Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.

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Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.
L.Alvarez-Consul and O. Garcıa-Prada, Dimensional reduction, SL(2,C)-equivariant bundles and stable holomorphic chains, Internat. J. Math. 12 (2001) 159–201. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy.Soc. London Ser. A 308 (1982) 523–615. O. Biquard, Fibres paraboliques stables et connexions singulieres plates, Bull. Soc. Math. Fr. 119 (1991) 231–257. O. Biquard and O. Garcıa-Prada, Parabolic vortex equations and instantons of infinite energy, J.Geom. Physics 21 (1997) 238–254. I. Biswas and S. Ramanan, An infinitesimal study of the moduli of Hitchin pairs, J. London Math.Soc. (2) 49 (1994) 219–231. H.U. Boden and K. Yokogawa, Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves. I, Internat. J. Math. 7 (1996) 573–598. F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. Ec. Norm. Super.IV. S´er. 28 (1995) 391–433. S.B. Bradlow and O. Garcıa-Prada, Stable triples,equivariant bundles and dimensional reduction,Math. Ann. 304 (1996) 225–252. S.B. Bradlow, O. Garcıa-Prada and P.B. Gothen, Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann. 328 (2004) 299–351. S.A. Cherkis and A. Kapustin, Hyper-Kahler metrics from periodic monopoles, Physical Review D, 65 (2002) 084015. K. Corlette, Flat G-bundles with canonical metrics, J.Differential Geom. 28 (1988) 361–382. S.K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc.55 (1987) 127–131. T. Frankel, Fixed points and torsion on K¨ahler manifolds, Ann. Math. 70 (1959) 1–8. V. Ginzburg, The global nilpotent variety is Lagrangian, Duke Math. J. 109 (2001) 511–519. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons (1978). P.B. Gothen, The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface, Internat. J. Math. 5 (1994) 861–875. P.B. Gothen, Components of spaces of representations and stable triples, Topology 40 (2001) 823–850. P.B. Gothen and A.D. King, Homological algebra of twisted quiver bundles, J. London Math. Soc.(2) 71 (2005) 85–99. T. Hausel, Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998) 169–192. T. Hausel, Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve,in Geometric Methods in Algebra and Number Theory, Progress in Mathematics, Vol.235 F. Bogomolov,Y. Tschinkel (Eds.) 2005. T. Hausel and M. Thaddeus, Examples of mirror partners arising from integrable systems, C. R.Acad. Sci. Paris Ser. 1 333 (2001) 313–318. T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent.Math. 153 (2003) 197–229. N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55(1987) 59–126. N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91–114. Y.I. Holla, Poincar´e polynomial of the moduli space of parabolic bundles, Proc. Indian Acad. Sci 110 (2000) 233–261. H. Konno, Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45 (1993) 253–276. H. Lange, Universal families of extensions, J. Algebra 83 (1983) 101–112. G. Laumon, Un analogue global du cˆone nilpotent, Duke Math. J. 57(1988) 647–671. I.G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319–343. E.Markman, Spectral curves and integrable systems, Comp. Math. 93 (1994) 255–290. M. Maruyama and K. Yokogawa,Moduli of parabolic stable sheaves, Math. Ann. 293 (1992) 77–99. H. Nakajima, Hyper-Kahler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, Moduli of vector bundles. Marcel Dekker. Lect. Notes Pure Appl. Math. 179 (1996) 199–208. Edited by Maruyama and Masaki. B. Nasatyr and B. Steer, Orbifold Riemann surfaces and the Yang–Mills–Higgs equations, Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995) 595–643. N. Nitsure, Cohomology of the moduli space of parabolic vector bundles, Proc. Indian Acad. Sci 95 (1986) 61–77. C.T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713–770.