Katharina Neusser

A cone structure on a complex manifold M is a closed submanifold C ⊂ PTM of the projectivized (holomorphic) tangent bundle of M which is submersive over M. Such structures arise naturally in differential and algebraic geometry, and when they do, they are typically equipped with a conic connection that specifies a distinguished family of curves on M in direction of C. In differential geometry, a classical example is the null cone bundle of a holomorphic conformal structure with the conic connection given by the null-geodesics. In algebraic geometry, one has the cone structures consisting of varieties of minimal rational tangents (VMRT) induced by minimal rational curves on uniruled projective manifolds. In this talk we will discuss various examples of cone structures and will introduce two invariants for conic connections. As an application of the study of these invariants, we will present a local-differential-geometric version of the global algebraic-geometric rigidity theorem of Mok and Hong–Hwang, which recognizes certain generalized flag varieties from their VMRT-structures. This talk is based on joint work with Jun-Muk Hwang.