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Symplectic stability, analytic stability in non-algebraic complex geometry. (English) Zbl 1089.53058

Since Mumford’s work and the appearance of Geometric Invariant Theory, the factorization problem for group actions in both complex and algebraic geometry has been a fundamental subject. In the classical algebraic framework, the stability condition depends on the choice of the linearization (of the action) in an ample line bundle. The question investigated in the paper is to find an analogy in the Kähler non-algebraic framework.
The author considers two notions of stability for actions of a reductive group \(G\) on Kählerian manifolds. The first one, called symplectic Hamiltonian stability [see P. Heinzner and A. Huckleberry, Analytic Hilbert quotients. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 37, 309–349 (1999; Zbl 0959.32013)], is defined using the position of the \(G\)-orbit with respect to the vanishing locus of an associated moment map with respect to a maximal compact subgroup of \(G\). For the second [see I. Mundet i Riera, J. Reine Angew. Math. 528, 41–80 (2000; Zbl 1002.53057)], one uses a Kählerian version of the Hilbert numerical criterion. The relation between both semistabilities is delicate for general Hamiltonian actions on non compact manifolds, i.e., one certainly needs to assume a completeness condition.
The author proves that for a large class of actions (including all Hamiltonian actions on compact Kähler manifolds and all linear representations) the analytic semistabiliy is \(G\)-invariant, has a pure complex geometric character and is an open condition. He deduces some new comparison results for this class relating symplectic semistability (polystability) and analytic semistability (polystability) and identifying the corresponding quotients.
More precisely, let \(\alpha: G\times X\to X\) be the holomorphic action of a reductive group \(G\) on the manifold \(X\). A symplectisation \(\sigma\) of \(\alpha\) is an equivalence class of triples \((K,g,\mu)\), where \(K\subset G\) is a maximal compact subgroup, \(g\) is a \(K\)-invariant Kähler metric on \(X\) and \(\mu\) is a moment map for the \(K\)-action on \((X,g)\); the group \(G\) acts diagonally on the set of such triples (the action of the first element is given by conjugation), and two triples are equivalent if they belong to the same orbit. Fixing a symplectisation \(\sigma\) represented by a triple \((K,g,\mu)\), a point \(x\in X\) is called
symplectically \(\sigma\)-stable if \(Gx\cap \mu^{-1}(0)\neq\emptyset\) and \(Lie(G)_x=0\) where \(Lie(G)_x\) is the infinitesimal stabiliser of \(x\),
symplectically \(\sigma\)-semistable if \(\overline{Gx}\cap \mu^{-1}(0)\neq\emptyset\),
symplectically \(\sigma\)-polystable if \(Gx\cap \mu^{-1}(0)\neq\emptyset\).
For any \((s,x)\in Lie(K)\times X\), one defines the maximal weight of \(x\) in the direction of \(s\) as \(\lambda^ s(x)=\lim_{t\to\infty}\langle \mu(e^{ts}x),-is\rangle\). Let \(H(G)\) be the set of all \(s\in{Lie(G)}\) such that \(s \in Lie(K)\) where \(K\) is a compact subgroup of \(G\). Then one can define \(\lambda^{s}(x)\) as before. The induced map \(\lambda: H(G)\times X\to \mathbb{R}\cup\{\infty\}\) only depends on the symplectisation \(\sigma\) and is invariant under the action of \(G\). The symplectisation \(\sigma\) is energy complete if for any \((s,x)\in H(G)\times X\) such that \(\lambda^{s}(x)<\infty\) the limit of \(e^{ts}x\) as \(t\) goes to \(\infty\) does exist.
Now, a point \(x\in X\) is called
analytically \(\sigma\)-semistable if \(\lambda^ s(x)\geq 0\) for all \(s\in H(G)\),
analytically \(\sigma\)-stable if it is semistable and \(\lambda^{s}(x)>0\) for any \(s\in H(G) \setminus {0}\),
analytically \(\sigma\)-polystable if semistable with \({Lie(G)}_ x\) reductive and \(\lambda^ s(x)>0\) if \(s\) is not equivalent to an element of \({Lie(G)}_{x}\).
The last section of the paper proves that if \(\sigma\) is energy complete, then the notions of symplectic \(\sigma\)-(poly, semi)stability are equivalent to analytic \(\sigma\)-(poly, semi)stability. In particular, for an energy complete symplectisation, the (analytic or symplectic) semi-stability are equivalent to \(\inf_{g \in G} \| \mu(gx) \| =0\) where the norm is computed with respect to any \(ad\)-invariant inner product on \(Lie(K)\).
The promised sequel for the infinite dimensional gauge setting gives a nice application of this work [see L. Bruasse and A. Teleman, Ann. Inst. Fourier 55, No. 3, 1017–1053 (2005; Zbl 1093.32009) and M. Lübke and A. Teleman, “The universal Kobayashi-Hitchin correspondence on Hermitian manifolds”, math.DG/0402341 (2004), to appear in Memoirs of the AMS].

MSC:

53D20 Momentum maps; symplectic reduction
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
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References:

[1] DOI: 10.1142/S0129167X01000897 · Zbl 1111.32303
[2] DOI: 10.1007/BF01446594 · Zbl 0728.32010
[3] Heinzner P., MSRI Publ. 37 pp 309–
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[5] DOI: 10.1007/BF01896243 · Zbl 0816.53018
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[7] DOI: 10.1007/978-3-642-96676-7
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[9] DOI: 10.1016/S0040-9383(98)00006-8 · Zbl 0981.14007
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