Recognizing G / P by Varieties of Minimal Rational Tangents

Abstract

We give an overview of the problem of biholomorphic equivalence of germs of free unbendable rational curves on complex manifolds, describing varieties of minimal rational tangents as a key invariant of the equivalence problem. Major examples are provided by lines on a rational homogeneous space G / P with a simple Lie group G and a maximal parabolic subgroup P. We review the previous works when P is associated to a long root and discuss recent progress when G / P is a symplectic Grassmannian, which is the most prominent example of the case when P is associated to a short root.