Optimal destabilizing vectors in some Gauge theoretical moduli problems
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1805-1826.

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.

Nous montrons que, du point de vue de la théorie de jauge, la filtration de Harder-Narasimhan d’un fibré vectoriel complexe au-dessus d’une courbe et la notion de sous-groupe déstabilisant optimal à un paramètre coïncident.

En utilisant l’approche de la GIT, la filtration de Harder-Narasimhan apparaît comme un objet limite pour l’action du groupe de jauge, dans la direction d’un vecteur déstabilisant optimal. Ce vecteur est un extremum de la “fonction de poids maximal”. Nous donnons une description complète de ces vecteurs déstabilisants optimaux. Nous montrons que le même principe s’applique à un autre problème de modules : celui des paires holomorphes (un fibré vectoriel complexe couplé avec un morphisme) sur une courbe complexe. On obtient dans ce contexte une nouvelle version du théorème de filtration de Harder-Narasimhan pour la notion de τ-stabilité. Ces résultats suggèrent que le principe reste valable en toute généralité en théorie de jauge.

DOI: 10.5802/aif.2228
Classification: 32M05,  53D20,  14L24,  14L30,  32L05,  32G13,  53C55
Keywords: GIT, optimal 1-parameter subgroup, gauge theory, maximal weight map, complex moduli problem, stability, Harder-Narasimhan filtration, moment map.
Bruasse, Laurent 1

1 CMI, LAPT UMR 6632 39, rue Frédéric Joliot Curie 13453 Marseille Cedex 13 (France)
@article{AIF_2006__56_6_1805_0,
     author = {Bruasse, Laurent},
     title = {Optimal destabilizing vectors in some {Gauge} theoretical moduli problems},
     journal = {Annales de l'Institut Fourier},
     pages = {1805--1826},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     doi = {10.5802/aif.2228},
     mrnumber = {2282676},
     zbl = {1112.32008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2228/}
}
TY  - JOUR
AU  - Bruasse, Laurent
TI  - Optimal destabilizing vectors in some Gauge theoretical moduli problems
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 1805
EP  - 1826
VL  - 56
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2228/
UR  - https://www.ams.org/mathscinet-getitem?mr=2282676
UR  - https://zbmath.org/?q=an%3A1112.32008
UR  - https://doi.org/10.5802/aif.2228
DO  - 10.5802/aif.2228
LA  - en
ID  - AIF_2006__56_6_1805_0
ER  - 
%0 Journal Article
%A Bruasse, Laurent
%T Optimal destabilizing vectors in some Gauge theoretical moduli problems
%J Annales de l'Institut Fourier
%D 2006
%P 1805-1826
%V 56
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2228
%R 10.5802/aif.2228
%G en
%F AIF_2006__56_6_1805_0
Bruasse, Laurent. Optimal destabilizing vectors in some Gauge theoretical moduli problems. Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1805-1826. doi : 10.5802/aif.2228. http://archive.numdam.org/articles/10.5802/aif.2228/

[1] Atiyah, M.; Bott, R. The Yang-Mills equations over a Riemann surface, Phil. Trans. R. Soc., Volume 308 (1983), pp. 523-615 | DOI | MR | Zbl

[2] Bradlow, S. B. Special metrics and stability for holomorphic bundles with global sections, J. Diff. Geom., Volume 33 (1991), pp. 169-213 | MR | Zbl

[3] Bradlow, S. B.; Daskalopoulos, G. D.; Wentworth, R. A. Birational equivalences of vortex moduli, Topology, Volume 35 (1996) no. 3, pp. 731-748 | DOI | MR | Zbl

[4] Bruasse, L. Harder-Narasimhan filtration on non Kähler manifolds, Int. Journal of Maths, Volume 12 (2001) no. 5, pp. 579-594 | DOI | MR | Zbl

[5] Bruasse, L. Stabilité et filtration de Harder-Narasimhan, Université Aix-Marseille I, CMI, dec (2001) (Ph. D. Thesis)

[6] Bruasse, L. Filtration de Harder-Narasimhan pour des fibrés complexes ou des faisceaux sans-torsion, Ann. Inst. Fourier, Volume 53 (2003) no. 2, pp. 539-562 | DOI | Numdam | MR | Zbl

[7] Bruasse, L.; Teleman, A. Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry, Ann. Inst. Fourier, Volume 55 (2005) no. 3, pp. 1017-1053 | DOI | Numdam | MR | Zbl

[8] Daskalopoulos, G. D. The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geometry, Volume 36 (1992), pp. 699-746 | MR | Zbl

[9] Harder, G.; Narasimhan, M. On the cohomology groups of moduli spaces, Math. Ann., Volume 212 (1975), pp. 215-248 | DOI | MR | Zbl

[10] Kempf, G. R. Instability in invariant theory, Ann. of Mathematics, Volume 108 (1978), pp. 299-316 | DOI | MR | Zbl

[11] Kirwan, F. C. Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, Volume 31, Princeton University Press, 1984 | MR | Zbl

[12] Lübke, M.; Teleman, A. The universal Kobayashi-Hitchin correspondance on Hermitian manifolds (2004) (preprint arXiv: math.DG/0402341, to appear in Memoirs of the AMS)

[13] Maruyama, M. The theorem of Grauert-Mülich-Spindler, Math. Ann., Volume 255 (1981), pp. 317-333 | DOI | MR | Zbl

[14] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, 1982 | MR | Zbl

[15] Mundet i Riera, I. Yang-Mills-Higgs theory for symplectic fibrations, Universad Autonoma de Madrid (1999) (Ph. D. Thesis)

[16] Mundet i Riera, I. A Hitchin-Kobayashi correspondence for Kähler fibrations, J. reine angew. Maths (2000) no. 528, pp. 41-80 | DOI | MR | Zbl

[17] Ramanan, S.; Ramanathan, A. Some remarks on the instability flag, Tôhoku Math. Journ., Volume 36 (1984), pp. 269-291 | DOI | MR | Zbl

[18] Shatz, S. The decomposition and specialization of algebraic families of vector bundles, Composito. Math., Volume 35 (1977), pp. 163-187 | Numdam | MR | Zbl

[19] Teleman, A. Symplectic stability, analytic stability in non-algebraic complex geometry, Int. Journal of Maths, Volume 15 (2004) no. 2, pp. 183-209 | DOI | MR | Zbl

Cited by Sources: