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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

Citations:

Zbl 0567.14027
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References:

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