de Mathématiques de Luminy
Optimal destabilizing vectors in some Gauge theoretical moduli problems.
We show that the "classical" Harder-Narasimhan filtration associated to a non semistable vector bundle $E$ can be viewed as a limit object for the action of the gauge group $Aut(E)$ in the direction of an optimal Hermitian endomorphism. We give a complete description of these optimal destabilizing endomorphisms. Then we show that this principle holds for another important moduli problem: holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with fixed source). We get a generalization of the Harder-Narasimhan filtration theorem for the associated notion of $\tau $-stability. These results suggest that the principle holds in the whole gauge theoretical framework.
Keywords : Gauge theory, complex moduli problem, stability, Harder-Narasimhan filtration, moment map.