## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Algebraic Geometry in East Asia — Hanoi 2005, K. Konno and V. Nguyen-Khac, eds. (Tokyo: Mathematical Society of Japan, 2008), 217 - 236

### Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents

Jaehyun Hong and Jun-Muk Hwang

#### 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.

#### Article information

**Source***Algebraic Geometry in East Asia — Hanoi 2005*, K. Konno and V. Nguyen-Khac, eds. (Tokyo: Mathematical Society of Japan, 2008), 217-236

**Dates**

Received: 3 February 2006

First available in Project Euclid:
16 December 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1545001307

**Digital Object Identifier**

doi:10.2969/aspm/05010217

**Mathematical Reviews number (MathSciNet)**

MR2409558

**Zentralblatt MATH identifier**

1186.14044

**Subjects**

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 32M10: Homogeneous complex manifolds [See also 14M17, 57T15] 14J45: Fano varieties

**Keywords**

rational homogeneous space Cartan connection minimal rational curve

#### Citation

Hong, Jaehyun; Hwang, Jun-Muk. Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents. Algebraic Geometry in East Asia — Hanoi 2005, 217--236, Mathematical Society of Japan, Tokyo, Japan, 2008. doi:10.2969/aspm/05010217. https://projecteuclid.org/euclid.aspm/1545001307