## Optimal destabilizing vectors in some gauge theoretical moduli problems.(English)Zbl 1112.32008

Summary: 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.

### MSC:

 32G13 Complex-analytic moduli problems 32M05 Complex Lie groups, group actions on complex spaces 53D20 Momentum maps; symplectic reduction 14L24 Geometric invariant theory 14L30 Group actions on varieties or schemes (quotients) 32L05 Holomorphic bundles and generalizations 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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### References:

 [1] Atiyah, M.; Bott, R., The Yang-Mills equations over a Riemann surface, Phil. Trans. R. Soc., 308, 523-615, (1983) · Zbl 0509.14014 [2] Bradlow, S. B., Special metrics and stability for holomorphic bundles with global sections, J. Diff. Geom., 33, 169-213, (1991) · Zbl 0697.32014 [3] Bradlow, S. B.; Daskalopoulos, G. D.; Wentworth, R. A., Birational equivalences of vortex moduli, Topology, 35, 3, 731-748, (1996) · Zbl 0856.32019 [4] Bruasse, L., Harder-Narasimhan filtration on non Kähler manifolds, Int. Journal of Maths, 12, 5, 579-594, (2001) · Zbl 1111.32303 [5] Bruasse, L., Stabilité et filtration de Harder-Narasimhan, (2001) [6] Bruasse, L., Filtration de Harder-Narasimhan pour des fibrés complexes ou des faisceaux sans-torsion, Ann. Inst. Fourier, 53, 2, 539-562, (2003) · Zbl 1114.32010 [7] Bruasse, L.; Teleman, A., Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry, Ann. Inst. Fourier, 55, 3, 1017-1053, (2005) · Zbl 1093.32009 [8] Daskalopoulos, G. D., The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geometry, 36, 699-746, (1992) · Zbl 0785.58014 [9] Harder, G.; Narasimhan, M., On the cohomology groups of moduli spaces, Math. Ann., 212, 215-248, (1975) · Zbl 0324.14006 [10] Kempf, G. R., Instability in invariant theory, Ann. of Mathematics, 108, 299-316, (1978) · Zbl 0406.14031 [11] Kirwan, F. C., Mathematical Notes, 31, Cohomology of quotients in symplectic and algebraic geometry, (1984), Princeton University Press · Zbl 0553.14020 [12] Lübke, M.; Teleman, A., The universal Kobayashi-Hitchin correspondance on Hermitian manifolds, (2004) · Zbl 1103.53014 [13] Maruyama, M., The theorem of grauert-Mülich-spindler, Math. Ann., 255, 317-333, (1981) · Zbl 0438.14015 [14] Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory, (1982), Springer-Verlag · Zbl 0504.14008 [15] Mundet i Riera, I., Yang-Mills-Higgs theory for symplectic fibrations, (1999) [16] Mundet i Riera, I., A Hitchin-Kobayashi correspondence for Kähler fibrations, J. reine angew. Maths, 528, 41-80, (2000) · Zbl 1002.53057 [17] Ramanan, S.; Ramanathan, A., Some remarks on the instability flag, Tôhoku Math. Journ., 36, 269-291, (1984) · Zbl 0567.14027 [18] Shatz, S., The decomposition and specialization of algebraic families of vector bundles, Composito. Math., 35, 163-187, (1977) · Zbl 0371.14010 [19] Teleman, A., Symplectic stability, analytic stability in non-algebraic complex geometry, Int. Journal of Maths, 15, 2, 183-209, (2004) · Zbl 1089.53058
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