##
**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

Now, a point \(x\in X\) is called

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].

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 \(Gx\cap \mu^{-1}(0)\neq\emptyset\).

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 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].

Reviewer: Julien Keller (London)

### 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) |

### Keywords:

symplectic stability; analytic stability; quotient; geometric invariant theory; Kähler non-algebraic; energy complete### 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– |

[4] | Heinzner P., J. Reine Angew. Math. 455 pp 123– |

[5] | DOI: 10.1007/BF01896243 · Zbl 0816.53018 |

[6] | Kirwan F. C., Math. Notes, in: Cohomology of quotients in symplectic and algebraic geometry (1984) |

[7] | DOI: 10.1007/978-3-642-96676-7 |

[8] | Mundet i Riera I., J. Reine Angew. Math. 528 pp 41– |

[9] | DOI: 10.1016/S0040-9383(98)00006-8 · Zbl 0981.14007 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.