Journal of Mathematics of Kyoto University

On parabolic geometry, II

Indranil Biswas

Full-text: Open access


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

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

First available in Project Euclid: 22 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation