Optimal destabilizing vectors in some gauge theoretical moduli problems

Details Optimal destabilizing vectors in some gauge theoretical moduli problems Optimal destabilizing vectors in some gauge theoretical moduli problems

2004-03-16

arxiv.org/abs/math/0403264v2

Mathematics

Differential Geometry

18 pages, 6 figures; new introduction, new references added, minor modifications

Scientific paper

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 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 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.

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