Andreas Höring, Projective Manifolds with Pseudoeffective (Co)tangent Bundle, III, IV

Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and Paun show that if the cotangent bundle is big, then X is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the (co)tangent bundle it leads to surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.