AN EXPLICIT CYCLE REPRESENTING THE FULTON-JOHNSON CLASS, I

AN EXPLICIT CYCLE REPRESENTING THE FULTON-JOHNSON CLASS, I

by

Jean-Paul Brasselet, Jose Seade & Tatsuo Suwa

Abstract. — For a singular hypersurfaceX in a complex manifold we prove, under certain conditions, an explicit formula for the Fulton-Johnson classes in terms of obstruction theory. In this setting, our formula is similar to the expression for the Schwartz-MacPherson classes provided by Brasselet and Schwartz. We use, on the one hand, a generalization of the virtual (or GSV) index of a vector field to the case when the ambient space has non-isolated singularities, and on the other hand a Proportionality Theorem for this index, similar to the one due to Brasselet and Schwartz.

Résumé (Une description explicite de la classe de Fulton Johnson, I). — Pour une hyper- surface singuli`ereXd’une vari´et´e complexe, et dans certaines conditions, nous mon- trons une formule explicite pour les classes de Fulton-Johnson en termes de th´eorie d’obstruction. Dans ce contexte notre formule est similaire `a l’expression des classes de Schwartz-MacPherson donn´ee par Brasselet et Schwartz. Nous utilisons, d’une part, une g´en´eralisation de l’indice virtuel (ou GSV-indice) d’un champs de vecteurs au cas o`u l’espace ambiant a des singularit´es non-isol´ees et, d’autre part, un Th´eor`eme de Proportionnalit´e pour cet indice, similaire `a celui dˆu `a Brasselet et Schwartz.

1. Introduction

There are several different ways to generalize the Chern classes of complex man- ifolds to the case of singular varieties. Among them are the Schwartz-MacPherson classes [5, 16, 20] and the Fulton-Johnson classes [8, 9]. Each one of them is defined in a relevant context and has its own interest and advantages. The construction in [5, 20] provides a geometric interpretation of the Schwartz-MacPherson classes via

2000 Mathematics Subject Classification. — 14C17, 32S55, 57R20, 58K45.

Key words and phrases. — Schwartz-MacPherson class, Fulton-Johnson class, Milnor class, radial vec- tor fields, virtual indices, Milnor fiber.

Research partially supported by the Cooperation Programs France-Japan, CNRS-JSPS (authors 1 and 3), France-M´exico CNRS/CONACYT (1 and 2), by CONACYT Grant G36357-E (2) and by JSPS Grant 14654010 (3).

obstruction theory. This approach is very useful for understanding what these classes measure.

The motivation for this work is to give such a geometric interpretation of the Fulton- Johnson classes, in the spirit of [5, 20]. Here we prove that if X ⊂M is a singular complex analytic hypersurface of dimension n, defined by a holomorphic function on a manifold M, then the Fulton-Johnson classes can be regarded as “weighted”

Schwartz-MacPherson classes.

In order to explain our result more precisely, let us consider a complex analytic manifold M of dimension m, and a compact singular analytic subvariety X ⊂ M. Let us endow M with a Whitney stratification adapted to X [24], and consider a triangulation (K) of M compatible with the stratification. We denote by (D) a cellular decomposition of M dual to (K). Let us notice that if the 2q-celld_α of (D) meetsX, it is dual of a 2(m−q)-simplexσα of (K) in X.

We recall that in her definition of Chern classes, M.H. Schwartz considers particular stratified r-frames v^r tangent to M, called radial frames. They have no singularity on the (2q−1)-skeleton of (D), withq=m−r+ 1, and isolated singularities on the 2q-cellsdα, at their barycenter{bσα}=dα∩σα. Let us denote byI(v^r,bσα) the index of ther-framev^rat bσα.

