Illinois Journal of Mathematics

Equivariant principal bundles over the complex projective line

Indranil Biswas

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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.

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

First available in Project Euclid: 19 December 2012

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Primary: 53B35: Hermitian and Kählerian structures [See also 32Cxx] 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]


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

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