Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action $G\times F\to F$ of a complex reductive Lie group $G$ on a finite dimensional (possibly non-compact) K\"ahler manifold $F$. Using a Hilbert type criterion for the (semi)stability of symplectic actions, we associate to any non semistable point $f\in F$ a unique optimal destabilizing vector in $\g$ 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 $\EE$ can be recovered as the limit object in the direction given by the optimal destabilizing vector of $\EE$. 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.