Characterizing singularities of varieties and of mappings

Invent. math. 81,427-447 (1985) I~l ve f l t io f les matbematicae �9 Springer-Verlag 1985

Characterizing singularities of varieties and of mappings

Terence Gaffney*'l and Herwig Hauser **'2

Department of Mathematics, Northeastern University, Boston MA 02115, USA 2 Institut fiir Mathematik, Universitfit Innsbruck, A-6020 Innsbruck, Austria

Introduction

In the two parts of this paper we propose to study singularities of complex-analytic varieties and mappings. The first part investigates singularities of germs of complex-analytic varieties in (~", 0) under analytic isomorphism and their relation to the singular subspace of the variety. The second part presents a characterization of analytic mappings under ~'-equivalence, i.e., coordinate changes in source and target.

Part I

Let (X, 0) and (Y, 0) be two germs of complex-analytic varieties in (ff~", 0). Assume they are smooth; then they are isomorphic, if and only if they have the same dimension. Assume they are singular; then they are isomorphic, if and only if... !

The purpose of Part I is to provide the dots with a syntactic, and the singular locus of (X, 0) with an analytic structure, such that the first characterizes via the second the initial structure of the variety (X, 0).

For an embedded hypersurface (X, 0) ___ (~", 0) of equation f (x )=0 , let Sing(X, 0) denote the singular subspace of (X, 0) defined by the ideal (f) +j(f) of (9, (j(f) =jacobian ideal of f) . Sing(X, 0) being an analytic invariant of the variety, it is natural to ask to what extent it determines the variety (X, 0), i.e., whether Sing(X, 0) ~ Sing(Y, 0) implies (X, 0) ~ (Y, 0)?

Answers to this question have been found in special cases: Benson and independently Shoshitaishvili could affirm the question for homogeneous (resp. quasihomogeneous) hypersurfaces with isolated singularity [B], [Sh]. Then Mather and Yau extended these results to arbitrary isolated hypersurface singularities [M-Y].

In Part I we propose the study and complete answer of the question for arbitrary varieties (X, 0) in (I~", 0).

* Supported in part by a grant from the NSF ** Supported in part by a grant from the Austrian Ministery of Science while visiting Brandeis and Northeastern University

428 T. Gaffney and H. Hauser

The answer is "Yes" for varieties of isolated singularity type. By definition, (X,0) has isolated singularity type (I.S.T.), if Sing(X,0)~Sing(X,a) for a + 0 varying in a sufficiently small neighborhood of 0 �9 IE".

I S T not I S T

The answer is "No" for varieties which are not of isolated singularity type: we shall give the example of a family of hypersurfaces (X~, 0), t e (~, 0), with constant singular subspace Sing(X~, 0)=Sing(Xo, 0) but varying analytic type (w 4). This shows in particular that even the continuous version of the question, i.e., (X, 0) varying in a local family, has a negative answer.

However, if one replaces Sing(X, 0) by a slightly bigger subspace of(X, 0), the answer turns again to the affirmative: more precisely, let S be a submanifold of~" and Z____~" a manifold whose coordinates form a set of equations for S. Let js( f ) and Jz(f) denote the ideal of partial derivatives in direction of S and Z, and Sings(X,0) the subvariety of (X, 0) defined by ( f )+ ~ . ' J s ( f )+ Jz ( f ) (~. = maximal ideal of (9.)

Theorem. For all manifolds So=C" containin9 the stratum Sc=~" alon 9 which Sings(X, 0) is trivial (i.e., a product), the analytic type of(X, O) is determined by Sings (X, 0).

By the example ofw 4, the assertion is sharp. As a special case one obtains, that Singe.(X, 0) defined by T K ( f ) = ( f ) + ~ , . j ( f) always characterizes (X, 0).

The Theorem will be generalized to the non hypersurface case defining Sings(X, 0) by the analytic module

CP,/(f) �9 (gP, + ~ . "Js(f) +Jz(f)

( f = (fl, ..-, fv) �9 (9. ~ defining equations of (X, 0)) and introducing a category of generalized analytic varieties defined by analytic modules.

Yau has informed us, that he and Mather have found that TK( f ) determines (X, 0) for (X, 0) complete intersection with isolated singularity. In the same case, Martin [ M a l l shows that Singo(X,0)determines (X,0). On the other hand, Dimca proved an analogous result using the ideal of minors of the jacobian matrix in the case of 0-dimensional or homogeneous complete intersections [D].

The first part concludes with the mentioned example and, as an application of the techniques used, an extension of Saito's result [Sa] characterizing quasihomogeneous polynomials.

Part II

In Part II we prove analogues of the main theorems of Part I for map germs. Two map germs f, 0 e r are d-equivalent if there are coordinate changes in

Characterizing singularities of varieties and of mappings 429

source and target, say r and l such that lofor=9. We let d denote the group formed from pairs of coordinate changes on source and target. The object that plays the part of the tangent space to the d -o rb i t o f f at f is ~ , Jc - ( f ) + f*(~p(9~), denoted Td(f). Given Part I, it is natural to conjecture that Td(f)

Td(g) ~ f ~ g. I f f has finite singularity type (i.e., V(f) is a complete intersection with isolated

singularity or V(f) is an isolated point) then this conjecture is true, and is proved in Part B of the Theorem.

Just as there is a geometric interpretation of CP,/TK(f) in Part I, so there is also a geometric interpretation of (gP,/Td(f). This (gp-module (via f ) can be viewed as the stalk of a sheaf on the target tI~P (the construction of a related sheaf is given by the proof of Part A of the theorem). The support of this sheaf is just the locus of instability in the target. This consists of all y ~ II2P such that the germ o f f at S=f-l(y) is unstable. Since two stable germs are d-equivalent if they are K-equivalent, it is precisely over the locus of instability that the more complicated behavior of the map takes place.

If the behavior of a map germ at the origin is different from its behavior at nearby points, following Part I it is natural to conjecture that if Tde(f)=Jc-(f) +f*((9~) = Tde(9), then f and g are d-equivalent. We prove this in Part A of the Theorem with the additional hypothesis that f is finitely determined i.e., Tde(f) ~= k(gp for some k, but f not stable.

Unfortunately, it is not clear how to prove the precise analogue of the corresponding theorem of Part I, because the sheaves in the d case are more difficult to work with (they involve direct images). In actually classifying germs, the finitely determined case is the most important and the one in which our theorem will be most useful.

The proofs of our theorems also illustrate some of the advances in technique of recent years. In their paper, Mather and Yau used the notion of finite determinacy to reduce to lie groups acting on jet spaces. The advantage of this method is that algebraic problems are reduced to ones involving finite dimensional vector spaces.

However, this limits the generality of the results that can be obtained. This paper shows that sufficient algebraic techniques exist to work with germs which are not finitely determined so that jets can be dispensed with. Thus, we can handle arbitrary varieties and map germs which are of finite singularity type. It even turns out that the algebra is strong enough to reduce everything to questions about finite dimensional vector spaces and linear maps between them again.

