Abstract
We construct a Harder–Narasimhan filtration for rank 2 tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank 2 tensors over a smooth projective complex variety, does not depend on a certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder–Narasimhan filtration for rank 2 tensors as this unique filtration coming from GIT. Symmetric rank 2 tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder–Narasimhan filtration in terms of intersection theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gieseker D, Geometric invariant theory and the moduli of bundles, Gauge theory and the topology of four-manifolds (eds) Robert Friedman and John W Morgan, IAS/Park City Mathematics Series, v. 4 (1994)
Gómez T and Sols I, Stable tensors and moduli space of orthogonal sheaves, Preprint 2001, math.AG/0103150
Gómez T, Sols I and Zamora A, A GIT characterization of the Harder–Narasimhan filtration, Rev. Mat. Complut. 28(1) (2015) 169–190
Gómez T, Langer A, Schmitt A H W and Sols I, Moduli spaces for principal bundles in arbitrary characteristic, Adv. Math. 219 (2008) 1177–1245
Hartshorne R, Algebraic Geometry, Grad. Texts in Math. 52 (1977) (Springer Verlag)
Hesselink W H, Uniform instability in reductive groups, J. Reine Angew. Math. 304 (1978) 74–96
Huybrechts D and Lehn M, Framed modules and their moduli, Int. J. Math. 6 (2) (1995) 297–324
Huybrechts D and Lehn M, The geometry of moduli spaces of sheaves, Aspects of Mathematics E31, Vieweg, Braunschweig/Wiesbaden (1997)
Harder G and Narasimhan M S, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975) 215–248
Kempf G, Instability in invariant theory, Ann. Math. (2) 108 (1) (1978) 299–316
Mumford D, Fogarty J and Kirwan F, Geometric invariant theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34 (1994) (Berlin: Springer-Verlag)
Pustetto A, Mehta–Ramanathan for ε and k-semistable decorated sheaves, to appear in Geom. Dedicata (2015), doi: 10.1007/s10711-015-0132-2
Simpson C, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. I.H.E.S. 79 (1994) 47–129
Zamora A, On the Harder–Narasimhan filtration of finite dimensional representations of quivers, Geom. Dedicata 170 (1) (2014) 185–194
Zamora A, GIT characterizations of Harder–Narasimhan filtrations, Ph.D. thesis. Universidad Complutense de Madrid (2013)
Acknowledgements
The author wishes to thank Usha Bhosle, Tomás L Gómez and Peter Newstead for useful discussions, and to the Isaac Newton Institute for Mathematical Sciences in Cambridge, United Kingdom, where part of this work was done, for hospitality. This work has been supported by project MTM2010-17389 and ICMAT Severo Ochoa project SEV-2011-0087 granted by Spanish Ministerio de Economía y Competitividad. The author was also supported by a FPU grant from the Spanish Ministerio de Educación.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: Parameswaran Sankaran
Rights and permissions
About this article
Cite this article
ZAMORA, A. Harder–Narasimhan filtration for rank 2 tensors and stable coverings. Proc Math Sci 126, 305–327 (2016). https://doi.org/10.1007/s12044-016-0283-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-016-0283-6