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APPEARED IN BULLETIN OF THE

arXiv:math/9204232v1 [math.CV] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 276-279ANALYTIC VARIETIES VERSUS INTEGRAL VARIETIESOF LIE ALGEBRAS OF VECTOR FIELDSHerwig Hauser and Gerd M¨ullerAbstract. We associate to any germ of an analytic variety a Lie algebra of tangentvector fields, the tangent algebra.Conversely, to any Lie algebra of vector fieldsan analytic germ can be associated, the integral variety.The paper investigatesproperties of this correspondence: The set of all tangent algebras is characterizedin purely Lie algebra theoretic terms.And it is shown that the tangent algebradetermines the analytic type of the variety.Local analytic varieties, defined as zero sets of complex analytic functions, canequally be considered as integral varieties associated to certain Lie algebras of vectorfields.

This is the theme of the present note. As a consequence one obtains a newway of studying singularities of varieties by looking at their Lie algebra.

It turns outthat the Lie algebra determines completely the variety up to isomorphism. Thusone may replace, to a certain extent, the local ring of functions on the variety bythe Lie algebra of vector fields tangent to the variety.We shall give a brief account of these observations.

Details will appear else-where, see [HM1, HM2]. The paper of Omori [O], which treats the same topic in aspecial case, served us as a valuable source of inspiration.

Various ideas are alreadyapparent there.Consider a germ X of a complex analytic variety embedded in some smoothambient space, X ⊂(Cn, 0). In this note, germ of variety shall always mean reducedbut possibly reducible complex space germ.

We associate to X the Lie algebra DXof vector fields on (Cn, 0) tangent to X. To do so let D denote the Lie algebraof germs of analytic vector fields on (Cn, 0).

We identify D with Der On, the Liealgebra of derivations of the algebra On of germs of analytic functions (Cn, 0) →C.We then setDX = {D ∈D, D(IX) ⊂IX},where IX ⊂On is the ideal of functions vanishing on X. This is a subalgebra ofD.

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 maycoincide.

This limits our interest to the reduced case. In this context, two mainproblems arise:1991 Mathematics Subject Classification.

Primary 13B10, 14B05, 17B65, 32B10, 57R25, 58A30.Received by the editors April 16, 1991 and, in revised form, August 28, 1991This work was done during a visit of the second author at the University of Innsbruck. Hethanks the members of the Mathematics Department for their hospitalityc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2HERWIG HAUSER AND GERD M¨ULLER• Characterize all Lie subalgebras A ⊂D that are of the form A = DX for asuitable X.• Find out to what extent the abstract Lie algebra DX determines the varietyX.1. Tangent algebras were characterized by Lie algebra propertiesIn order to discuss the first problem let us fix some notation.

A subalgebra A ofa Lie algebra B will be called balanced (in B) if A contains no ideal ̸= 0 of B butan element a ̸= 0 such that[a, B] ⊂Aand[[a, B], B] ⊂A.A visible subalgebra of B is a subalgebra A that admits a chain of subalgebrasA = Am ⊂Am−1 ⊂· · · ⊂A0 = Bsuch that Ak is maximal balanced in Ak−1 for k = 1, . .

. , m. In case m = 1, i.e., ifA is a maximal balanced subalgebra of B, A is called maximal visible.

Note thatthese notions are of a purely Lie algebra theoretic nature.For a finite family X = {X1, . .

. , Xp} of germs Xi ⊂(Cn, 0) let DX = Ti DXi bethe Lie algebra of vector fields tangent to all Xi (the Xi may be contained in eachother).

Our first result may be considered as a variation of the classical FrobeniusTheorem in the singular case (see e.g., [N, 2.11]).Theorem 1. Let A ⊂D be a subalgebra.

(a) There is a set of germs X as above such that A = DX if and only if A is avisible subalgebra of D.(b) There is a smooth germ X ⊂(Cn, 0) different from ∅and (Cn, 0) such thatA = DX if and only if A is a maximal visible subalgebra of D.(c) There is an irreducible germ X ⊂(Cn, 0) with an isolated singularity at 0such that A = DX if and only if A is a maximal visible subalgebra of the algebra D0of vector fields vanishing at 0. (d) There is an analytic germ X ⊂(Cn, 0) such that A = DX if and only if A isgeometric in D, i.e.

by definition, A is visible in every subalgebra B of D containingA.Comments. (i) It is easy to see that the family X of germs Xi is not unique.

Forexample, if X is the set of irreducible components Xi of some germ X = S Xione has DX = DX. Moreover, DX = DX,Sing X where Sing X denotes the singularsubspace of X.But in case X is an irredundant set of irreducible germs, i.e.,deleting any germ from X alters DX, the family X is uniquely determined by DX.In particular, the variety X of a maximal geometric subalgebra as in (b) and (c) isunique.

(ii) There is a relative version of Theorem 1 where D is replaced by DZ for someset of germs Z and where all varieties X associated to visible subalgebras of DZ aredetermined. Namely, a subalgebra A of DZ is visible if and only if there is a setof irreducible germs X with Xi ̸⊂Zj for all i, j such that A = DZ,X.

Theorem 1represents the cases Z = ∅, resp. Z = {0}.

ANALYTIC VARIETIES VERSUS INTEGRAL VARIETIES32. Singularities are determined by their tangent algebraWe now turn to the second problem, the characterization of the isomorphismtype of germs via their Lie algebra.

If X, Y ⊂(Cn, 0) are isomorphic then theassociated Lie algebras DX and DY are isomorphic. In fact, every isomorphism X →Y can be extended to an automorphism φ of (Cn, 0) with algebra automorphismφ∗: On →On.

