Analytic varieties versus integral varieties of Lie algebras of vector fields. (English) Zbl 0762.32004
Let \(X\) be a reduced (but possibly reducible) germ of a complex analytic variety. A Lie algebra of vector fields \(\mathbb{D}_ X\) (called the tangent algebra) is associated to \(X\) (a subalgebra of \(\mathbb{D}=\text{Der} {\mathcal O}_ n\)). Conversely, to any Lie algebra of vector fields an analytic germ \(X\) is associated (called the integral variety). In the paper under review, the properties of this correspondence are investigated. The main results are:
1. The set of all tangent algebras is characterized in purely Lie algebra theoretic terms (given explicitly); this result can be considered as a Frobenius-type theorem in the singular case.
2. The tangent algebra determines the analytic type of the variety (i.e. singularities are determined by their tangent algebra).
The paper is mainly informal with sketch of proofs (details will appear elsewhere).
Previously H. Omori [J. Differ. Geom. 15, 493-512 (1980; Zbl 0476.32010)] studied the same problems in a special case, inspiring this general result.
1. The set of all tangent algebras is characterized in purely Lie algebra theoretic terms (given explicitly); this result can be considered as a Frobenius-type theorem in the singular case.
2. The tangent algebra determines the analytic type of the variety (i.e. singularities are determined by their tangent algebra).
The paper is mainly informal with sketch of proofs (details will appear elsewhere).
Previously H. Omori [J. Differ. Geom. 15, 493-512 (1980; Zbl 0476.32010)] studied the same problems in a special case, inspiring this general result.
Reviewer: O.Păsărescu (Bucureşti)
MSC:
| 32B10 | Germs of analytic sets, local parametrization |
| 13B10 | Morphisms of commutative rings |
| 14B05 | Singularities in algebraic geometry |
| 17B15 | Representations of Lie algebras and Lie superalgebras, analytic theory |
| 57R25 | Vector fields, frame fields in differential topology |
| 58A30 | Vector distributions (subbundles of the tangent bundles) |
Citations:
Zbl 0476.32010References:
| [1] | H. Hauser and G. Müller, Analytic varieties and Lie algebras of vector fields. Part I: The Gröbner correspondence, preprint 1991. To be published. |
| [2] | -, Analytic varieties and Lie algebras of vector fields. Part II: Singularities are determined by their tangent algebra (to appear). |
| [3] | Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. · Zbl 0188.25803 |
| [4] | Hideki Omori, A method of classifying expansive singularities, J. Differential Geom. 15 (1980), no. 4, 493 – 512 (1981). · Zbl 0476.32010 |
| [5] | Hugo Rossi, Vector fields on analytic spaces, Ann. of Math. (2) 78 (1963), 455 – 467. · Zbl 0129.29701 · doi:10.2307/1970536 |
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