Illinois Journal of Mathematics

Equivariant principal bundles over the complex projective line

Indranil Biswas

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1355927036

Digital Object Identifier
doi:10.1215/ijm/1355927036

Mathematical Reviews number (MathSciNet)
MR3006688

Zentralblatt MATH identifier
1256.53022

Subjects
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]

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


Export citation

References

  • L. Álvarez-Cónsul and O. García-Prada, Dimensional reduction and quiver bundles, J. Reine Angew. Math. 556 (2003), 1–46.
  • I. Biswas, Holomorphic Hermitian vector bundles over the Riemann sphere, Bull. Sci. Math. 132 (2008), 246–356.
  • A. I. Bondal and M. M. Kapranov, Homogeneous bundles, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 45–55.
  • N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. No. 1285, Hermann, Paris, 1960.
  • A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138.
  • S. Helgason, Differential geometry, lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence, RI, 2001.
  • L. Hille, Homogeneous vector bundles and Koszul algebras, Math. Nachr. 191 (1998), 189–195.
  • G. Ottaviani and E. Rubei, Quivers and the cohomology of homogeneous vector bundles, Duke Math. J. 132 (2006), 459–508.