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Principal co-Higgs bundles on ℙ1

Part of: Families, fibrations Curves

Published online by Cambridge University Press:  05 March 2020

Indranil Biswas ,
Oscar García-Prada ,
Jacques Hurtubise  and
Steven Rayan
Indranil Biswas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai400005, India (indranil@math.tifr.res.in)
Oscar García-Prada
Affiliation:
ICMAT (Instituto de Ciencias Matemáticas), Calle Nicolás Cabrera, no. 13–15, Campus Cantoblanco, 28049Madrid, Spain (oscar.garcia-prada@icmat.es)
Jacques Hurtubise
Affiliation:
Department of Mathematics & Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal, QC, CanadaH3A 0B9 (jacques.hurtubise@mcgill.ca)
Steven Rayan
Affiliation:
Centre for Quantum Topology and Its Applications (quanTA) and Department of Mathematics & Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, CanadaS7N 5E6 (rayan@math.usask.ca)
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Abstract

For complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

Keywords

co-Higgs bundleprincipal bundleprojective linestabilitysimple rootsmodulistratification

MSC classification

Primary: 14H60: Vector bundles on curves and their moduli
Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 512 - 530
Copyright
Copyright © Edinburgh Mathematical Society 2020

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References

1.Anchouche, B., Azad, H. and Biswas, I., Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323 (2002), 693–712.CrossRefGoogle Scholar
2.Anchouche, B. and Biswas, I., Einstein–Hermitian connections on polystable principal bundles over a compact Kähler manifold, Amer. J. Math. 123 (2001), 207–228.CrossRefGoogle Scholar
3.Babbitt, D. G. and Varadarajan, V. S., Local moduli for meromorphic differential equations, Astérisque 169–170 (1989).Google Scholar
4.Balaji, V. and Parameswaran, A. J., Tensor product theorem for Hitchin pairs – an algebraic approach, Ann. Inst. Four. 61 (2011), 2361–2403.CrossRefGoogle Scholar
5.Birkhoff, G. D., Singular points of ordinary linear differential equations, Trans. Amer. Math. Soc. 10 (1909), 436–470.CrossRefGoogle Scholar
6.Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
7.Bottacin, F., Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. École Norm. Sup. 28 (1995), 391–433.CrossRefGoogle Scholar
8.Bradlow, S. B., García-Prada, O. and Mundet i Riera, I., Relative Hitchin–Kobayashi correspondences for principal pairs, Quart. J. Math. 54 (2003), 171–208.CrossRefGoogle Scholar
9.Friedman, R. and Morgan, J. W., Holomorphic principal bundles over elliptic curves. II. The parabolic construction, Jour. Diff. Geom. 56 (2000), 301–379.CrossRefGoogle Scholar
10.Gualtieri, M., Generalized complex geometry, Ann. Math. 174 (2011), 75–123.CrossRefGoogle Scholar
11.Grothendieck, A., Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138.CrossRefGoogle Scholar
12.Hitchin, N. J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.CrossRefGoogle Scholar
13.Hitchin, N. J., Lie groups and Teichmüller space, Topology 31 (1992), 449–473.CrossRefGoogle Scholar
14.Hitchin, N. J., Generalized Calabi–Yau manifolds, Quart. J. Math. 54 (2003), 281–308.CrossRefGoogle Scholar
15.Hitchin, N. J., Generalized holomorphic bundles and the B-field action, J. Geom. Phys. 61 (2011), 352–362.CrossRefGoogle Scholar
16.Humphreys, J. E., Linear algebraic groups, Graduate Texts in Mathematics, Volume 21 (Springer-Verlag, New York, 1975).CrossRefGoogle Scholar
17.Maruyama, M., Openness of a family of torsion free sheaves, J. Math. Kyoto Univ. 16 (1976), 627–637.CrossRefGoogle Scholar
18.García-Prada, O., Gothen, P. B. and Mundet i Riera, I., The Hitchin–Kobayashi correspondence, Higgs pairs, and surface group representations, preprint (arXiv.org/abs/0909.4487, 2012).Google Scholar
19.Markman, E., Spectral curves and integrable systems, Compos. Math. 93 (1994), 255–290.Google Scholar
20.Nitsure, N., Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275–300.CrossRefGoogle Scholar
21.Rayan, S., Geometry of co-Higgs bundles, D.Phil. Thesis (Oxford, 2011).Google Scholar
22.Rayan, S., Co-Higgs bundles on ℙ1, New York J. Math. 19 (2013), 925–945.Google Scholar
23.Rayan, S., Constructing co-Higgs bundles on ℙ2, Quart. J. Math. 65 (2014), 1437–1460.CrossRefGoogle Scholar
24.Schmitt, A. H. W., Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zurich, 2008).CrossRefGoogle Scholar