The result of [5] tells us that the Schwartz-MacPherson class c_r−1(X) of X of degree (r−1) is represented inH_2(r−1)(X) by the cycle

X

σα⊂X, dimσα=2(r−1)

I(v^r,bσα)·σα

In this article we prove:

Theorem 1.1. — Let us assume thatX ⊂M is a hypersurface, defined byX=f⁻¹(0), wheref :M →Dis a holomorphic function into an open disc around0inC. For each point a∈ X let Fa denote a local Milnor fiber, and let χ(Fa) be its Euler-Poincar´e characteristic. Then the Fulton-Johnson class c^{F J}_r−1(X) of X of degree (r−1) is represented inH_2(r−1)(X)by the cycle

(1.1) X

σα⊂X, dimσα=2(r−1)

χ(F_bσα)I(v^r,bσα)·σα

On the other hand, the question of understanding the difference between the Schwartz-MacPherson and the Fulton-Johnson classes has been addressed by sev- eral authors, and this led to the concept of Milnor classes, defined by µ∗(X) = (−1)ⁿ⁺¹ c∗(X)−c^{F J}∗ (X)

,n= dimX, see for instance [1, 3, 19, 25]. Let us define the local Milnor number ofX at the pointa∈X byµ(X, a) = (−1)ⁿ⁺¹(1−χ(Fa));

it coincides with the usual Milnor number of [17] when a is an isolated singularity ofX. It is non zero only on the singular set Σ ofX. We have the following immediate consequence of Theorem 1.1:

Corollary 1.2. — Under the assumptions of Theorem 1.1, the Milnor class µ_r−1(X) inH_2(r−1)(X) is represented by the cycle

(1.2) X

σα⊂Σ dimσα=2(r−1)

µ(X,bσα)I(v^r,σbα)·σα

One of the key ingredients we use for proving the Theorem 1.1 is aProportionality Theoremfor the index of vector fields and frames on singular varieties, similar to the one given in [5]. In order to establish it we were led to definingthe local virtual index at an isolated zero of a smooth vector field on a complex hypersurface with (possibly) non-isolated singularities. This is a generalization of the indices defined previously in [4, 12, 15]. We call it “local” virtual index to distinguish it from the “global” virtual index at a whole component of the singular set, as studied in [4]

We notice that for hypersurfaces with isolated singularities one also has the homo- logical index of [11], which coincides with the index in [12]. It would be interesting to know whether our generalized virtual (or GSV) index coincides with the generalized homological index in [10] when the ambient space has non-isolated singularities.

Our formulae can also be obtained in another way, using the MacPherson mor- phism c∗ (see [16]) together with the Verdier specialization map of constructible functions [23], since one knows (see for instance [19]) that the Fulton-Johnson and the Milnor classes are image by the morphismc∗ of certain constructible functions.

The advantage of our construction here is to provide a geometric and explicit point of view, which can be used to study the general case. This is being done in [6].

2. The local virtual index of a vector field

Let (X,0) be a hypersurface germ in an open setU ⊂Cⁿ⁺¹, defined by a holomor- phic functionf : (U,0)→(C,0). Let us endowU with a Whitney stratification{Vi} compatible withX and let us consider the subspaceEof the tangent bundleTU ofU consisting of the union of the tangent bundles of all the strata.:

(2.3) E=[

Vi

T Vi

A section ofTU whose image is inE is called astratified vector fieldonU.

Let v be a stratified vector field on (X,0) with an isolated singularity (zero) at 0 ∈ X. We want to define an index ofv at 0 ∈X which coincides with the GSV- index of [12] (or the virtual index in [4]) when 0 is also an isolated singularity ofX. For this, let us consider a (sufficiently small) ballBε around 0∈ U and denote byT the Milnor tubef⁻¹(Dδ)∩B_ε, whereD_δ is a (sufficiently small) disc around 0∈C. We let∂T be the “boundary”f⁻¹(Cδ)∩Bε ofT, Cδ =∂Dδ.

Letr be the radial vector field inCwhose solutions are straight lines converging to 0. It can be lifted to a vector field er in T, whose solutions are arcs that start in

∂T and finish in X; since the corresponding trajectories in Care transversal to all the circles (Cη) around 0∈Cof radiusη∈]0, δ[, it follows that the solutions oferare transversal to all the tubes f⁻¹(Cη). This vector field er defines aC^∞ retraction ξ ofT intoX, withX as fixed point set. The restriction ofξto any fixed Milnor fibre F =f⁻¹(t0)∩B_ε, t0∈C_δ, provides a continuous mapπ:F→X, which is surjective and it isC^∞over the regular part ofX. We call such mapξ, or alsoπ, adegenerating mapforX (this was called a “collapsing map” in [14]). Since the singular set Σ ofX is a Zariski closed subset of X, we notice that we can choose the lifting er so that π⁻¹(Xreg) is an open dense subset ofF, whereXregis the regular partXreg=XrΣ.

