## Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry.(English)Zbl 1093.32009

The object of this paper is to find analogues of the Harder-Narasimhan filtration for other moduli problems in complex geometry, in particular allowing the underlying manifold to be non-algebraic and non-compact. The underlying idea is that the Harder-Narasimhan result can be interpreted as an assignment which associates to a non-semistable object a semistable object for a different moduli problem. The tool for constructing these objects is the theory of optimal one-parameter subgroups (OPS), which is well-established in classical GIT (geometric invariant theory). In fact (see, for example, [S. Ramanan and A. Ramanathan, Tôhoku Math. J., II. Ser. 36, 269–291 (1984; Zbl 0567.14027)]), if $$\tau$$ is an optimal destabilising OPS of a non-semistable point $$x\in{\mathbb P}^n$$ for a linear action of a reductive group $$G$$, then $$\tau(t)x$$ converges to a point $$x_0$$ which is semistable with respect to an induced action of the reductive centralizator $$Z(\tau)$$ of $$\tau$$ as a $$Z(\tau)$$-stable subvariety of $${\mathbb P}^n$$.
In this paper, the authors generalise this result to certain holomorphic actions $$G\times F\to F$$, where $$F$$ is now a Kähler manifold (not necessarily compact). There are technical problems which require them to impose a certain completeness condition in order to construct a $$G$$-equivariant maximal weight function. Using this, they then prove the existence and uniqueness (up to a suitable equivalence) of an optimal destabilising element $$\xi$$ in the Lie algebra $$G$$ of any non-semistable point $$f\in F$$. Following the principle explained above, they show that $$e^{t\xi}f$$ converges to a point $$f_0$$ which is semistable with respect to a natural action of $$Z(\xi)$$ on a certain submanifold of $$F$$.
The paper is completed by studying the optimal destabilising vectors of the non-semistable objects of two important gauge theoretical moduli problems: holomorphic bundles and holomorphic pairs.

### MSC:

 32M05 Complex Lie groups, group actions on complex spaces 32L05 Holomorphic bundles and generalizations 32Q15 Kähler manifolds 14L24 Geometric invariant theory 53D20 Momentum maps; symplectic reduction

Zbl 0567.14027
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### References:

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