Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

Details Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

2003-09-19

arxiv.org/abs/math/0309315v3

Mathematics

Complex Variables

Latex, 30 pages, comments are welcome; Some modifications; To appear in Annales de l'Institut Fourier (2005) vol. 55

Scientific paper

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.

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