Optimal destabilizing vectors in some Gauge theoretical moduli problems

Bruasse, Laurent

Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1805-1826.

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 $\tau$-stabilité. Ces résultats suggèrent que le principe reste valable en toute généralité en théorie de jauge.

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.

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.
Mot clés : GIT, sous-groupe à un paramètre, théorie de jauge, application de poids maximal, problème de module, stabilité, application moment, filtration de Harder-Narasimhan.

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