## Journal of Mathematics of Kyoto University

### On parabolic geometry, II

Indranil Biswas

#### 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

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

Digital Object Identifier
doi:10.1215/kjm/1256219163

Mathematical Reviews number (MathSciNet)
MR2571848

Zentralblatt MATH identifier
1185.53028

#### 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