Lemma 3 is of independent interest to singularists for it shows that the module structure of Td(f) is "convex" in some sense. It implies that if Td(f)= Td(g) then Td(f) is an ft*(gp-module for all t, where f t = f + t (g-f) .

1. Deformations

Let K denote the group K = G/p((9,)xAut(C",0) acting on (9, ~ via (L, qJ).f = L- ( f o if). For f e (9,P let ( f ) denote the ideal of (9, generated by the components fi of f and (X, 0) the complex-analytic variety in (IE", 0) of local ring (gx, o = (gn/(f)"


Lemma 1. ([M], p. 136) The fibers of the map

n " (gP~ ~{isomorphism classes of analytic varieties}

f ~ [(X, 0)]

(respectively z : (gP-o {analytic varieties}, f -o(X, 0)) are the K-orbits (respectively Glp((gn)-orbits ) in OPn. In particular, the fibers of z contain with any two elements f , g ~ (gP~ all elements except a finite number of the complex line l = f + C( f - g) c_ (pv through f and g. As the fibers of n are connected, any two isomorphic varieties can be joined by a family of isomorphic varieties.

Definition 1. A map from an analytic manifold T to a free (?~-module, f : (T, 0) -}((PV, fo), is called analytic at a point 0E T, if the induced map-germ F : (T• r defined by F ( t , x ) = f ( t ) ( x ) is analytic. For given foC(9~ v, f : (T, 0)~((9~ v, fo) is called a deformation o f f o; f is K-trivial, if its images f : = f ( t ) lie for t close to 0 ~ T in the K-orbit of fo, i.e., for any t ~ (T, 0) there exists (L,, g~,)~ K with L,. (fr ~ ~t)=fo. Geometrically speaking, a deformation h: (X , O) --}(T, 0) of (Xo, 0)= (h-1(0), 0) is trivial, if all its fibers (X,, 0) are isomorphic to (Xo, 0). Proposition 1. (Ephraim, [E2]) I f f : (T, O)-}((gv~,fo) is trivial, there exists an analytic map (L, q)) : (T, 0)~ (K, e), e = (1, Idr ~ K, such that L t �9 (ft ~ ~t) = fo. Equivalently, a trivial deformation h : (X, O)~(T, O) of (X o, O) is a product

(X, O) > (X o x T, O)

(r,o)

Proposition 2. Let f : (T, 0)--}((9, v, fo) be a deformation�9 (a) f is K-trivial, if and only if for any coordinate s on (T, 0) and any point

t ~ (T, 0) the tangent-vector O=f,: = ~ (t) ~ (gP, to the path ft belongs analytically to

the tangent-space T K ( f t ) : = ( f ) . ( g P . + ~ . . j ( f t ) of the K-orbit of ft, i.e., by definition, there exist analytic mappings M : (T, O)~(Mp((9,), Mo) and ~p : (T, 0) ~((9~, ~o), ~0,(0) = 0 E ~2", with

O=ft = M,. f + ~Pt" 0~ f, (x = (x ~, ..., xn) coordinates).

(b) Let G denote the group G = Gln((gn); then f is G-trivial, if and only if @=f belongs analytically to (f~). 0~ for t E (T, 0).

Proof. To simplify notation, we assume (T, O)=(~E, 0). For given deformations L: (T, O)--}(Glp((gn),l ) and ~:(T, 0)~(Aut(~",0) , Idc . ) the equation f o = L , �9 ( f o ~,) is equivalent to 0--- @=(L,. (f, o ~,) for some coordinate s on T. By chain and product-rule we obtain

0 = @=L,. (ft o~,) + L,. @=q),. (@=ft ~ fb,) + L t �9 (@=fro @,) ~ (P~. (*)


Multiplication from the left by/~- 1 and composition from the right with ~t- ~ give

Osft e (fO" (9~ + ~,, .j(f~) = r K ( f t ) (**)

analytically in t e (T, 0). Conversely, assume (**): then

~?~ft = M, " ft + ho, " C?xft e TK(f t ) ***)

with deformations M: (T, O)~(Mp((9.), Mo) = 0 E C ". The differential equations

O,L, = L t . (Mr o ~,)

and ho :(T, 0)~((9~, hog) with hot(O)

~o = Idr

Lo = 1

admit analytic solutions ~b : (T, 0)--,(Aut (IE", 0), Id, .) and L: (T, O)--*(Glp((9,,), 1) by integration of vectorfields. By substitution in (***) we get (*) and hence the K-triviality of f. This proves (a) and (b) follows similarly.

Remark. Note that in the above proof, h0t(0 ) = 0 implies ~bt(0 ) = 0. This assures, that the varieties X t and X o defined by f and fo are actually isomorphic at the origin, (X t, 0) ~ (X o, 0). The weaker assumption

O.ft e (f). OP. +J(fO

might allow ~t(0) = a t 4 = 0 etE" and would only imply (Xt, at) ~ (X0, 0) but not necessarily (Xt, 0) - (Xo, 0).

The following result shows that in certain cases, the hypothesis of Proposit- ion 2, gsf te TK(f~) analytically in t e T can be weakened to d s f e TK(fO pointwise:

Lemma 2. Let gt : ( T, 0)--,(((9~) q, go) be an analytic map and Mt be the submodule of (9~ generated by the q components of gt.

I f {Mt}t~(r,o ) is a f iat family over (T, 0), then for any analytic map h~ :(T, 0) ~ ( (9~, ho)

ht E M t pointwise for t ~ (T, O)

implies

ht E M t analytically in t e (T, 0).

This holds in particular for families Mt = Mo constant.

Proof. By [Ha], Th. 1, ch. 1.2, (cf. also [Ga] and [Hi]), {Mt},~(r.o)is flat if and only if

Mt@A(Mo)=(9~ for t~ (T ,O) , (*)

where A(Mo) is a complement of M o in (9~ given by the Division Theorem for modules:

M o O A ( M o ) = (9~.


Note that the equality (*) holds analytically in t (loc. cit. Th. 2, AII). Division of ht by Mt gives

ht= ~t mod Mt

analytically in t with hteA(Mo). As h t e M t pointwise, ~ e M t n A ( M o ) = O , i.e., = 0 and therefore h t ~ M t analytically in t.