We want to useπ to lift the stratified vector fieldv on X to a vector field onF. Firstly, let us consider the case whereX has an isolated singularity at 0. The mapπ is a diffeomorphism restricted to a neighbourhoodN ⊂F ofF ∩∂B_ε. Then v can be lifted to a non-singular vector field on N and extended to the interior of F with finitely many singularities, by elementary obstruction theory. By definition [12], the total Poincar´e-Hopf index of this vector field onF is the GSV-index ofv onX.

We want to generalize this construction to the case when the singularity ofX at 0 is not necessarily isolated. Let us consider (X,0) as above, a possibly non-isolated germ. We fix a Milnor fibre F = f⁻¹(to)∩Bε for some to ∈ Cδ. Given a point x ∈ F, we let γ_x be the solution of rethat starts atx. The end-point of γ_x is the point π(x) ∈X. We parametrize this arc γ_x by the interval [0,1], with γ_x(0) =x and γx(1) =π(x). We assume that this interval [0,1] is the straight arc in Cgoing from to to 0, so that for each t ∈ [0,1[, the point γx(t) is in a unique Milnor fibre F_t= f⁻¹(t)∩B_ε. The family of tangent spaces to F_t at the pointsγ_x(t) defines a 1-parameter family ofn-dimensional subspaces of Cⁿ⁺¹,{T Ft}_γx(t). By [18] we may assume that the Whitney stratification{Vi} satisfies the strict Thom wf-condition.

This implies that for each trajectoryγ_x(t) the corresponding family{T Ft}γx(t)has a well defined limit space Λπ(x),i.e.it converges to ann-plane Λπ(x)⊂T_π(x)(U) when t →1. Hence one has an identification TxF ∼= Λπ(x) which defines an isomorphism of vector spaces. Moreover, sincewf implies the Thomaf-condition one has that the limit space Λπ(x)contains the spaceT_π(x)V_itangent to the stratum that containsπ(x).

Therefore the vectorv(π(x)) can be lifted to a vectorev(x)∈T_xF. This vector fieldev is non-singular over the inverse image ofXreg, which is open and dense inF. Alsoev is non-zero on a neighbourhood of F ∩∂Bε, since v is assumed to have an isolated singularity at 0. Furthermore, by thew_f-condition the vector fieldveis continuous, so it has a well defined Poincar´e-Hopf index in F. The wf-condition also implies that the angle betweenv(π(x)) andv(x) is small. That is, given anye α >0 small, we can chooseδ sufficiently small with respect to αso that the angle between v(π(x)) and e

v(x) is less thanα. This implies that if we replaceev by some other lifting of v, the induced vector fields onF are homotopic. Since f induces a locally trivial fibration

over the punctured discD_δr0, then the homotopy class ofevdoes not depend on the choice of the Milnor fibre. So we obtain:

Proposition 2.1. — The Poincar´e-Hopf index of ev in F depends only on X ⊂ U and the vector field v. It is independent of the choices of the Milnor fibre F as well as the liftings involved in its definition. We call this integer the local virtual indexof v on X at0, and we denote it byIv(v,0, X).

In other words, the index Iv(v,0, X) is the obstruction Obs(v, Te ^∗F, π⁻¹(Bε)) to the extension of the liftingev as a section ofT F without singularity onπ⁻¹(Bε(0)).

Let us consider now the case wherewis a stratified vector field transversal to the boundarySε=∂(Bε) of every small ballBε,pointing outwards; it has a unique sin- gular point (insideBε) at 0. The Poincar´e-Hopf index ofw at the point 0, denoted byI(w,0), is equal to 1, computed either inM or in the stratumV_i(0) ofX contain- ing 0 (if the dimension ofVi(0) is more than 0). The liftingwe is a section ofT F on π⁻¹(Sε) =F∩Sε, pointing outwardsπ⁻¹(Bε) =F ∩Bε.

Let us denote byT^∗F the fiber bundle overF which isT F minus the zero section.

The obstruction to the extension ofweas a section ofT^∗F insideπ⁻¹(Bε) is equal to the Euler-Poincar´e characteristic of the Milnor fiber,i.e.

(2.4) Obs(w, Te ^∗F, π⁻¹(Bε)) =χ(F).