Then Φ(D) := φ∗◦D◦(φ∗)−1 defines an automorphism Φ of D withΦ(DY ) = DX. By abuse of notation we write again Φ = φ∗.

This map is continuousif D is provided with the topology induced from the coefficientwise topology on On.Conversely we haveTheorem 2. Let X and Y be germs of analytic varieties in (Cn, 0) different from∅.

Assume that n ≥3. For every isomorphism Φ: DY →DX of topological Liealgebras there is a unique automorphism φ of (Cn, 0) sending X onto Y and suchthat Φ = φ∗.Thus the analytic isomorphism type of X is entirely given by the abstract topo-logical Lie algebra DX.Omori [O] proved this in the special case of weightedhomogeneous varieties.We indicate some ideas appearing in the proofs of Theorems 1 and 2.3.

Proof of Theorem 1In order to study visible subalgebras of D we associate to any A ⊂D the germX(A) in (Cn, 0) defined by the idealpI(A) of On whereI(A) = {g ∈On, g · D ⊂A}.Here the On-module structure of D is used. The germ X(A) will be called theintegral variety of X.Note that every germ X ⊂(Cn, 0) different from ∅and(Cn, 0) can be recovered from DX as X = X(DX): The inclusion X(DX) ⊂X isobvious from the definition.

For the converse, assume that some g ∈I(DX) doesnot belong to IX. Consider the vector fields g∂x1, .

. .

, g∂xn. In every point outsidethe zero set of g in X they are linearly independent.

As they are tangent to X bydefinition of I(DX) a Theorem of Rossi [R, Theorem 3.2] implies that the germ ofX taken in such a point is isomorphic to (Cn, 0). But these points are dense in Xand we get a contradiction.Let us now consider assertion (b) of Theorem 1.

The proof that DX is a balancedsubalgebra of D is a bit involved and will be left out.Concerning maximality,assume that DX is contained in a balanced subalgebra A ⊂D. Then in fact DX ⊂A ⊂DX(A).

One shows that A balanced implies X(A) ̸= ∅and (Cn, 0). MoreoverX(A) = X(DX(A)) ⊂X(DX) = X.

Now if X is smooth one deduces from DX ⊂DX(A) that X(A) = X. This shows DX = A and proves necessity in (b).For sufficiency, start with a maximal visible subalgebra A ⊂D.

Similarly asabove A ⊂DX(A) with X(A) ̸= ∅and (Cn, 0). As DX(A) is balanced, maximalityof A gives A = DX(A).

Write X = X(A). If Sing X ̸= ∅then DSing X is balanced.Again by maximality, the inclusion DX ⊂DSing X is actually an equality.

Thisimplies X = Sing X, which is impossible. Therefore X is smooth.Part (a) of Theorem 1 is proved by induction.

Here one proves and uses at oncethe relative version of the theorem mentioned earlier. To illustrate, let X = {X}consist of one singular germ X.

Choose k ∈N maximal with Z = Singk X :=

4HERWIG HAUSER AND GERD M¨ULLERSing(· · · (Sing(X)) ̸= ∅. The inclusion DX ⊂D is split into DX = DZ,X ⊂DZ andDZ ⊂D.

The first is visible by induction and the second is maximal visible by (b)since Z is smooth.Conversely, if A is a visible subalgebra of D use induction on the length of thechain and the relative version of part (b) to find X.4. Proof of Theorem 2We conclude with some remarks on the proof of Theorem 2.

For f ∈On considerthe C-linear map λf : DX →DX defined byλf(D) = Φ(f · Φ−1(D)).If Φ: DY →DX is induced from an automorphism φ of (Cn, 0), say Φ = φ∗, onechecks by computation that the equality λf(D) = φ∗(f ) · D holds for all D in DX.If Φ is an arbitrary continuous Lie algebra isomorphism, we are led to establish thesame equality in order to recover a map φ that could be an appropriate candidateto induce Φ and to define an isomorphism between X and Y .Thus the first thing to do is to check whether any vector field D is mapped byλf into the On-module (D) generated by D. This can be seen for all D of a certaindense subset U of IX · D by writing (D) as an intersection of subalgebras of DXof form DX,Z. This is the key step in the proof and it is here that we need theassumption n ≥3.

Once this is accomplished, the relative version of Theorem 1and the fact that Φ is a Lie algebra isomorphism guarantee that λf maps DX,Z intoDX,Z. Hence the module (D) is mapped into itself.

This impliesλf(D) = φ∗(f, D) · Dwith suitable factor φ∗(f, D) ∈On. Then the continuity of Φ is used to show thatφ∗(f, D) is actually independent of D, say φ∗(f, D) = φ∗(f ).

Therefore, again bycontinuity,λf(D) = φ∗(f ) · Dwill hold for all D ∈IX · D. Finally we deduce from this equality that the map φthus obtained is an automorphism of (Cn, 0) mapping X to Y and inducing Φ.References[HM1] H. Hauser and G. M¨uller, Analytic varieties and Lie algebras of vector fields. Part I: TheGr¨obner correspondence, preprint 1991.

To be published. [HM2], Analytic varieties and Lie algebras of vector fields.

Part II: Singularities aredetermined by their tangent algebra (to appear).[N]R. Narasimhan, Analysis on real and complex manifolds, North Holland, Amsterdam,1968.[O]H.

Omori, A method of classifying expansive singularities, J. Differential Geom. 15 (1980),493–512.[R]H.

Rossi, Vector fields on analytic spaces, Ann. of Math.

(2) 78 (1963), 455–467.Mathematisches Institut der Universit¨at Innsbruck, A-6020 Innsbruck, AustriaFachbereich Mathematik der Universit¨at Mainz, D-6500 Mainz, Germany

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