Analytic Varieties versus Integral Varieties of Lie Algebras of Vector Fields

We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the tangent algebra. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the integral variety. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent algebra determines the analytic type of the variety. Local analytic varieties, defined as zero sets of complex analytic functions, can equally be considered as integral varieties associated to certain Lie algebras of vector fields. This is the theme of the present note. As a consequence one obtains a new way of studying singularities of varieties by looking at their Lie algebra. It turns out that the Lie algebra determines completely the variety up to isomorphism. Thus one may replace, to a certain extent, the local ring of functions on the variety by the Lie algebra of vector fields tangent to the variety. We shall give a brief account of these observations. Details will appear elsewhere, see [HM1, HM2]. The paper of Omori [O], which treats the same topic in a special case, served us as a valuable source of inspiration. Various ideas are already apparent there. Consider a germ X of a complex analytic variety embedded in some smooth ambient space, X ⊂ (C, 0). In this note, germ of variety shall always mean reduced but possibly reducible complex space germ. We associate to X the Lie algebra DX of vector fields on (C, 0) tangent to X . To do so let D denote the Lie algebra of germs of analytic vector fields on (C, 0). We identify D with Der On, the Lie algebra of derivations of the algebra On of germs of analytic functions (C, 0) → C. We then set DX = {D ∈ D, D(IX) ⊂ IX}, where IX ⊂ On is the ideal of functions vanishing on X . This is a subalgebra of D. It will be called the tangent algebra of X . In case X is a nonreduced germ, simple examples show that the tangent algebra of X and of its reduction Xred may coincide. This limits our interest to the reduced case. In this context, two main problems arise: 1991Mathematics Subject Classification. Primary 13B10, 14B05, 17B65, 32B10, 57R25, 58A30. Received by the editors April 16, 1991 and, in revised form, August 28, 1991 This work was done during a visit of the second author at the University of Innsbruck. He thanks the members of the Mathematics Department for their hospitality c ©1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page