We obtain:

Proposition 2.2. — Ifwis a stratified vector fieldpointing outwardsthe ball Bε along its boundary Sε=∂(Bε), then its local virtual index equals the Euler-Poincar´e char- acteristic of the Milnor fiber:

Iv(w,0, X) =χ(F) = 1 + (−1)ⁿµ(X,0).

In the sequel, for any vector bundle ξ over a space B, we will denote by ξ^∗ the bundle overB which isξminus its zero section.

3. Proportionality Theorems

Let us consider again a stratified vector field v defined on the ball Bε ⊂ U, with a unique singularity at 0. We assume further that v is constructed by the radial extension process of M. H. Schwartz [20]. This means, essentially, that ifV_j is any stratum containing Vi(0) in its closure, then the vector fieldv is transversal to the boundary of every tubular neighbourhood of Vi(0) in X, pointing outwards. The Poincar´e-Hopf index ofv, computed inVi(0) and denotedI(v,0), can be any integer, and the fact that v is constructed by radial extension implies thatI(v,0) equals the Poincar´e-Hopf index of v computed in U. We shall call v a vector field constructed by radial extension, or simply a radial vector fieldif this does not lead to confusion, as in Theorem 3.1 below. If the stratumV_i(0) has dimension 0, this implies thatv is

actually radial and its local virtual index equalsχ(F), by the proposition above. In the next section we will show that, more generally, we have:

Theorem 3.1 (Proportionality Theorem for vector fields). — Let v be a radial vector field. Then the local virtual index of v in X, Iv(v,0, X), is proportional to the Poincar´e-Hopf indexI(v,0) ofv in the ambient space B_ε:

Iv(v,0, X) =χ(F)·I(v,0).

Let us recall some basic facts about the notion of radial frames, as defined by M.H. Schwartz [21], in order to generalize the notion of radial vector fields. A radial r-frame is a set v^r = (v1, v2, . . . , v_r) of r stratified vector fields constructed by the M.H. Schwartz method of radial extension.

Let us consider a Whitney stratification ofUcompatible withX and a triangulation (K) ofU compatible with the stratification. Let us consider a cell decomposition (D) ofU dual of (K). Each cell of (D) meets the strata transversally. The union of cells which meetX is a tubular neighbourhood of X in U. Ak-celldαmeeting X is dual of a (2(n+ 1)−k)-dimensionalK-simplexσα inX. Let us denote byT^rU the fiber bundle associated toTU whose fiber atx∈ U is the set of (complex)r-frames inT_xU. A section ofT^rU on a subsetAof U is anr-frame tangent toU overA.

The general obstruction theory (see [22]) tells us that the obstruction dimension to the construction of anr-frame tangent toU is equal to 2q= 2((n+ 1)−r+ 1). In the same way, onXreg, the obstruction dimension is 2p= 2(n−r+ 1) and onV_i^2sit is equal to 2e= 2(s−r+ 1). This implies that we can construct a stratifiedr-framev^r with isolated singularities on the 2q-cells d^2q_α of a cellular decomposition (D) of U, with indexI(v^r, T^rU, d^2q_α) in the barycenter{σb_α}=d^2q_α ∩σ_α.

Since ther-frame is stratified, we can also consider the indexI(v^r|_Vi, T^rV_i, d^2q_α ∩V_i) of its restriction to the stratumVi containingσbα. The main property of the radial frames [21] is that these two indices are equal:

I(v^r, T^rU, d^2q_α) =I(v^r|_Vi, T^rV_i, d^2q_α ∩V_i).

We denote this common index by I(v^r,σbα). The method above for lifting a vector field fromX to a local Milnor fiber works for frames and we have:

Theorem 3.2 (Proportionality Theorem for frames). — Let v^r be a radialr-frame with isolated singularities on the2q-cellsd^2q_α with indexI(v^r,σb_α)in the barycenter{bσ_α}= d^2q_α ∩σ_α. Then the obstruction to the extension ofev^r as a section of T^rF onβe^2p = π⁻¹(d^2q_α ∩X)is

Obs(ev^r, T^rF,βe^2p) =χ(Fσ_bα)·I(v^r,bσ_α).