The singular subspace of a variety

Definition 2. For a manifold S in IE" and f e (9 p we denote j s ( f ) the submodule of (9. p generated by the partial derivatives of f in direction S:

Js( f ) = (t?slf, .... t?skf) c__ (gp

for a coordinate system s = (sx, ..., sk) on S. Note t ha t j s ( f ) does not depend on the choice of coordinates. We then define

Ts f = ( f ) . C)~ + ~ , " Js( f ) +Jz(f) =c (.0~

where Z__C C" denotes any complement of S in C", S x Z ~ r whose coordinates form a set of equations of S. The module Ts f does not depend on the choice of Z. In the special cases S=r and S=O we get T e , f = T K ( f ) and T o f = ( f ) �9 (9~+j(f). Let (X, O) be the variety defined by f. We denote by Sings(X, O) and call the singular subspace in direction S of (X, O) the set of singular points of (X, O) provided with the analytic structure of the module (9~/Tsf. Even though Sings(X, O) is not a variety in the strict sense, if (X, O) is not a hypersurface, we consider it as a subvariety of (X, 0). In the appendix we shall indicate how one could extend the notion of analytic varieties and morphisms between them to varieties defined by analytic modules instead of local rings.

For two varieties (X, O) and (Y, O) in (C", O) and two manifolds S, S'=c~" we say that Sings (X, O) and Sings, (Y, O) are isomorphic, if there exist a coordinate change ~ : (~", O)~(~E",O) sending S to S' and an (9.-linear isomorphism L: (P~(~P such that

Ts,g = Tr ( f o r (9~.

where f and g s (P~ denote defining equations for (X, 0) and (Y, 0).

Lemma 3. The singular subspace Sings(X, 0) is an analytic invariant of (X, 0). More explicitely, if ~b : (112",0)--.(112",0) is an isomorphism sending (X, O) to (Y, 0), then Sings(X, 0) and Sing,(s)(X, 0) are isomorphic in the above sense.

The proof follows immediately from the definitions.

Remarks. For S = 0 =c IE" we mostly write Sing(X, 0) instead of Sing 0 (X, 0). If (X, 0) is a hypersurface, Sing(X, 0) is the usual singular subspace defined by the ideal ( f ) +j ( f ) . If (X, 0) is not a hypersurface, the singular subspace defined by the minors of the jacobian matrix of f and the singular subspace defined here are actually different. As it turns out, our proof needs the module structure of (9~/Tsf on the singular locus of (X, 0) to be able to recover (X, 0) from it.

Note that for S c= S' one has Sings(X, 0)=c Sings, (X, 0).


The analytic and the singular stratum of a variety

Definition 3. The analytic stratum of a variety (X, 0) in (C", 0) is the set-germ A of points in (1~", 0) along which a representative X of (X, 0) is trivial:

A = {a ~ (C", 0), (X, a) ~ (X, 0)} ~ (~", 0).

We say that (X, 0) has isolated analytic type, if A = 0, i.e., (X, a) ~r (X, 0) for a in a neighborhood of 0 e rE".

Theorem. (Ephraim, [E 2]) Let (X, O) be a variety in (IF.", 0). (1) The analytic stratum A of (X, O) is the germ of a submanifold of (~", 0). (2) The variety (X,O) is a product alon9 A:

( x , o) ~- (Xo, o) • A ,

for some variety (X o, O) in (ff~", 0).

Definition 4. For a manifold S____ C" we define the singular stratum in S-direction of (X, 0) as the stratum S = S s in (C", 0) along which Sings(X , 0) is trivial. This is again a manifold as Ephraim's Theorem extends to analytic modules (cf. appendix). We say that (X, 0) has isolated singularity type, if 2; o = 0, i.e.,

Sing(X, a) ~ Sing(X, 0)

for a in a small neighborhood of 0 ~ r Obviously, (X, 0) being of isolated singularity type implies (X, 0) of isolated analytic type. The converse is not true in general: in w we shall construct a variety (X, 0) in (~"x ~, 0) with constant singular type along (0 • ~ , 0) but varying analytic type.

Examples. (1) Whitney Umbrella: (X, 0) c_ (C 3, 0) : X 2 Z - - y2 = O.

Sing(X, 0) : (x 2, xz, y) =C t~3, I.S.T. (2) Cone over the Tacnode: (X, 0)__c (C3, 0) : x 4 + y4 _ xyz 2 = O.

Sing(X, 0) : (4x 3 - yz 2, 4y 3 - xz 2, xyz) ____ 0 3, I.S.T.

\ \

! Y t i i ! ~ t


(3) Vanishing node: x 3 "~- X2Z 2 - y2 =0. Sing(X, 0) : (3x 2 + 2XZ 2, y , X2Z), I.S.T.

I Y

I L

I

- - - - J - ~ - ~ t - - - m _

Here some other examples of varieties with isolated singularity type:

(4) The product of two cusps with perpendicular tangentlines at 0. (5) Cusp deforming along a cusp to a parabola.

(6) The Peak: Cusp walking along cusp with same tangentline, stretching as 0 approaches.

(7) The Thorn: Product of two cusps with same tangentline at 0: X : S2 + t 2, y = s 3, z : t 3.


Remark. For different S___~" the stratum Z s along which Sings(X, 0) is trivial, might vary. However, one should expect that the stratum becomes smaller as S increases:

Proposition 3. Let S c= S' be two manifolds in fly and Z, Z' the corresponding singular strata of (X, 0). Assume (X, 0) is a hypersurface. I f Z 3= S', then Z ~ Z'.

Remark. We don't know, if Z ~ Z' in general. For hypersurfaces, setting S = 0 and S '= Z, it follows that S'~ Z'. This will be useful in applications of the theorem of the next section.

Proof. Let g = (gl . . . . . gq) and g '= (g~, ..., g~) denote systems of defining equations of Sings(X,0 ) and Sings,(X,0 ). Choosing a decomposition S ' x Z ~ - C " and coordinates x of C", s of S, (s, s') of S' and z of Z we can write

g = (f , x ~ f , x ~ , f , ~ f , t?s,f) and g'= (f, xOsf, x~ , f , azf, O)

with the obvious notation. For t ~ C", we set ht(x) = g(x + t), h;(x) = g'(x + t). After a suitable coordinate change in IE", we can assume by Ephraim's Theorem that Sings(X, 0) is constant along Z. As S'c=Z, Proposition 2(b) gives

js,(ht) r (h,) . (9 q (1)

analytically in t e S'. By part (a) of Proposition 2:

js,(h3 c= (h~). (9~ + ~ , .jr (2)

analytically in t s Z'. We have to show that Sings(X, 0) is trivial along Z', i.e.,

jz,(h 0 c= (hi). (9~ + ~ , .jc.(ht) (3)

analytically in t ~ Z'. This will follow from (2) if one can prove that

jv(ht) - jv (h~ mod(ht) �9 (9, q (4)

analytically in t ~ S" and for any submanifold V of ~". Let v be a coordinate on V. Comparing ht and h~ componentwise, it becomes

clear that it is sufficient to show

O~O~, f t - 0 mod (hi)

analytically in t ~ Z', where ft(x) = f ( x + t). For v = s or v = s', this follows directly from (1). For v=z, write O~O~,f=O~,O~f and again (1) applies. This proves the Proposition.

