  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.

Complex Variables;Algebraic Geometry;Differential Geometry;Symplectic Geometry