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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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