Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry
Résumé
We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action G×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 Lie(G) and then a naturally defined point f_0 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 Shatz stratification for holomorphic vector bundles. In the last chapter we show that our results can be generalized to the gauge theoretical framework: first we show that the system of semistable quotients associated with the classical Harder-Narasimhan filtration of a non-semistable bundle E can be recovered as the limit object in the direction given by the optimal destabilizing vector of E. Second, we extend this principle to holomorphic pairs: we give the analogue of the Harder-Narasimhan theorem for this moduli problem and we discuss the relation between the Harder-Narasimhan filtration of a non-semistable holomorphic pair and its optimal destabilizing vector.