Optimal destabilizing vectors in some Gauge theoretical moduli problems

Laurent Bruasse[1]

  • [1] CMI, LAPT UMR 6632 39, rue Frédéric Joliot Curie 13453 Marseille Cedex 13 (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 6, page 1805-1826
  • ISSN: 0373-0956

Abstract

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We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing 1 -parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called “maximal weight function”. We give a complete description of these optimal destabilizing endomorphisms. Then we show how this principle may be applied to an other complex moduli problem: holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with fixed source) over a complex curve. We get here a new version of the Harder-Narasimhan filtration theorem for the notion of τ -stability. These results suggest that the principle holds in the whole gauge theoretical framework.

How to cite

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Bruasse, Laurent. "Optimal destabilizing vectors in some Gauge theoretical moduli problems." Annales de l’institut Fourier 56.6 (2006): 1805-1826. <http://eudml.org/doc/10192>.

@article{Bruasse2006,
abstract = {We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing $1$-parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called “maximal weight function”. We give a complete description of these optimal destabilizing endomorphisms. Then we show how this principle may be applied to an other complex moduli problem: holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with fixed source) over a complex curve. We get here a new version of the Harder-Narasimhan filtration theorem for the notion of $\tau $-stability. These results suggest that the principle holds in the whole gauge theoretical framework.},
affiliation = {CMI, LAPT UMR 6632 39, rue Frédéric Joliot Curie 13453 Marseille Cedex 13 (France)},
author = {Bruasse, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {GIT; optimal $1$-parameter subgroup; gauge theory; maximal weight map; complex moduli problem; stability; Harder-Narasimhan filtration; moment map; optimal 1-parameter subgroup; geometric invariant theory},
language = {eng},
number = {6},
pages = {1805-1826},
publisher = {Association des Annales de l’institut Fourier},
title = {Optimal destabilizing vectors in some Gauge theoretical moduli problems},
url = {http://eudml.org/doc/10192},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Bruasse, Laurent
TI - Optimal destabilizing vectors in some Gauge theoretical moduli problems
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 6
SP - 1805
EP - 1826
AB - We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing $1$-parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called “maximal weight function”. We give a complete description of these optimal destabilizing endomorphisms. Then we show how this principle may be applied to an other complex moduli problem: holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with fixed source) over a complex curve. We get here a new version of the Harder-Narasimhan filtration theorem for the notion of $\tau $-stability. These results suggest that the principle holds in the whole gauge theoretical framework.
LA - eng
KW - GIT; optimal $1$-parameter subgroup; gauge theory; maximal weight map; complex moduli problem; stability; Harder-Narasimhan filtration; moment map; optimal 1-parameter subgroup; geometric invariant theory
UR - http://eudml.org/doc/10192
ER -

References

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