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A GIT interpretation of the Harder–Narasimhan filtration

Abstract

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a “maximally destabilizing” 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder–Narasimhan filtration.

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References

  1. Bruasse, L.: Optimal destabilizing vectors in some gauge theoretical moduli problems. Ann. Inst. Fourier (Grenoble) 56(6), 1805–1826 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bruasse, L., Teleman, A.: Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry. Ann. Inst. Fourier (Grenoble) 55(3), 1017–1053 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math. 106, 45–60 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hoskins, V., Kirwan, F.: Quotients of unstable subvarieties and moduli spaces of sheaves of Fixed Harder-Narasimhan type. Proc. Lond. Math. Soc. 105(3), 852–890 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Huybrechts, D., Lehn, M.: Framed modules and their moduli. Int. J. Math. 6(2), 297–324 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics E31. Vieweg, Braunschweig (1997)

    Book  Google Scholar 

  7. Kempf, G.: Instability in invariant theory. Ann. Math. 108(1, 2), 299–316 (1978)

    Google Scholar 

  8. Maruyama, M.: Moduli of stable sheaves, I. J. Math. Kyoto Univ. 17, 91–126 (1977)

    Google Scholar 

  9. Maruyama, M.: Moduli of stable sheaves, II. J. Math. Kyoto Univ. 18, 557–614 (1978)

    Google Scholar 

  10. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd edn., vol. 2, p 34. Springer, Berlin (1994)

  11. Newstead, P.E.: Lectures on Introduction to Moduli Problems and Orbit Spaces, Published for the Tata Institute of Fundamental Research. Bombay. Springer, Berlin (1978)

  12. Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tôhoku Math. J. 36, 269–291 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. I.H.E.S. 79, 47–129 (1994)

    Google Scholar 

  14. Zamora, A.: Harder-Narasimhan filtration for rank 2 tensors and stable coverings. arXiv:1306.5651, (2013) (submitted preprint)

  15. Zamora, A.: GIT characterizations of Harder-Narasimhan filtrations, Master Thesis, Universidad Complutense de Madrid, Available at e-prints UCM server, http://www.mat.ucm.es/invesmat/wp-content/uploads/2011/12/trabajo-master-curso-2008-09-alfonso-zamora (2009)

  16. Zamora, A.: GIT characterizations of Harder-Narasimhan filtrations, Ph.D. Thesis, Universidad Complutense de Madrid (2013)

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Acknowledgments

We thank Francisco Presas for discussions. This work was funded by the Grant MTM2010-17389 and ICMAT Severo Ochoa project SEV-2011-0087 of the Spanish Ministerio de Economía y Competitividad. A. Zamora was supported by a FPU Grant from the Spanish Ministerio de Educación. Finally A. Zamora would like to thank the Department of Mathematics at Columbia University, where part of this work was done, for hospitality. This work is part of A. Zamora’s Ph.D. thesis (c.f. [16]).

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Correspondence to Tomás L. Gómez.

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Gómez, T.L., Sols, I. & Zamora, A. A GIT interpretation of the Harder–Narasimhan filtration. Rev Mat Complut 28, 169–190 (2015). https://doi.org/10.1007/s13163-014-0149-3

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  • DOI: https://doi.org/10.1007/s13163-014-0149-3

Keywords

  • Vector bundles
  • Moduli space
  • Stability
  • Geometric invariant theory

Mathematics Subject Classification (2010)

  • Primary 14D20
  • Secondary 14L24