4. Proof of the Proportionality Theorems

The proofs of Theorems 3.1 and 3.2 are analogous to the proof of Th´eor`eme 11.1 in [5]. We first give some topological properties of the Milnor fiber. Then we prove independence and proportionality properties for the obstructions in question. We will prove Theorem 3.1 in section 4.4 and Theorem 3.2 in section 4.5.

4.1. Topological properties of the Milnor fiber. — We will denote by{Vi}the strata of a stratification ofX∩Bε, restriction of a Whitney stratification ofU toX, and we denote by{Wj}a Whitney stratification ofF such that:

(i)π:F →X∩B_εis a stratified map,

(ii) for everyj, the restriction ofπtoW_j is a map of constant rank fromW_j to a stratumVi ofX.

Such stratifications exist by [13]. We notice that eachπ⁻¹(Vi) is union of strata {W_j}.

In the case of isolated singularities, the construction of “poly`edres d’effondrement”

by Lˆe [14] allows us to prove that there are triangulations ofU andFcompatible with the previous stratifications, and such thatπ is a simplicial map. For non necessarily isolated singularities, let us consider a triangulation (K) of X compatible with the stratification {Vi}; as the restriction of π to each stratum {Wj} of F has constant rank, the intersection of the inverse image of a simplex of (K) with the strataWjcan be decomposed into cells eσ_β satisfying the following proposition:

Proposition 4.1. — There is a simplicial triangulation (K) of U compatible with the stratification {Vi} and a cellular decomposition (K)e of F compatible with the strati- fication {Wj}, such that for each cell eσβ of (K), there is a simplexe σα of (K) such that π(σe_β) =σ_α and the restriction of πto each open cellσe_β has constant rank.

Let us denote by (∆) a barycentric subdivision of (K) and by (D) the cell decom- position dual of (K) defined by (∆). The intersection of a (D)-cell d<sup>`</sup><sub>α</sub> with X is a (∆)-subcomplex of dimension`−2, denoted byδ^`−2_α . Using [5] one can construct a cell decomposition (D) ofe F dual of (K) satisfying the following property:e

Proposition 4.2 ([5], Proposition 3). — Let us consider a (K)-simplex σ_α, its dual cell d^{`</sup><sub>α</sub> and δ<sub>α</sub><sup>`−2} = d^`_α∩X. Let us denote by {eσβ}β∈Bα the set of (K)-cells such thate π(eσ_β) =σ_αanddimπ(σe_β) = dim(σα). Let us denote byde_β the dual cell ofeσ_β in(D).e One has:

Closure of π⁻¹(δ_α^`−2) =Closure of [

β∈Bα

deβ

We can suppose that the barycenterσbαof the celld²ⁿ⁺²_α in the cellular decompo- sition (D) corresponds to the point 0 inU, open subset inCⁿ⁺¹. Let us denote by 2s the dimension ofV_i, byb^2sa small euclidean ball centered at 0 inV_i and byD^2n+2−2s

a small disc, transverse tob^2s. The tube b²ⁿ⁺²=b^2s×D^2n+2−2sis homeomorphic to a (2n+ 2)-ball, neighbourhood of 0 in the dual celld²ⁿ⁺²_α . The intersection

β²ⁿ=b²ⁿ⁺²∩X

is not always homeomorphic to a ball, but it is contractible to 0. One defines

∂β²ⁿ =∂b²ⁿ⁺²∩X.

Let us denote

βe²ⁿ=π⁻¹(β²ⁿ) and ∂βe²ⁿ=π⁻¹(∂β²ⁿ) in the Milnor fiberF.

Proposition 4.3. — Let x ∈ V_i^2s. Then dimπ⁻¹(x) 6 2(n−s−1) for all x ∈ b^2s. More precisely:

dimπ⁻¹(x) =

(0 if s=n

2d62(n−s−1) if s6n−1.

Proof. — Using the stratifications ofF and X∩Bε, we see that π⁻¹(Vi) is a union of strata ofF such that on each of them the restriction ofπhas constant rank. The strata ofπ⁻¹(Vi) of maximal dimension have dimension dim(Vi) + 2d. Moreover, as π⁻¹(Vi) is an analytic subspace ofF contained in the closure ofπ⁻¹(Xreg), one has

dimπ⁻¹(Vi) = dim(Vi) + 2d <dimπ⁻¹(Xreg) = 2n and the result follows.

