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VOL. 50 | 2008 Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents
Jaehyun Hong, Jun-Muk Hwang

Editor(s) Kazuhiro Konno, Viet Nguyen-Khac

Abstract

Let $S = G / G'$ be a rational homogeneous space defined by a complex simple Lie group $G$ and a maximal parabolic subgroup $G'$. For a base point $s \in S$, let $\mathcal{C}_s \subset \mathbb{P}T_s (S)$ be the variety of minimal rational tangents at $s$. In the study of rigidity of rational homogeneous spaces, the following question naturally arises. Let $X$ be a Fano manifold of Picard number 1 such that the variety of minimal rational tangents at a general point $x \in X$, $\mathcal{C}_x \subset \mathbb{P}T_x (X)$, is isomorphic to $\mathcal{C}_s \subset \mathbb{P}T_s (S)$. Is $X$ biholomorphic to $S$? An affirmative answer has been given by Mok when $S$ is a Hermitian symmetric space or a homogeneous contact manifold. Extending Mok's method further and combining it with the theory of differential systems on $S$, we will give an affirmative answer when $G'$ is associated to a long simple root.

Information

Published: 1 January 2008
First available in Project Euclid: 16 December 2018

zbMATH: 1186.14044
MathSciNet: MR2409558

Digital Object Identifier: 10.2969/aspm/05010217

Subjects:
Primary: 14J45 , 32M10 , 53C15

Keywords: Cartan connection , minimal rational curve , rational homogeneous space

Rights: Copyright © 2008 Mathematical Society of Japan

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