# Characterizing unipolar flag manifolds by their varieties of minimal rational tangents

It is well known that rational algebraic curves play a key role in the geometry of complex projective varieties, especially of Fano manifolds. In particular, on Fano manifolds of Picard number (= the 2nd Betti number) one, which are sometimes called "unipolar", one may consider rational curves of minimal degree passing through general points. Tangent directions of minimal rational curves through a general point $x$ in a unipolar Fano manifold $X$ form a projective subvariety $\mathcal{C}_{x,X}$ in the projectivized tangent space $\mathbb{P}(T_xX)$, called the variety of minimal rational tangents (VMRT).

In 90-s J.-M. Hwang and N. Mok developed a philosophy declaring that the geometry of a unipolar Fano manifold is governed by the geometry of its VMRT at a general point, as an embedded projective variety. In support of this thesis, they proposed a program of characterizing unipolar flag manifolds in the class of all unipolar Fano manifolds by their VMRT. In the following decades a number of partial results were obtained by Mok, Hwang, and their collaborators.

Recently the program was successfully completed (J.-M. Hwang, Q. Li, and the speaker). The main result states that a unipolar Fano manifold $X$ whose VMRT at a general point is isomorphic to the one of a unipolar flag manifold $Y$ is itself isomorphic to $Y$. Interestingly, the proof of the main result involves a bunch of ideas and techniques from "pure" algebraic geometry, differential geometry, structure and representation theory of simple Lie groups and algebras, and theory of spherical varieties (which extends the theory of toric varieties).