de Mathématiques de Luminy
Bruasse Laurent, Teleman Andrei.
Optimal destabilizing vectors in KŠhler geometry and gauge theory
We generalize the classical Harder-Narasimhan filtration theorem for a large class of complex geometric moduli problems. First we consider the finite dimensional framework : a holomorphic action G x F > F of a complex reductive Lie group G on a finite dimensional (possibly non-compact) Kähler manifold F. Using a Hilbert type criterion for the (semi)stability of symplectic actions, we associate to any non semistable point f F a unique optimal destabilizing vector in g and then a naturally defined point f0 which is semistable for the action of a certain reductive subgroup of G on a submanifold of F. We get a natural stratification of F which is the analogue of the Schatz stratification for holomorphic vector bundles. Last we give two examples which show that our results can be generalized to the gauge theoretical framework.
Keywords : symplectic actions, hamiltonian actions, stability, Harder-Narasimhan filtration, Schatz stratification, gauge theory.