## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 49, Number 2 (2009), 381-387.

### On parabolic geometry, II

#### Abstract

Let $G$ be a simple linear algebraic group defined over $\mathbb{C}$ and $P$ a parabolic subgroup of it. Let $(M, E_P, \omega)$ be a holomorphic parabolic geometry of type $G/P$ over a smooth complex projective variety $M$. We prove that $(M, E_ , \omega)$ is holomorphically isomorphic to the standard parabolic geometry $(G/P, G, \omega_0)$ whenever $M$ is rationally connected. We then show that this is indeed the case if $M$ has Picard number one and contains a (possibly singular) rational curve. This last result is a generalization of the main result of [3], where we dealt with the case $G = PGL(d, \mathbb{C})$, $G/P = \mathbb{P}^{d-1}_{\mathbb{C}}$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 49, Number 2 (2009), 381-387.

**Dates**

First available in Project Euclid: 22 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1256219163

**Digital Object Identifier**

doi:10.1215/kjm/1256219163

**Mathematical Reviews number (MathSciNet)**

MR2571848

**Zentralblatt MATH identifier**

1185.53028

**Subjects**

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]

#### Citation

Biswas, Indranil. On parabolic geometry, II. J. Math. Kyoto Univ. 49 (2009), no. 2, 381--387. doi:10.1215/kjm/1256219163. https://projecteuclid.org/euclid.kjm/1256219163