2. Characterization of varieties by their singular subspace

Theorem. Let (X, O) and ( Y, O) be two germs of complex-analytic varieties in (•", 0). Let S c=ffy be a manifold containing the stratum Z c=O]" along which the singular subspace Sings(X, 0) of (X, O) is trivial.


The varieties (X, 0) and (Y, O) are isomorphic if and only if their singular subspaces Sings(X,0 ) and Singr are isomorphic via an isomorphism h-- (L, ~).

Remarks. (1) The theorem fails over ~-~ or a field of positive characteristic. It also fails if the manifold S does not contain the stratum Z (cf. the example, w 4).

(2) The assertion becomes stronger the smaller a manifold S satisfying S =3 Z can be chosen.

(3) For practical computations, the most interesting case occurs, when (X, 0) and (Y, 0) are reduced hypersurfaces: the singular subspaces are then varieties of smaller dimension and have usually simpler equations.

Example. The Whitney umbrella (X, 0)~(~F3,0) defined by x E - - y E z = 0 is of isolated singularity type and hence determined by Sing(X, 0) = Sing 0 (X, 0), i.e., the ideal (x, yz, y2)= (93. Therefore in order to check that a variety (Y, 0)_c(I123, 0) is isomorphic to (X, 0), it is sufficient to show that the ideal defining Sing(Y, 0) is isomorphic to (x, yz, y2).

Particular cases. (i) Take S =112": The analytic type of(X, 0) is always determined by the largest singular subspace Singc,(X, 0).

(2) Assume S____tE" contains both the strata along which Sings(X,0 ) and Sings (Y, 0) are trivial: then (X, 0) --- (Y, 0) if and only if Sings (X, 0) ~ Sings (Y, 0).

(3) The geometrically most interesting case occurs perhaps when (X, 0) and (Y, 0) are of isolated singularity type (def. 4): (X, 0) ~ (Y, 0) if and only if Sing(X, 0) -~ Sing(Y, 0) (S = 0).

(4) Let S be now the stratum of Sing(X, 0)-triviality of a hypersurface (X, 0). Then by Proposition 3 and the remark following it, S contains the stratum of Sings(X , 0)-triviality. Let S' denote the corresponding stratum for the hypersurface (Y,0), then (X,0)=(Y,0) if and only if Sings(X,O)~Sings,(Y,O). This provides a way to find a manifold S strictly smaller than C" such that Sings(X, 0) characterizes (X, 0).

(5) Suppose (X, 0) is of isolated singularity type. If (X, 0) is also of finite singularity type (dimc((9,P/Tof< oe), then the analytic type of (X, 0) is determined by the finite-dimensional vectorspace (gsing(x,o)=(9Pn/Tof and its (9.-modulestructure. Example: (X,0) is a complete intersection with isolated singularity.

(6) Let (X, 0) be a hypersurface. As Sing(X, 0) is then defined by an ideal, we can consider Sing(Sing(X, 0)) (cf. appendix for the non hypersurface case and the iteration of passing to the singular subspace). If for instance Sing(X, 0) has isolated singularity type (and consequently (X, 0) as well), (X, 0) is characterized by Sing(Sing(X, 0)), which for Sing(X, 0) finite singularity type, would be again finite dimensional.

(7) Let (X,0) be arbitrary and S_c__C" containing the Sings(X,0)-stratum of triviality. Then (X,0) is a product (Yx Z, 0) with (Z, 0) smooth, if and only if Sings(X, 0) is a product along (Z, 0).

(8) The theorem also provides an extension of Shoshitaishvili's result on right- equivalence [Sh]: Let f and 9 ~ (9. define hypersurfaces of isolated singularity type. Assume f weakly quasihomogeneous (cf. Prop. 4, w 4). Then f and # are right equivalent if and only if (9,/j(f) and (9./J(9) are isomorphic as IE-algebras.


Remark. Let i : (~n ~ ( ~ be a C-linear map such that the subvariety C(X, 0) of(X, 0) = ( f - 1(0), 0), f e On, defined by the components of i( f) is an analytic invariant of (X, 0). Then if Sing s (X, 0)_c_ C(X, 0)__c (X, 0) with S ~= S s = Sing s (X, 0)-stratum, C(X, 0) determines the analytic type of (X, 0). This combined with the example of w 4 illustrates, that in general Sings (X, 0) with S ~ Zs is the "smallest" subvariety of the hypersurface (X, 0) which characterizes the analytic type of (X, 0).

Proof of the Theorem. If (X,0)~(Y,0), then Sings(X,O)~Sing,{s~(Y,O ) by Lemma 3. To prove the inverse implication, we shall proceed in several steps:

(1) Reduction to the case Sings(X,O)=Sings(Y,O): The isomorphism h = (L, r between Sings(X, 0) and Sing,~s)(Y, 0) is induced from ~b ~ Aut(~", 0) and L~ Glp(@n). Replacing (Y, 0) by ~b-a(y, 0) and the system of equations y e (9~ of r 0) by (L-lo ~b)- g, one checks immediately that Sings(X, 0)= Sings(Y, 0).

(2) Set f = f + t(g - f ) e (9 p, t ~ I~, f , g e (9~ equations for (X, 0) and (Y, 0): Then the family {(Xt, 0)}t~c defined by f~ describes a deformation from (X, 0) to (Y, 0).

(3) Sings(X t, 0)=Sings(Xo,0 ) for t e T_c_C Zariski-open: Let h and k E (@~)~ be defining equations for Sings(X, 0) and Sings(Y, 0) derived from f and # in the obvious way. Then ht:= h + t ( k - h ) defines Sing s (X, 0) and Lemma 1 proves the equality (cf. also Lemma 1' of appendix).

(4) Reduction to the assertion: For any t o e T, the deformation {(X, 0)},~r of (Xto, 0) is trivial: If {(Xt, 0)} is locally trivial at any t o e T, it will be globally trivial over any compact subset of the connected set T, hence (X o, 0) = (X, 0) and (X1,0) =(Y, 0) will be isomorphic. (This argument fails over ~, or for char>0, as then T need not to be connected.)

(5) The deformation {(Xt, O)}t~(T, to) of(Xto, 0) is trivial for any t o e T: By (3), 3tft = g - f e Tsf~ = Ts f = Tsg for any t e T. By Lemma 2, this inclusion is analytic in t. We can therefore apply Proposition 2 with TK(f~) replaced by Tsft to the deformation f~ and obtain isomorphisms ~b, :(Cn, 0 ) ~ ( ~ n, a3, a~=~bt(0), sending (Xt, at) to (Xto , 0). As mentioned in the remark following Proposition 2, ~b, may not send origin to origin:

j f

We claim that the initial hypothesis on the manifold S, namely that S contains the Sings (X, 0)-triviality stratum, will insure that fit actually does preserve the origin. Hence at = 0 ~ ~n and (X,, 0)--- (Xto, 0) for all t e (T, to).