One obtains that dimπ⁻¹(b^2s)62(n−s−1) + 2s= 2(n−1). On the other hand, Proposition 4.1 implies that dimβe²ⁿ 6 2n. As β²ⁿ ∩Xreg is not empty, one gets dimβe²ⁿ= 2n.

4.2. The obstruction depends only on the index. — In this section, we show that Obs(ev, T^∗F, π⁻¹(Bε)) depends only on the Poincar´e-Hopf indexI(v,0) ofvat 0 as a section ofT V_i and not on the vector fieldv itself. Moreover, ifI(v,0) = 0, then Obs(ev, T^∗F, π⁻¹(Bε)) = 0.

A non-zero sectionv ofT b^2sover∂b^2sdetermines a cycleγof T^∗b^2swhose index I(γ) is, by definition, the class of γ in H2s−1(T^∗b^2s) ∼= Z. One can extend v as a section of T b^2s inside b^2s with an isolated zero at 0, by a homothety centered at 0, along the rays ofb^2s. This section can now be extended by the radial extension process [20] as a section ofE(see (2.3)) overb²ⁿ⁺². One obtains a section ofE, still denoted by v, without zero over b²ⁿ⁺²r{0}, in particular over ∂b²ⁿ⁺². Let us consider the restriction of v on ∂β²ⁿ = ∂b²ⁿ⁺² ∩X, one denotes by ev the section of T^∗F over

∂βe²ⁿ=π⁻¹(∂β²ⁿ) defined by a lifting ofv.

Since working in the ballB_εis equivalent to working in the tubeb²ⁿ⁺², one has Obs(ev, T^∗F, π⁻¹(Bε)) = Obs(v, Te ^∗F,βe²ⁿ).

Proposition 4.4. — Letv0 andv1 be two sections of T^∗b^2sover∂b^2s. They define two cycles γ0 and γ1 of T^∗b^2s. Let ev0 and ve1 be liftings of v0 and v1 respectively, over

∂βe²ⁿ.

(a)If I(γ0) =I(γ1), thenObs(ev₀, T^∗F,βe²ⁿ) = Obs(ev₁, T^∗F,βe²ⁿ), (b)If I(γ0) = 0, then, Obs(ve0, T^∗F,βe²ⁿ) = 0.

Proof

a) IfI(γ0) =I(γ1), thenv0 and v1 are homotopic over∂b^2s. The same holds for their extensions over b^2s andb²ⁿ⁺². The liftingsev0 and ev1 over∂βe²ⁿ are homotopic as sections of T F, so the obstructions Obs(ev0, T^∗F,βe²ⁿ) and Obs(ev1, T^∗F,βe²ⁿ) are equal.

b) IfI(γ0) = 0, then by a) one can take for v0 the restriction to∂b^2s of a vector fieldv1without singularities inb^2s. The lifting ofv1 inF is a section ofT F without singularities overβe²ⁿ. One has Obs(ev1, T^∗F,βe²ⁿ) = 0 and the result follows by a).

4.3. The obstruction is proportional to I(γ).— In this section, we prove the proportionality itself, i.e.we show that there is a constant C such that Obs(ev, T^∗F, π⁻¹(Bε)) =C·I(v,0).

Proposition 4.5. — Let v be the radial vector field previously defined, γ the cycle in T^∗b^2sdefined by the restriction ofv to∂b^2sandev a lifting ofvover∂βe²ⁿ. Then there is a constant C such that

Obs(ev, T^∗F,βe²ⁿ) =C·I(γ).

Proof. — Proposition 4.4 shows that Obs(ev, T^∗F,βe²ⁿ) does not depend on the cycleγ defined by a section v of T^∗b^2s over ∂b^2s and whose index is I(γ). Let us consider two cycles inT^∗b^2sdefined in the following way:

i) The cycleγis defined by a smooth map ψ1:∂b^2s−→T^∗b^2s,

such thatψ1(ξ) =v(ξ) for the unitary vector fieldvtangent tob^2salong the boundary

∂b^2s, defining a smooth section ofT ∂b^2s, i.e.γ=ψ1(∂b^2s).

ii) The cycleγ0 is defined by the smooth map

(4.5) ψ0:∂b^2s−→T^∗b^2s

such thatψ0(ξ) is the unitary vector inT0b^2sparallel tov(ξ) and with origin 0. Then, γ0 =ψ0(∂b^2s) is a cycle in the fiber T0b^2s andψ0 is a map with rank 2s−1 nearly everywhere and it preserves the orientations.

