conjectures and recent results (joint work with A. Höring).