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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.52 no.1 Bogotá Jan./June 2018
https://doi.org/10.15446/recolma.v1n52.74525
Original articles
Stabilization of the Homotopy Groups of the Moduli Spaces of k-Higgs Bundles
Estabilización de los Grupos de Homotopía de los Espacios Móduli de los k-Fibrados de Higgs
1 Universidad de Costa Rica UCR, San José - Costa Rica
The work of Hausel proves that the Białynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli spaces of k-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of M k (2, d), and generalize it to M k (3, d), the moduli spaces of k-Higgs bundles of degree d, and ranks two and three respectively, over a compact Riemann surface X, using the results from the works of Hausel and Thaddeus, among other tools.
Keywords: Moduli of Higgs Bundles; Variations of Hodge Structures; Vector Bundles
El trabajo de Hausel prueba que la estratificación de Białynicki-Birula del espacio moduli de fibrados de Higgs de rango dos coincide con su estratificación de Shatz. Él usa este hecho para calcular algunos grupos de homotopía del espacio moduli de k-fibrados de Higgs de rango dos. Desafortunadamente, estas dos estratificaciones no coinciden en general. Aquí, el objetivo es presentar una prueba diferente de la estabilización de los grupos de homotopía de M k (2, d), y generalizarla a Mk(3, d), los espacios moduli de k-fibrados de Higgs de grado d, y rangos dos y tres respectivamente, sobre una superficie de Riemann compacta X, usando los resultados de los trabajos de Hausel y Thaddeus, entre otras herramientas.
Palabras clave: Moduli de Fibrados de Higgs; Variaciones de Estructuras de Hodge; Fibrados Vectoriales
References
1. Atiyah, M. F., Vector Bundles and Künneth Formula, Topology 1 (1962), 245-248. [ Links ]
2. Atiyah, M. F., K-Theory, W. A. Benjamin, New York-Amsterdam, 1967. [ Links ]
3. Atiyah, M. F. and Bott, R., Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 523-615. [ Links ]
4. Bento, S., Topologia do Espaco Moduli de Fibrados de Higgs Torcidos, Tese de Doutoramento, Universidade do Porto, Porto, Portugal, 2010. [ Links ]
5. Bradlow, S. B. and García-Prada, O., Stable triples, equivariant bundles and dimensional reduction, Math. Ann. 304 (1996), 225-252. [ Links ]
6. Bradlow, S. B., García-Prada, O. and Gothen, P. B., Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann. 328 (2004), 299-351. [ Links ]
7. Bradlow, S. B., García-Prada, O. and Gothen, P. B., Homotopy groups of moduli spaces of representations, Topology 47 (2008), 203-224. [ Links ]
8. Frankel, T., Fixed points and torsion on Kähler manifolds, Ann. Math. 70 (1959), 1-8. [ Links ]
9. Fulton, W., Algebraic Topology, A First Course, Springer, New York, 1995. [ Links ]
10. Gothen, P. B., The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface, Int. J. Math. 5 (1994), no. 1, 861-875. [ Links ]
11. Gothen, P. B. and Zúñiga-Rojas, R. A., Stratifications on the moduli space of Higgs bundles, Portugaliae Mathematica, to appear. [ Links ]
12. Griffiths, P. and Harris, J., Principles of Algebraic Geometry, Wiley, New York, 1978. [ Links ]
13. Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002. [ Links ]
14. Hausel, T., Geometry of the moduli space of Higgs bundles, Ph. D. thesis, Cambridge, 1998. [ Links ]
15. Hausel, T. and Thaddeus, M., Mirror symmetry, Langlads duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229. [ Links ]
16. Hausel, T. and Thaddeus, M., Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles, Proc. London Math. Soc. 88 (2004), 632-658. [ Links ]
17. Hitchin, N. J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126. [ Links ]
18. Husemoller, D., Fibre Bundles, third edition, Graduate Texts in Mathematics 20, Springer, New York , 1994. [ Links ]
19. James, I. M. and ed., Handbook of Algebraic Topology, (1995). [ Links ]
20. Macdonald, I. G., Symmetric products of an algebraic curve, Topology 1 (1962), 319-343. [ Links ]
21. Markman, E., Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, Adv. Math. 208 (2007), 622-646. [ Links ]
22. Muñoz, V., Ortega, D. and Vázquez-Gallo, M. J., Hodge polynomials of the moduli spaces of pairs, Int. J. Math. 18 (2007), 695-721. [ Links ]
23. Nitsure, N., Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275-300. [ Links ]
24. Simpson, C. T., Constructing variations of Hodge structures using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918. [ Links ]
25. Simpson, C. T., Higgs bundles and local systems, Publ. Math. IHÉS (1992), 5-95. [ Links ]
26. Zúñiga-Rojas, R. A., Homotopy groups of the moduli space of Higgs bundles, Ph. D. thesis, Porto, Portugal, 2015. [ Links ]
Received: May 15, 2017; Accepted: October 07, 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
