Illinois Journal of Mathematics

Equivariant principal bundles over the complex projective line

Indranil Biswas

Abstract

Let $G$ be a connected complex reductive linear algebraic group, and let $K \subset G$ be a maximal compact subgroup of it. Let $E_G$ be a holomorphic principal $G$-bundles over the complex projective line ${\mathbb C}{\mathbb P}^1$ and $E_K \subset E_G$ a $C^\infty$ reduction of structure group of $E_G$ to $K$. We consider all pairs $(E_G ,E_K)$ of this type such that the total space of $E_K$ is equipped with a $C^\infty$ lift of the standard action of $\operatorname{SU}(2)$ on ${\mathbb C}{\mathbb P}^1$ which satisfies the following two conditions: the actions of $K$ and $\operatorname{SU}(2)$ on $E_K$ commute, and for each element $g \in \operatorname{SU}(2)$, the induced action of $g$ on $E_G$ is holomorphic. We give a classification of the isomorphism classes of all such objects.

Article information

Source
Illinois J. Math., Volume 55, Number 1 (2011), 261-285.

Dates
First available in Project Euclid: 19 December 2012

https://projecteuclid.org/euclid.ijm/1355927036

Digital Object Identifier
doi:10.1215/ijm/1355927036

Mathematical Reviews number (MathSciNet)
MR3006688

Zentralblatt MATH identifier
1256.53022

Citation

Biswas, Indranil. Equivariant principal bundles over the complex projective line. Illinois J. Math. 55 (2011), no. 1, 261--285. doi:10.1215/ijm/1355927036. https://projecteuclid.org/euclid.ijm/1355927036

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