## Illinois Journal of Mathematics

### Higgs bundles over elliptic curves

#### Abstract

In this paper, we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres.

#### Article information

Source
Illinois J. Math., Volume 58, Number 1 (2014), 43-96.

Dates
First available in Project Euclid: 1 April 2015

https://projecteuclid.org/euclid.ijm/1427897168

Digital Object Identifier
doi:10.1215/ijm/1427897168

Mathematical Reviews number (MathSciNet)
MR3331841

Zentralblatt MATH identifier
06428021

#### Citation

Franco, Emilio; Garcia-Prada, Oscar; Newstead, P. E. Higgs bundles over elliptic curves. Illinois J. Math. 58 (2014), no. 1, 43--96. doi:10.1215/ijm/1427897168. https://projecteuclid.org/euclid.ijm/1427897168

#### References

• M. Aparicio Arroyo and O. Garcia-Prada, Higgs bundles for the Lorentz group, Illinois J. Math. 55 (2013), 1299–1326.
• M. F. Atiyah, Vector bundles over elliptic curves, Proc. Lond. Math. Soc. (3) 7 (1957), 414–452.
• M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 308 (1982), 523–615.
• J. Bernstein and O. Schwarzman, Chevalley's theorem for the complex crystallographic groups, J. Nonlinear Math. Phys. 13 (2006), no. 3, 323–351.
• I. Biswas and S. Ramanan, An infinitesimal study of the moduli of Hitchin pairs, J. Lond. Math. Soc. (2) 49 (1994), 219–231.
• K. Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361–382.
• S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc. (3) 55 (1987), 127–131.
• R. Friedman and J. Morgan, Holomorphic principal bundles over elliptic curves I, available at \arxivurlarXiv:math/9811130.
• R. Friedman and J. Morgan, Holomorphic principal bundles over elliptic curves II, J. Differential Geom. 56 (2000), 301–379.
• R. Friedman, J. Morgan and E. Witten, Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), 97–118.
• W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics, vol. 129, Springer, New York, 1991.
• O. Garcia-Prada, P. Gothen and I. Mundet i Riera, The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, available at \arxivurlarXiv:0909.4487v2.
• R. Hartshorne, Algebraic geometry, Springer, New York, 1977.
• N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 59–126.
• N. J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), no. 1, 91–114.
• F. Hreinsdottir, A case where choosing a product order makes the calculations of a Groebner basis much faster, J. Symbolic Comput. 18 (1994), 373–378.
• Y. Laszlo, About $G$-bundles over elliptic curves, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 2, 413–424.
• J. Le Potier, Lectures on vector bundles, Cambridge University Press, Cambridge, 1997.
• E. Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976), 17–32.
• D. Mumford, Projective invariants of projective structures and applications, Proc. internat. congr. mathematicians, Inst. Mittag-Leffler, Djursholm, 1962, pp. 526–530.
• D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd ed., Springer, Berlin, 1994.
• M. S. Narasimhan and C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567.
• P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research, Bombay, 1978. Reprinted by Narosa Publishing House, 2012.
• N. Nitsure, Moduli space of semistable pairs on a curve, Proc. Lond. Math. Soc. (3) 62 (1991), 275–300.
• V. L. Popov, Irregular and singular loci of commuting varieties, Transform. Groups 13 (2008), 819–837.
• A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), 653–683.
• S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 151–166.
• A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152.
• A. Ramanathan, Moduli for principal bundles over algebraic curves: I, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 3, 301–328.
• A. Ramanathan, Moduli for principal bundles over algebraic curves: II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 421–449.
• R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compos. Math. 38 (1979), no. 3, 311–327.
• C. Schweigert, On moduli spaces of flat connections with non-simply connected structure group, Nuclear Phys. B 492 (1997), 743–755.
• C. T. Simpson, Higgs bundles and local systems, Publ. Math. Inst. Hautes Etudes Sci. 75 (1992), 5–95.
• C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. Inst. Hautes Etudes Sci. 79 (1994), 47–129.
• C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety II, Publ. Math. Inst. Hautes Etudes Sci. 80 (1995), 5–79.
• L. Tu, Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), 1–26.