Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry
[Filtration de Harder-Narasimhan et vecteurs déstabilisants optimaux en géométrie complexe]
Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 1017-1053.

Nous généralisons ici la théorie des sous-groupes déstabilisants optimaux à un paramètre dans un cadre non algébrique : celui des actions holomorphes de groupes de Lie complexes réductifs sur une variété kählerienne de dimension finie (compacte ou non). Dans une seconde partie, nous montrons comment ces résultats peuvent s'étendre dans le cadre de la théorie de jauge, nous explorons la relation entre filtration de Harder-Narasimhan et vecteur déstabilisant optimal d'un objet non semistable.

We give here a generalization of the theory of optimal destabilizing 1-parameter subgroups to non algebraic complex geometry : we consider holomorphic actions of a complex reductive Lie group on a finite dimensional (possibly non compact) Kähler manifold. In a second part we show how these results may extend in the gauge theoretical framework and we discuss the relation between the Harder-Narasimhan filtration and the optimal detstabilizing vectors of a non semistable object.

DOI : 10.5802/aif.2120
Classification : 32M05, 53D20, 14L24, 14L30, 32L05, 32Q15
Keywords: symplectic actions, Hamiltonian actions, stability, Harder Narasimhan filtration, Shatz stratification, gauge theory.
Mot clés : action symplectique, action hamiltonienne, stabilité, filtration de Harder-Narasimhan, stratification de Shatz, théorie de jauge
Bruasse, Laurent 1 ; Teleman, Andrei 

1 IML, CNRS UPR 9016, 163 avenue de Luminy, 13288 Marseille cedex 09 (France), CMI, LATP UMR 6632, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13 (France)
@article{AIF_2005__55_3_1017_0,
     author = {Bruasse, Laurent and Teleman, Andrei},
     title = {Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry},
     journal = {Annales de l'Institut Fourier},
     pages = {1017--1053},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     doi = {10.5802/aif.2120},
     mrnumber = {2149409},
     zbl = {1093.32009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2120/}
}
TY  - JOUR
AU  - Bruasse, Laurent
AU  - Teleman, Andrei
TI  - Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 1017
EP  - 1053
VL  - 55
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2120/
DO  - 10.5802/aif.2120
LA  - en
ID  - AIF_2005__55_3_1017_0
ER  - 
%0 Journal Article
%A Bruasse, Laurent
%A Teleman, Andrei
%T Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry
%J Annales de l'Institut Fourier
%D 2005
%P 1017-1053
%V 55
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2120/
%R 10.5802/aif.2120
%G en
%F AIF_2005__55_3_1017_0
Bruasse, Laurent; Teleman, Andrei. Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry. Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 1017-1053. doi : 10.5802/aif.2120. http://www.numdam.org/articles/10.5802/aif.2120/

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

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

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

[4] L. Bruasse Optimal destabilizing vectors in some gauge theoretical moduli problems (2004) (IML, ref arxiv math.DG/0403264, http://arxiv.org/abs/math.DG/0403264)

[5] L. Bruasse; A. Teleman Harder-Narasimhan filtrations and optimal destabilizing vectors in gauge theory (2003) (article in preparation)

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

[7] P. Heinzner Geometric invariant theory on Stein spaces, Math. Ann., Volume 289 (1991), pp. 631-662 | DOI | MR | Zbl

[8] P. Heinzner; A. Huckleberry Analytic Hilbert Quotient (Several complex variables), Volume 37 (1999), pp. 309-349 | Zbl

[9] P. Heinzner; F. Loose Reduction of complex Hamiltonian G-spaces, Geometric and Functional Analysis, Volume 4 (1994) no. 3, pp. 288-297 | DOI | MR | Zbl

[10] N. J. Hitchin; A. Karlhede; U. Lindström; M. Rouk Hyperkähler metrics and supersymmetry, Commun. Math. Phys., Volume 108 (1987), pp. 535-589 | DOI | MR | Zbl

[11] F. C. Kirwan Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, Volume 31 (1984) no. Princeton University Press | MR | Zbl

[12] M. Lübke; A. Teleman The universal Kobayashi-Hitchin correspondance (2003) (article in preparation)

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

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

[15] I. Mundet i Riera A Hitchin-Kobayashi correspondence for Kähler fibrations, J. reine angew. Maths, Volume 528 (2000), pp. 41-80 | MR | Zbl

[16] Ch. Okonek; A. Schmitt; A. Teleman Master spaces for stable pairs, Topology, Volume 38 (1999) no. 1, pp. 117-139 | DOI | MR | Zbl

[17] Ch. Okonek; A. Teleman Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces, Comm. Math. Phys., Volume 227 (2002) no. 3, pp. 551-585 | DOI | MR | Zbl

[18] P. Orlik Seifert Manifold, Lectures Notes in Maths, 291, Springer Verlag, 1972 | MR | Zbl

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

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

[21] P. Slodowy Die Theorie der optimalen einparameteruntergruppen für instabile vektoren, Algebraische Transformationsgruppen und Invariantentheorie (DMV Seminar, 13 DMV Seminar), Volume 13 (1989), pp. 115-131 | Zbl

[22] A. Teleman Analytic stability, symplectic stability in non-algebraic complex geometry (2003) (preprint, math.CV/0309230, http://arxiv.org/abs/math.CV/0309230)

Cité par Sources :