Notice first that from the way Tsf is defined and enters in the proof of Proposition 2, the S-coordinates of at must be 0. Therefore, if at moves with t, it must move off S.

Second, the established isomorphism between (X, a,) and (Xto, O) implies S i n g s ( X . at) ~ Sings(X, o, 0) and by (3): S i n g s ( X . at) -~ Sings(X. 0). This suggests that at belongs to the Sings(X . 0)-stratum St = X. As X __c S by assumption, at = 0 would follow. However, one might imagine the following phenomena:

Q t / /

Z=0

xt 0 xt I

T t o

Here, S i n g s ( X , aO ~- Sings(X,, 0) does not imply at e S. In order to show that this type of coalescing cannot arise for the family {(X t, 0)}, represent the situation on a small open neighborhood U of 0 e ~". By Lemma l'(b) of the appendix, there exists D__c T open, to e D, such that for U sufficiently small and representatives X t of (X,, O) on U, SingsX , = SingsX,o holds on U for t 6 D. This implies that the strata X, coincide on U, X t = Z,o for t ~ D. Furthermore, reducing eventually U again, one can assume that Z t c= S on U. But then Sings(X . at) _~ Sings(X . 0) implies for at ~ U and t e D that at e St c= S. As a t was not allowed to move inside of S, at = 0, and (X,, O) ~ (Xto, 0). This establishes (5) and concludes the proof of the theorem.

3. Appendix: Generalized analytic varieties

In this section we describe briefly a category of generalized analytic varieties in which the singular subspace of a variety as defined in w 1 is again a variety and an extension of the theorem of w 2 holds.

Definition 1. A germ of a (generalized) analytic variety (X, 0) in (IE", 0) is a couple

(x , o) = ( (x , o), ex, o)

such that m d m l , . , m d (1) (gx, o = (9'2/1 is a quotient module of (9.': = (9'2'| ... | =(9. for

some d-tuple m = (m 1 . . . . . md) E N d. (2) (X, 0)= {(X. 0)}r~N is a collection of germs of subsets (X. 0) of (t~", 0)

defined by

for some presentation

(X. O) = {x e (IE", 0), rk ( f ( x ) ) <= r}

0~ Y, (9~-~ ~x,o-~ 0 (peN)


of Cx, 0. The rank of an element A = ( A 1 , . . . , Ap) �9 (IEm) v is the maximal number of ll;-linearly independent components Ai of A. The germs (X r, 0) do not depend on the choice of the presentation f .

Remark�9 As (9, =(9, |174174 one can identify varieties with analytic modules (92/I and (92//-, where n~ is obtained from m by adding l's, r~ = (m~ . . . . . rod, 1 . . . . . 1), a n d / i s the image o f / i n (92.

On the other hand, for m, m' �9 N d with m___< m' component by component, we shall identify (92/I and (9"'/I x 0, where I x 0 denotes the image of I under the canonical injection (92 ~ (92'.

This signifies that we won't distinguish between varieties, whose modules differ only by free components (we are interested in the image of the presentation f but not in the cokernel). Therefore, whenever we compare two varieties (X, 0) and

t _ _ h i ' t ( X , 0), (gx, o = (92/1, (gx, ,o- ( 9 , / I , we may tacitly assume that m = m ' � 9 N d. Obviously, the underlying sets (X,, 0) are not affected by these identifications.

Morphisms. Any analytic map-germ ~b'(IE",0)~(IE",0) induces for arbitrary m �9 N d a morphism

~ * m . . . ~ m �9 (9. (9.

sending g �9 (92 to ~*(g) = g o q~. Second, any matrix L � 9 Mm((9,): = Mmj(9,) x ... x Mm((9,) defines an (9,-linear map

m _ . ~ tn L: (9, (9.

multiplying component by component.

Definition 2. A morphism h: (92~(92 is a couple h = (L, ~) with underlying map Lo ~* "(9~-~(92. As ~*o M = q~*(M)o ~b* for any M �9 Mm((9.), the composition of morphisms is well defined.

Let (X,0) and (Y, 0) be two varieties in (II;",0) with (9x,o=(92/I and (gr,o=(92"/J, m � 9 N a, m' �9 N d'. As described above, we may assume m = m ' �9 N a.

r . l ' _ 4 " l ' r d We say that two morphisms h' (9~(9t, and h . (9. (9.(l, l �9 N ) are equivalent on (X, 0), if for any k �9 N d with k > l, l', m (by components) the image of the induced map

lies in I = (_9~ (more precisely, if Im((h, 0) - (h', 0)) ___ I x 0__c (9~). A morphism h" (X, 0) (9.-~(9. (k >m) ~(Y, O) is the equivalence class over (X, O) of a morphism H: k k

satisfying H(J) c= I c= (9~.

In particular, h induces a map h: (gr, o~(gx, o . The composition of two morphisms is well defined- we can talk of isomorphic

analytic varieties: (X, 0) - (Y, 0) if and only if there exist r :(I~",O)~(IE",0) and m._~ m L�9 , such that h=Lo~*:(92--*(92 induces a bijection L: (9. (9.,

h: (gr, o-§ o-

Definition 3. Let (X, 0) and (X', 0) be two varieties in (IE", 0), (gx, o = (gin~ I and (gx', o = 02'/I'. We may assume m = m'. We say that (X', 0) is a subvariety of(X, 0) if (gx', o is a quotient of (gx, o, say , I___ I'.


A morphism h : (X, 0)~(Y, 0) is an embedding, if it induces an isomorphism from (X, 0) onto a subvariety of (Y, 0).

In the so defined category one may carry out the usual operations, in particular, show that the fiber-product of two varieties over a third exists.

We shall now indicate, how one has to modify the statements and proofs of the preceeding sections in order to obtain the analogue results for generalized analytic varieties.

For m ~ N n and p ~ N, let K denote the group K = Glmp((Pn) x Aut(~ n, 0) with Glmp((~,) = Glm(C.) x Glp((9.) = Glml(Cg.) x ... x Glm~(C,) x GIp(C.), acting on

ra p O,"P: = (9. | component by component. We have:

Lemma 1'. (a) The fibers of the map

zc : (9~"P-o { isomorphism classes of analytic varieties}

f--* [(X, 0)]

(respectively z : (_9~"P~{analytic varieties}, f - o ( X , 0)) given by

O~ I , (9~"--* (9x, o---'0

are the K-orbits (respectively Glp((9,)-orbits) in (tim, ~. (b) For any family {(Xt, 0)}t~(r,0 ) of varieties such that (Xt, O)~-(Xo, O )

(respectively (Xt, O) = (X o, 0)), there exist open neighborhoods D of 0 ~ T and U of 0 ~ IE" with representatives X t of (Xt, O) on U such that Xt~-Xo (respectively X t = Xo) on U for t ~ D.

Proposition 1 holds as it stands.