In the case of the radial vector field wpointing outwardsb^2s along the boundary

∂b^2s, the cycleγ0 is a cycle of index 1 inH2s−1(T₀^∗b^2s). We denote it by [c0].

4.3.1. Homotopy between ψ0 and ψ1 over ∂b^2s. — Let us construct a homotopy ψ between ψ0 and ψ1 in T^∗b^2s. In order to do that, one extends on b^2s the vector fieldv previously defined on∂b^2s, by a homothety of center 0. One denotes byv⁰ the extension; it has an isolated singularity at the point 0. One defines a map

J : ]0,1]×b^2s−→T b^2s

such thatJ(ρ, ξ) is the unitary vector parallel tov⁰(ξ) at the pointρξ; we will denote it byv_ρ(ρξ).

The mapψis the restriction ofJ to∂b^2s, it is a diffeomorphism over its image. Let us defineψρbyψρ(ξ) =ψ(ρ, ξ). Ifρgoes to 0, then the limit ofψρcoincides with the mapψ0defined in (4.5). Let us denote bySthe unit sphere of the fiberT0b^2s. Asψ0

andψ1are homotopic,ψ0is a (C²-differentiable) mapψ0:∂b^2s∼=S^2s−1→ S ∼=S^2s−1 with topological degreeI(γ).

One has, at the level of chains and cycles inH2s−1(T^∗b^2s):

(4.6) ∂Imψ= Imψ1−I(γ)·[c0].

The proof of Proposition 4.5 consists of showing that one has still a formula of type (4.6) at the level of the radial extension ofv, still denoted byv, over∂β²ⁿ (formula (iii) of Lemma 4.6) and at the level of the lifting ofvin F, over∂βe²ⁿ (formula (4.7)).

We will conclude the proof of Proposition 4.5 using the Transgression Lemma (Lemma 4.7).

4.3.2. Construction of the homotopy Ψ over ∂β²ⁿ. — Let us denote byβ^2n−2s = D^2n+2−2s∩X, the intersection of X with the transversal disc tob^2sin U, and byθ the piecewise differentiable homeomorphism

θ:b^2s×β^2n−2s−→β²ⁿ

such that θ(ξ, ζ) is the point of β²ⁿ whose barycentric coordinates, relative to the vertices of (∆)∩(∂b²ⁿ⁺²rV_i), are equal to those ofζand the others, corresponding to the vertices of ∂b^2s, are proportional to those ofξ. On the one hand, for ξ fixed, θ(ξ, ζ) is on a ray of D_ξ^2n+2−2s, on the other hand, ζ and θ(ξ, ζ) are in the same stratum.

The boundary∂β²ⁿ is

∂β²ⁿ=θ (∂b^2s×β^2n−2s)∪(b^2s×∂β^2n−2s) . Let us define a map

Ψ : ]0,1]×∂β²ⁿ−→E^∗

such that Ψ(ρ, y) = Ψ(ρ, θ(ξ, ζ)) is the vector at the pointy_ρ =θ(ρξ, ζ) obtained by radial extension, at this point, ofvρ(ρξ).

Let us denote, for ρ ∈]0,1], Ψρ(y) = Ψ(ρ, y). Then Ψ1(y) is the original vector field vdefined on∂β²ⁿ. One defines Ψ0 as the limit of Ψρ forρgoing to zero.

We define the cycle Γ inE^∗in the following way: one considers the radial extension, along β^2n−2s, of the radial vector field w constructed on b^2s. It defines a chain ofE(β²ⁿ), canonically oriented byb^2sandβ^2n−2sand whose oriented boundary is Γ.

One has Γ∩E₀^∗ = Γ∩T₀^∗V_i ∼=S ∼=∂b^2s. In fact Γ can be written Γ⁰∪Γ⁰⁰ where Γ⁰ is the union of radial extensions, alongβ^2n−2sof vectors ofS and Γ⁰⁰is the union of radial extensions, in∂β^2n−2s of vectors ofb^2s.

