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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bruasse, L.: Optimal destabilizing vectors in some gauge theoretical moduli problems. Ann. Inst. Fourier (Grenoble) 56(6), 1805–1826 (2006)
Bruasse, L., Teleman, A.: Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry. Ann. Inst. Fourier (Grenoble) 55(3), 1017–1053 (2005)
Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math. 106, 45–60 (1977)
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)
Huybrechts, D., Lehn, M.: Framed modules and their moduli. Int. J. Math. 6(2), 297–324 (1995)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics E31. Vieweg, Braunschweig (1997)
Kempf, G.: Instability in invariant theory. Ann. Math. 108(1, 2), 299–316 (1978)
Maruyama, M.: Moduli of stable sheaves, I. J. Math. Kyoto Univ. 17, 91–126 (1977)
Maruyama, M.: Moduli of stable sheaves, II. J. Math. Kyoto Univ. 18, 557–614 (1978)
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)
Newstead, P.E.: Lectures on Introduction to Moduli Problems and Orbit Spaces, Published for the Tata Institute of Fundamental Research. Bombay. Springer, Berlin (1978)
Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tôhoku Math. J. 36, 269–291 (1984)
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)
Zamora, A.: Harder-Narasimhan filtration for rank 2 tensors and stable coverings. arXiv:1306.5651, (2013) (submitted preprint)
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)
Zamora, A.: GIT characterizations of Harder-Narasimhan filtrations, Ph.D. Thesis, Universidad Complutense de Madrid (2013)
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]).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-014-0149-3