Proposition 2'. Let f : (T, O)-o((p~"v, fo) be a deformation. (a) f is K-trivial, if and only if for any coordinate s on (T, 0), Osft belongs

analytically in t ~ (T, O) to

T K ( f ) = Y~ M m,( (g , ) " f t + M p( (g n) . f t + ~r162 " J( f t) C= (9 mp"

(b) f is G-trivial ( G = Glp( (9,) ), if and only if O~ft ~ Mp( O.) . ft c= (9~"v analytically.

Definition 2". Set T s f = EMm,(~.). f + Mp((9,). f +~, .js(f)+jz(f)c__(_9~"P and Sings(X, 0) = the singular subspace of(X, 0) defined by Tsf. It is obvious from the definitions, that Sings(X, 0) is a generalized analytic variety and a subvariety of (X, 0).

Lemmas 2 and 3, Definitions 3 and 4, and Ephraim's Theorem remain valid in the generalized context as they are. We don't know if Proposition 3 extends.

Theorem. Let (X,0) and (Y,O) be two germs of generalized complex-analytic varieties in (112",0). Let S c=(E" be a manifold containing the stratum S c=~" along which the singular subspace Sings(X, 0) of (X, 0) is trivial.

The varieties (X, 0) and ( Y, O) are isomorphic if and only if the singular subspaces Sings(X, 0) and Sing~,(s)(Y, 0) of (X, 0) and (Y, O) are isomorphic via an isomorphism h = (L, 0).

The proof of the theorem can be translated word by word from the proof of the theorem of w 2.


Remarks. The procedure of passing to the singular subspace can now be iterated, looking at Sings(X, 0), Sings, (Sings (X, 0)) etc. It might happen that after a certain number of steps, the corresponding singular subspace is defined by a finite dimensional vectorspace, which, in turn, will therefore determine together with its (9.-module structure the analytic type of the original variety (X, 0).

Example. Sing(X, 0) is a complete intersection with isolated singularity.

4. Example

We shall construct a family of hypersurfaces (Xt, 0), t e 112, with constant singular subspace Sing (X t, 0): = Sing o (X t, 0) = Sing o (X o, 0) (for t + + 1) but varying analytic type (X,, 0) ~ (X o, 0).

This shows that the singular subspace Singo(Xo, 0) need not determine the analytic type of the variety, which, in turn, illustrates, that the assumptions made in the theorem, w 2, are actually necessary.

Let h: (C", 0)--*(II~, 0) be any function satisfying h Cj(h)C= (9,. Define a family f t : (C"xl l ; "x lE,0)~(( ; ,0) by f t ( x , y , z ) = h ( x ) + ( l + z + t ) . h ( y ) , and let (X~, 0) ____ (1122"+ ~, 0) be the hypersurface defined by ft.

We claim that Singo(Xt, 0)= Singo(Xo, 0) for t # +_ 1: Computation gives

j(ft) = [j(h(x)) + j(h(y)) + (h(y))] �9 (92, +~.

Hence

To f = [(h(x)) + (h(y)) +j(h(x)) +j(h(y))] . (gz, +1.

On the other hand, the family {(Xt, 0)},~c is not trivial: For, if {f,}t~c were trivial, we would have by Proposition 2, w 1,

Otf t = h(y) e (ft) + mz,+ 1 "J(ft)

= (ft) + ~2n + 1 .j(h(x)) + ~2. + 1 .j(h(y)) + ~2, +1" (h(y)).

Solving for h(y) implies either h(y)ej(h(y)) or h(x)ej(h(x)) contradicting the assumption on h.

Proposition 4. (cf. [Sa] for notations and the isolated singularity case). Let f e (9. define a hypersurface (X, O) which is of isolated analytic type (def. 3). Then f Ej ( f ) -=-jr and only if f is K-equivalent to a weakly quasihomogeneous g ~ (9,.

Proof. I f f is weakly quasihomogeneous with respect to weights dl . . . . . d. e Z on coordinates xl . . . . , x, on I12", Euler's formula says

deg(f) , f ( x ) = i~=t d,. x~.~x (X)~j ( f ) .

Assume conversely f ~ j ( f ) . Note that the proof of 4.1 in [Sa] is valid for any f ~ (9,, provided f e j ( f ) implies f E ~ . . j ( f ) by means of 3.3. It will be therefore sufficient to show that f isolated analytic type implies f 6 m. . j ( f ) . Write f = 52 a i �9 Ox,f with


aie (9,. If for at least one i, a i = c i - b~ with b~ e ~ , and c~ e C, c~ ~: 0, the vector e = (cl . . . . . c,) e C" would be non-zero. By a C-linear coordinate change, one can achieve c = (0, 1,0 . . . . . 0), i.e.,

By Proposition 2(a) it follows that (X, 0) is trivial along the Xz-axis and hence (X, 0) not of isolated analytic type.

5. Part II: ~-equivalenee theorems

One of the principal tools for our results on d-equivalence will be C*-level preserving maps. These are maps of the form F : C " x c k ~ c p X C*, F(0)=0, F(x, u)= ( f ( x , u), u). If f = f ( x , 0) has particular importance, we may call F an unfolding of f . We first define some objects for C-level preserving maps, analogous to the d- tangent spaces for map germs.

Definition 1. Let F: (C" x C ,0 )~ (C p x C,0) be a C-level preserving map germ.

Then je.(F) is the (9, + 1-module generated by ~x~'

a) 7"~(F) = m, .je,(F) + F*(mp(9 p p + 1) c= (9",+1 b) T~e(F) =je,(F) + F*((9;+ 1) ~ (gP+ 1"

In this section the exponent of an ideal indicates a power of the ideal, while the exponent on (gv + 1 indicates Cartesian product. The ideal generated by the p • p minors of the jacobian matrix of f is denoted J( f ) .

We say a level preserving map germ is trivial in a region B_c_ C if there exist C-level preserving biholomorphic germs H of (C " • 0 • and K of (C p x C, 0 x B) such that

K o F o H = f x l .

We now give some well known criteria for triviality.

Proposition 1. (Thom-Levine) a) Suppose B is a region of C. A C-level preserving map germ F : (C" x C, 0 x B) ~ (C p x C, 0 x B) is trivial iff OtF a ~ 7"d F" for all a E B. (Here F" denotes the germ of F at (0, a).)

b) Suppose B =0, then F is trivial iff OtF~ ~ Tale F~ (cf. [duP] p. 113 for a) and I-M] p. 8 for b)).

Throughout this section f : (C", 0)~(C p, 0) will be a complex analytic germ of finite singularity type. If g:(C", 0)~(CP,0) is any other holomorphic germ, let G(x, t) = ( f ( x ) + t (O(x)- f (x ) ) , t) be the level preserving germ linking f and 9. We wish to show under various hypotheses on f and g that G is trivial, using Proposition 1. Our method of attack is to compare T d ( G ) with 7"d(F), where F(x, t)= ( f (x ) , t). The key technique in this comparison is to study the various module structures of T d ( F ) and of Tale(F). We begin with a lemma which shows that the module structure of T d ( F ) has a certain convexity property.

