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On moduli spaces of Hitchin pairs

Published online by Cambridge University Press:  18 July 2011

INDRANIL BISWAS ,
PETER B. GOTHEN  and
MARINA LOGARES
INDRANIL BISWAS
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. e-mail: indranil@math.tifr.res.in
PETER B. GOTHEN
Affiliation:
Departamento de Matematica Pura, Facultade de Ciencias, Rua do Campo Alegre 687, 4169-007 PortoPortugal. e-mail: pbgothen@fc.up.pt
MARINA LOGARES
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Serrano 113 Bis 28006 Madrid. Spain. e-mail: mlogares@fc.up.pt
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Abstract

Let X be a compact Riemann surface X of genus at–least two. Fix a holomorphic line bundle L over X. Let be the moduli space of Hitchin pairs (E, φ ∈ H0(End0(E) ⊗ L)) over X of rank r and fixed determinant of degree d. The following conditions are imposed:

  1. (i) deg(L) ≥ 2g−2, r ≥ 2, and L⊗rKX⊗r;

  2. (ii) (r, d) = 1; and

  3. (iii) if g = 2 then r ≥ 6, and if g = 3 then r ≥ 4.

We prove that that the isomorphism class of the variety uniquely determines the isomorphism class of the Riemann surface X. Moreover, our analysis shows that is irreducible (this result holds without the additional hypothesis on the rank for low genus).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 441 - 457
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[AS]Arapura, D. and Sastry, P.Intermediate Jacobian and Hodge structures of moduli spaces. Proc. Indian Acad. Sci. (Math. Sci.) 110 (2000), 1–26.CrossRefGoogle Scholar
[AB]Atiyah, M. F. and Bott, R.The Yang-Mills equations over Riemann surfaces. Phil. Trans. Roy. Soc. Lond. A308 (1982), 523–615.Google Scholar
[BG]Biswas, I. and Gómez, T. L.A Torelli theorem for the moduli space of Higgs bundles on a curve. Quart. J. Math. 54 (2003), 159–169.CrossRefGoogle Scholar
[BR]Biswas, I. and Ramanan, S.An infinitesimal study of the moduli of Hitchin pairs. J. London Math. Soc. 49 (1994), 219–231.CrossRefGoogle Scholar
[Bo]Bottacin, F.Symplectic geometry on moduli spaces of stable pairs. Ann. Sci. École Norm. Sup. 28 (1995), 391–433.CrossRefGoogle Scholar
[BGG]Bradlow, S. B., García–Prada, O. and Gothen, P. B.Homotopy groups of moduli spaces of representations. Topology 47 (2008), 203–224.CrossRefGoogle Scholar
[BGM]Bradlow, S. B., García-Prada, O. and i Riera, I. MundetRelative Hitchin–Kobayashi correspondences for principal pairs. Quart. J. Math. 54 (2003), 171–208.CrossRefGoogle Scholar
[Ca]Carlson, J.Extensions of mixed Hodge structures, in: Journées de géométrie algébrique d'Angers Juillet 1979/Algebraic Geometry, Angers, 1979 (Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980), pp. 107–127.Google Scholar
[De1]Deligne, P.Theorie de Hodge I. Proc. Int. Cong. Math. 1 (1970), 425–430.Google Scholar
[De2]Deligne, P.Theorie de Hodge II. Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–58.CrossRefGoogle Scholar
[De3]Deligne, P.Theorie de Hodge III. Publ. Math. Inst. Hautes Études Sci. 44 (1974), 5–77.CrossRefGoogle Scholar
[GK]Gothen, P. B. and King, A. D.Homological algebra of twisted quiver bundles. J. London Math. Soc. 71 (2005), 85–99.CrossRefGoogle Scholar
[GH]Griffiths, P. and Harris, J.Principles of Algebraic Geometry. (Wiley Classics Library, John Wiley and Sons, Inc., 1994).CrossRefGoogle Scholar
[Ha]Hausel, T.Compactification of moduli of Higgs bundles. J. reine angew. Math. 503 (1998), 169–192.CrossRefGoogle Scholar
[Hi]Hitchin, N. J.The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55 (1987), 59–126.CrossRefGoogle Scholar
[Ki]Kirwan, F. C.Cohomology of quotients in symplectic and algebraic geometry. Math. Notes 31 (Princeton University Press, 1984).Google Scholar
[Li]Lin, T. R. Hermitian–Yang–Mills–Higgs metrics and stability for holomorphic vector bundles with Higgs fields over a compact Kähler manifold. Ph. D. Thesis UC–San Diego. (1989).Google Scholar
[Ma]Markman, E.Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces. Adv. Math. 208 (2007), 622–646.CrossRefGoogle Scholar
[MN]Mumford, D. and Newstead, P.Periods of a moduli space of bundles on curves, Amer. J. Math. 90 (1968), 1200–1208.CrossRefGoogle Scholar
[Mu]Muñoz, V.Torelli theorem for the moduli space of pairs. Math. Proc. Camb. Phil. Soc. 146 (2009), 675–693.CrossRefGoogle Scholar
[NR]Narasimhan, M. S. and Ramanan, S.Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. 101 (1975), 391–417.CrossRefGoogle Scholar
[Ni]Nitsure, N.Moduli space of semistable pairs on a curve. Proc. London Math. Soc. 62 (1991), 275–300.CrossRefGoogle Scholar
[Sch]Schmitt, A. H.Projective moduli for Hitchin pairs. Internat. J. Math. 9 (1998), no. 1, 107–118.CrossRefGoogle Scholar
[Sh]Shatz, S. S.The decomposition and specialization of algebraic families of vector bundles. Composition. Math. 45 (1977), 163–187.Google Scholar
[Si1]Simpson, C. T.The Hodge filtration on non-abelian cohomology. Algebraic Geometry—(Santa Cruz 1995) 217–281. Proc. Sympos. Pure Math., 62, Part 2, (Amer. Math. Soc., Providence, RI, 1997).CrossRefGoogle Scholar
[Si2]Simpson, C. T.Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.CrossRefGoogle Scholar
[Si3]Simpson, C. T.Katz's middle convolution algorithm. Pure Appl. Math. Q. 5 (2009), no. 2, Special Issue: in honor of Friedrich Hirzebruch. Part 1. 781–852.CrossRefGoogle Scholar
[Si4]Simpson, C. T.Moduli of representations of a smooth Projective Variety II. Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79.CrossRefGoogle Scholar