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International Mathematics Research Papers (2006) Vol. 2006 : article ID 73597, 82 pages, doi:10.1155/IMRP/2006/73597
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces

L. Álvarez-Cónsul, O. García-Prada and A. H. W. Schmitt

Departamento de Matemáticas, Instituto de Matemáticas y Física Fundamental (IMAFF), Consejo Superior de Investigaciones Científicas (CSIC) Serrano 123, 28006 Madrid, Spain E-mail address: lac{at}mat.csic.es
Departamento de Matemáticas, Instituto de Matemáticas y Física Fundamental (IMAFF), Consejo Superior de Investigaciones Científicas (CSIC) Serrano 121, 28006 Madrid, Spain E-mail address: oscar.garcia-prada{at}uam.es
Fachbereich Mathematik, Universität Duisburg-Essen Campus Essen, 45117 Essen, Germany E-mail address: alexander.schmitt{at}uni-essen.de

We study holomorphic (n + 1)-chains En -> En–1 -> ··· -> E0 consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on n real parameters was introduced by the first two authors and moduli spaces were constructed by the third author. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples). Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.



References

  1. Álvarez-Cónsul L., García-Prada O. Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains. International Journal of Mathematics (2001) 12(2):159–201.[CrossRef][Web of Science]
  2. Álvarez-Cónsul L., García-Prada O. Hitchin-Kobayashi correspondence, quivers, and vortices. Communications in Mathematical Physics (2003) 238(1-2):1–33.[Web of Science]
  3. Bradlow S. B. Special metrics and stability for holomorphic bundles with global sections. Journal of Differential Geometry (1991) 33(1):169–213.[Web of Science]
  4. Bradlow S. B., García-Prada O. Stable triples, equivariant bundles and dimensional reduction. Mathematische Annalen (1996) 304(2):225–252.[CrossRef][Web of Science]
  5. Bradlow S. B., García-Prada O., Gothen P. B. Surface group representations and U(p, q)-Higgs bundles. Journal of Differential Geometry (2003) 64(1):111–170.[Web of Science]
  6. Bradlow S. B., García-Prada O., Gothen P. B. Moduli spaces of holomorphic triples over compact Riemann surfaces. Mathematische Annalen (2004) 328(1-2):299–351.[CrossRef][Web of Science]
  7. Corlette K. Flat G-bundles with canonical metrics. Journal of Differential Geometry (1988) 28(3):361–382.[Web of Science]
  8. Donaldson S. K. Twisted harmonic maps and the self-duality equations. Proceedings of the London Mathematical Society. Third Series (1987) 55(1):127–131.
  9. Gabriel P. Représentations indécomposables. In: Séminaire Bourbaki, 26ème année (1973/1974), Exp. No. 444 (1975) 431. Berlin: Springer. 143–169. Lecture Notes in Math.
  10. García-Prada O. Dimensional reduction of stable bundles, vortices and stable pairs. International Journal of Mathematics (1994) 5(1):1–52.[CrossRef]
  11. García-Prada O., Gothen P. B., Muñoz V. Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. to appear in Memoirs of the American Mathematical Society.
  12. Gothen P. B. The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface. International Journal of Mathematics (1994) 5(6):861–875.[CrossRef]
  13. Gothen P. B. Topology of U(2, 1) representation spaces. The Bulletin of the London Mathematical Society (2002) 34(6):729–738.[CrossRef]
  14. Gothen P. B., King A. D. Homological algebra of twisted quiver bundles. Journal of the London Mathematical Society. Second Series (2005) 71(1):85–99.[Web of Science]
  15. Hitchin N. J. The self-duality equations on a Riemann surface. Proceedings of the London Mathematical Society. Third Series (1987) 55(1):59–126.
  16. Hoffmann N. Rationality and Poincaré families for vector bundles with extra structure on a curve. http://arxiv.org/abs/math.AG/0511656.
  17. King A. D. Moduli of representations of finite-dimensional algebras. The Quarterly Journal of Mathematics. Oxford. Second Series (1994) 45(180):515–530.
  18. Lange H. Universal families of extensions. Journal of Algebra (1983) 83(1):101–112.[CrossRef][Web of Science]
  19. Laumon G. Fibrés vectoriels spéciaux. Bulletin de la Société Mathématique de France (1991) 119(1):97–119.[Web of Science]
  20. Logares M. Betti numbers of parabolic U(2, 1)-Higgs bundle moduli spaces. http://arxiv.org/abs/math.AG/0601342.
  21. Mercat V. Le problème de Brill-Noether: présentation. preprint, January 2001, 22 pages, http://www.liv.ac.uk/~newstead/bnt.html.
  22. Schmitt A. H. W. Moduli problems of sheaves associated with oriented trees. Algebras and Representation Theory (2003) 6(1):1–32.[CrossRef][Web of Science]
  23. Schmitt A. H. W. Global boundedness for decorated sheaves. International Mathematics Research Notices (2004) 2004(68):3637–3671.[Abstract/Free Full Text]
  24. Schmitt A. H. W. Moduli for decorated tuples of sheaves and representation spaces for quivers. Proceedings of the Indian Academy of Mathematical Sciences (2005) 115(1):15–49.
  25. Simpson C. T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. Journal of the American Mathematical Society (1988) 1(4):867–918.[CrossRef]
  26. Sundaram N. Special divisors and vector bundles. The Tohoku Mathematical Journal. Second Series (1987) 39(2):175–213.
  27. Thaddeus M. Stable pairs, linear systems and the Verlinde formula. Inventiones Mathematicae (1994) 117(2):317–353.[CrossRef][Web of Science]
  28. Uhlenbeck K., Yau S.-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Communications on Pure and Applied Mathematics (1986) 39–S:257–293.[CrossRef]
  29. Uhlenbeck K., Yau S.-T. A note on our previous paper: "On the existence of Hermitian-Yang-Mills connections in stable vector bundles". Communications on Pure and Applied Mathematics (1989) 42(5):703–707.[CrossRef][Web of Science]

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