/02P + 1 In what follows, let H(x, t ) = ( f ( x ) , 9(x), t)~ ,~.+ 1 �9


Lemma 1. a) Suppose T ~ ( f ) = T ~ ( g ) , then T s t ( F ) ~ H*((9~p+ 1) and ~FsJ(F) is an H*((92v+ O-module.

b) Suppose T~4e( f )= Tale(g) then 7"de(F ) is an H*((gEp + 1)-module.

Proof. a) It is obvious thatje.(F) is closed under multiplication by H*((gEp + 1). It suffices to consider H*(~Ep + 1)' F*(~p(gg + 1)" Our method of proof will also show H*(c~2p " (9Pp + 1) ~ T d(F ) .

Note that T d ( f ) == ~ 'd (F) , and that this implies (F*(gv+ 1). (g*(mpC~))= TzC(F). In particular, gk.ei = ak, i o F + rk, i OxF, where ei is a canonical basis of ~P. Hence, Aki(X,y)=xkei--aki(y) has the property that A k , i ( g , f ) ~ ~ , . jr

Suppose B(x, y, t) z ~2v + a. Consider (B o H). (D o F) = F,(D o F) (B o H) et, DeOP+I . By the Division Theorem for modules we can write B(x ,y , t ) e t = Z Qk, i, l(x, y, t) . Aki(X, y) + Cl(y, t). Then (B o H) et = Cl o F rood ~,,. jr ,F. Hence,

k, i

Z(Dl ~ F) . (Bo H)et ~ Z4(F) . If B ~ ~2p, then Ct(y, t) ~ ~ p . (gv + a, thus (B o H) e t ~ T~C(F) also. This implies

H*(m2p. (9~p+ 1)---- T~C(F). The proof of b) is similar.

Remark. The above proof gives the following useful fact. Suppose h ~ (9, such that h.e i e T d ( f ) or T~ge(f) then, T s / ( f ) (resp. Ts /~( f ) ) is an (gp+ 1-module via (h , f ) .

A key assumption in what follows is that all our germs have finite singularity type, or F.S.T. (i.e. T ~ ( f ) ~ = ~ k . (~ for some k). It is obvious that if T~C(f ) = Txr and f has F.S.T. then so does g. Does the equality TsC~(f) = Td~(g) imply the same result?

Proposition 2. Suppose TsC~(f) = TsCe(g) and f has F.S.T. Then g has F.S.T. also.

Proof. Let R be the ring of all functions h such that h. T~g~(g) ( T ~ ( g ) . Let M be 1

the ideal of all such functions vanishing at zero. Then ifh ~ M, ] ~ is in R also. For

1 1 + h" T,~,(g) lies in h*((9 0 �9 TzC~(g), and the above remark shows that if h is in R,

TsCe(g) is an h*(9 a-module, so h*(9 a �9 Tsl~(g) is in TzJe(g). Since h e M implies 1 + h is a unit in R, Nakayama's lemma can be applied to R-modules. We have that

Tsge(g) is a finitely generated R-module, for R contains g*(gp. Further, it is easy to

check that TeuC(g) is a R-submodule of Tsar(g). Since dim T~l~(g) _ < n +p it ~ ( g ) ~,~(g) T d (g )

follows by a variant of Nakayama's lemma that M ~ +v + 1. T~cff~(g)C Tsff(g). This implies that T~ff(g) 3 T~C(g). (9, ~ M" + p + 1. (gp. We have

M "+p+I �9 (gv ,3 ( f ,g , J ( f ) ) "+p+I �9 (9.P 3 ~,'k (gp

for some k, since f has F.S.T., and M contains f~, O j, and the generators of J ( f ) . Thus, T:)ff(g) k v ~ , . (9, and g has F.S.T.

The next result shows that we can lift the hypotheses of Lemma 1 concerning the equality of tangent spaces from the level of germs to the level of unfoldings. We


compare T d ( G a) which may vary with a, to T d ( F ~) which is constant with a. This will allow us to show that for most a values "F~(G a) is constant, and will imply that atG" �9 Td (G ~) which we need to apply Proposition 1.

Proposition 3. Suppose f has finite singularity type. a) I f T d ( f ) = Tsl(g), then 7"o/(G ~) = 7"d(F ~) for all a ~ �9 except a finite

number of values. b) I f Tde( f ) = Tde(g), then TS~e(G a) = "F~e(F a) for all except a finite number

of values.

Proof. a) As noted before Td(f )=Td(g)~TJ~fr( f )=TA~(g) . Thus, g has F.S.T., and by our results on ~-equivalence, G a has F.S.T. for all except at most a finite number of values of a.

Further, it follows that if J(f)+f*(~p)C.~=~k. and j ( f ) + f , ( ~ p ) ( g z ~ k - 1 , then the analogous statements are true for G a except for the finite number of a values.

For we know the G a are JU-equivalent except for the odd a values; hence the ideals J(G a) + Ga*(~v+ 1)(9n+ 1 are right equivalent for all values of a, apart from the above finite number.

It is easy to see that Td(G a) z=Td(F), for by Lemma 1, G"*(~p(9~+ 1) = n*(~2p(9~p+ 1) __C Td(F). We also have that ~. . jc , , (G a) c= Td(F) for on, "Je.(g)c= T~t(f)c= 7"d(V).

We now must show that T d ( G a) = Td(F) for all except a finite number of values of a. The next lemma gives us an easy to apply criterion for equality.

Lemma 2. Suppose Td(G") + H*(c~C2p+ 1) Td(F) = Td(F), then Td(G ~) = T d ( F ) .

Proof. We first show that Td(G a) + Ga*(~p+ 1) Td(F) = Td(F). Consider

Td(F) 7"d~r Ga) + Ga*(c/~p + 1) T:/(F) (*)

this has the structure of an H*((92p+ 0-module. For,

H*((92p + ~)(~. . jr ~ + G"*(~p(9~+ 1) "q- Ga*(~p+ 1) ?'~r

c= Td(G") + G"*(~, + 1) Td(F) + H*((92, + 1) (Ga*(~p(PPp + 1))

c 'Td(G") + Ga*(~'~p + i) 'Fd(F) + H*(~Et,) (Ga*(c/gp(9~ + 1 )

C "F.5~(G a) + G~*(o.n, + 1) Td(F), since H*(~2,)e, c= Td(F).

Further Td(G") + Ga*(a~ep+ 1) ' ~ ( F ) contains r a) + Ga*(cgCp+ 0)jr and except for a finite number of a values, this contains ~,(~zk+ 0 jc ,F .

This is important, for F*(~p(9~ + 1) is always finitely generated as an H*((gzp + 1) module, but jr may not be. The above inclusion shows that all but a finite number of the elements of j r are already contained in the denominator of (*). Thus (*) is a f.g. H*((92p+ 1)-module, so by Nakayama's lemma (*) is zero.

Given that Td(F)= Td(G")+ Ga*(c~p+ 1) 7"d(F), we now want to show that T~t(F) (9. + 1 T ~ ) ~ is a finitely generated G~*((gp+ O-module. For good values of a, ~ j r

Character iz ing singularit ies o f varieties and of mapp ings 445

T d ( F ) is f.g. as a Ga*((gp+l)-module, hence ~njc.(G~ ) is finitely generated also since

Td(F) G~ is Noetherian. Consequently, ~ is finitely generated, and

Nakayama's lemma again applies. Returning to the proof of Proposition 3, we now show that checking the

hypothesis of Lemma 2 is a finite, algebraic problem. Let

T d ( F ) A= ~r162 7"d(F) where ~k~J( f )+ f*(~p)(9,,.

Notice that ~r is finite dimensional as a C-vector space, and ~ d ( G ~) + H*(~p + 1) Td(F) = k + ,(~,je,(F)).

(For k + ~(~.je.(F)) __c (G~*(~p + 1) "-[- J(G~)) ~dr

__c ~ , .jc.(G o) + 6o, 7"d(F)).

Further, k 2k ~,+l(~nJr Td(F) contains ~n+ l(~.'Jc-(Ga)) --l- G~*(~p+ l(~p(~nP+ 1))"

Consider the polynomial map of finite dimensional vector spaces induced by

which maps the direct sum ~"(9"+1 ~P(9~+1 1)into Dx( Ga) + G a*, 2k I zt~n . ( ~ - - - A . ~ n + l ~ n ~ n + 1] r l(~p(fipP+

The above remarks ensure this mapping is well defined. The map is linear for fixed a, and surjective at a iffthe hypothesis of Lemma 2 is satisfied. It is easy to check that the map is surjective at a = 0, 1. By the lemma ~ d ( G a) -- ~d(F") for all except a finite number of a values.

The proof of Part b) is similar. As before we have that T'de(F ) ~ "['de(G a) and that Tde(F) is an H*((92p+ 1)-module.

Claim. Suppose T~c~(G ~) + (G~*(~p+ 1) * P + 1 + H (~v~2p + 1)) 7"cleF = Td~F then T~/~(G ") = T~g~(F).

The proof is similar to the proof of 3.1 in [G 1]: The map that we consider in this case is induced by Dx(G a) + G a* with domain

(9," + ~ (9~,+ 1 and target T~g~(F) k+l,,,qn ~ ) p + l / o p l �9 * p + l -- . Here l

~r . , , , p+x~p+l ~,+a'Jc,(F)+H (~2p+I)T~e(F) refers to the power of ~ , + 1 which lies in (J(G ~) + Ga*~cp + 1) (_9. + 1 for generic a. (We can ensure 1 is in this generic set.)

It is easy to check that Dx(G") + G"* surjects iff the hypothesis of the claim holds, and that it surjects for a = 0, 1. Hence, except for a finite number of bad values, Td~(G")= Td~(F).

We now develop the notion of morphism for the d-case; in what follows we call h e (9, p a generating map for T d ( f ) if T~t(h) = Tsg(f). (Similarly for T~( f ) . )

Our notion of morphism must be modeled on the change that Tsl( f ) undergoes as we pass from Tsl( f ) to Td(l ofo r). At the same time, the notion of morphism ought to be independent of the mapping which gives rise to Td( f ) .

Definition 2. Suppose m = Tsg(f~' m , - T ~ ( f ) "


A morphism with domain M (resp. Me) is given by a triple (r, L, h) where re Aut(~", 0), L is a p • p invertible matrix with entries in (gp such that L = Dl for some leAut(lI2P,0), and h is a generating map of T d ( f ) (resp. Tde( f ) ) . The morphism is given by ~ ( L o h o r)(3 ~ r), where ~ E M. It is easy to see that this

( gives a well defined mixed module homorphism between M and T d ( l o h o r) resp.

M e and Ts~e(l o h r)) which is an isomorphism of mixed modules.

Before we give our main theorem, recall that f is a stable germ if T d e ( f ) = (9~, and f is finitely determined if T d e ( f ) ~ k p m,(9, for some k.

Theorem. Suppose f and 9 are complex analytic map germs f : (I12",0)~(I12P,0), 9 : (II~",0)--*((12P,0). Suppose f has F.S.T.

A) I f f is finitely determined, but not stable, then (gP" CP" T ~ ( f ) - Tale(g) iff f ~ g.

B) ~ •" T d ( f ) - T d ( 9 ) iff f ,~ 9.

Proof of B). If (9"p C"P T d ( f ~ ~ Td(g~ we can find h such that T~C(f) = Td(h) , T d ( g )

= T d ( l o h o r). It follows from the Propositions 1 and 3 that we can choose a path connecting

l o h o r with 9 and a path connecting h and f such that the family of mappings given by these paths are ~r to h and l o h o r. Hence f is d-equivalent to g.

Proof of A). By a preliminary coordinate change we may assume T~Ce(f) = Tde(9). Thus Tde(G a) = Tde(F) for all a values off some Z-closed subset of C.

By Proposition 1 and the compactness of our path connecting f and 9, we can cover this path by a finite set of open intervals Ii with distinguished points xi such that the unfolding G given by (li, xi) is trivial. We must show that the families of trivializing diffeomorphisms that occur are origin preserving in source and target.

Since f is finitely determined, Tde(f)z=~k, cP, for some k, hence ~Fde(F) 3=~R,(gP,§ 1. Thus "F,5~e(G a) D= ~nk(_gnP+ 1 for all a values along our path. This

implies that the target sheaf t~. j c~- / ,_ , ,~p+ ~ has support only along the t-taxis.

The assumption that f is not stable implies it does have support there. This implies

that the sheaves 9~, 9"(9~ on II; p have support only at the origin, for all t values

in a neighborhood of our curve in 112. This forces the trivializing diffeomorphisms to be origin preserving so f and 9 are equivalent. Acknowledgements. Discussions with Max Benson, Robert Ephraim, Karl Knight, and Frank-Olaf Schreyer have contributed to the development of this paper; the graphics were done by Daniel Wilson.

Bibliography [B] Benson, M.: Analytic equivalence of isolated hypersurface singularities defined by

homogeneous polynomials. Proc. Symp. Pure Math. Am. Math. Soc. 40, 111-118 (1983) [ B 1 ] Benson, M.: Analytic equivalence of isolated hypersurface singularities defined by

homogeneous polynomials. Thesis, Harvard University 1982


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[G 1]

[G 2]

[G-P]

[W]

[duP]

[IV[]

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Oblatum 26-VI- 1984 & 10-I- 1985

Characterizing singularities of varieties and of mappings

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