Lemma 4.6

i) Forρ > 0,Ψρ is a piecewise differentiable homeomorphism from ∂β²ⁿ onto its image,

ii) Ψ0 : ∂β²ⁿ → Γ is a piecewise differentiable homeomorphism, with topological degree I(γ),

iii) ∂Im Ψ = Im Ψ1−I(γ)·Γ.

Proof. — The only point to be proved is (ii). We show that the topological degrees of Ψ0 andψ0 are the same. Letζ∈Γ∩E₀^∗ such that ψ₀⁻¹(ζ) consists ofI(γ) points ξj∈∂b^2s, and at each of themψ0is differentiable of rank 2s−1. From the definition of the local radial extension (see [5] Proposition 7.4) one obtains that Ψ0 is still an homeomorphism in the neighbourhood of each pointξj, considered as in ∂β²ⁿ, and that Ψ0 respects the orientations of∂β²ⁿ and Γ. One has Ψ⁻¹₀ (ζ) =ψ⁻¹₀ (ζ), proving the Lemma.

4.3.3. Lifting of the homotopy over∂βe²ⁿ. — Let us define the map Ψ : ]0,e 1]×∂βe²ⁿ−→T F|_βe2n

such thatΨ(ρ,e ey) is the lifting ateyof Ψ(ρ, π(y)), fore π(y)e ∈∂β²ⁿ. We defineΨeρ(y) =e Ψ(ρ,e y).e

Ifρ= 1, thenΨe1 is the lifting of the radial extension ofv, along∂β²ⁿ,i.e.ev.

If ρ = 0, then the image of the map Ψe0 is the lifting of Γ, denoted by eΓ. It can be oriented with the orientation induced by the one of Γ|Xreg, and we claim that it is a (2n−1)-cycle. In fact, the dimension of Γ|e_π⁻¹_(Xreg) is the same as the dimension of Γ|Xreg, i.e.2n−1. If V_j^2h is a stratum whose dimension 2h is bigger than or equal to 2s, theneΓ|_π⁻¹_(Vj)=π⁻¹(Γ|Vj). Now, for transversality reasons, the dimension of Γ|Vj =ψ0(∂β²ⁿ∩V_j^2h) is 2h−1. By Lemma 4.3, one has, forx∈V_j^2h, dimπ⁻¹(x)62(n−h−1). One obtains dimΓ|e_π⁻¹_(Vj)62n−3, that proves the claim.

One has

(4.7) ∂ImΨ = Ime Ψe1−I(γ)·eΓ and ImΨe1=ev(∂βe²ⁿ).

4.3.4. End of the Proof of Proposition 4.5. — Let us recall the Transgression Lemma ([7], see also [5] and [21]):

Lemma 4.7. — Let p:T F →F be the projection of the tangent bundle toF. There are differential forms Ω²ⁿ andΠ²ⁿ⁻¹ onT F, and Ω²ⁿ₀ onF, such that:

(i) Π²ⁿ⁻¹ induces on each fiberT_eyF the fundamental form of H²ⁿ⁻¹(Ty_eF), (ii) Ω²ⁿ=p^∗(Ω²ⁿ₀ ) =−dΠ²ⁿ⁻¹.

Proof. — The differential forms are the transgression differential forms, induced from the classical Chern transgression differential forms [7] on the universal bundle over the Grassmanian, as classifying space. The induced transgression forms verify (i) and (ii).

Let us denote by ye_i the singularities of ev inside βe²ⁿ with Poincar´e-Hopf index I(ev,yei). Let us denote by eγi the cycle defined inT_ye^∗_iF in the same way as in (4.5).

By Lemma 4.7(i), one has Z

e γi

Π²ⁿ⁻¹=I(v,e eyi).

Let us apply the Stokes formula inT F to the differential forms−Π²ⁿ⁻¹ and Ω²ⁿ and to the variety defined byev(βe²ⁿ). One has

Z

ev(eβ²ⁿ)

Ω²ⁿ=− Z

ev(eβ²ⁿ)

dΠ²ⁿ⁻¹=− Z

∂ev(eβ²ⁿ)

Π²ⁿ⁻¹ Observing that

∂ev(βe²ⁿ) =ev(∂βe²ⁿ)∪ieγ_i one obtains:

(4.8) Obs(ev, T^∗F,βe²ⁿ) = Z

e v(∂βe²ⁿ)

Π²ⁿ⁻¹+ Z

βe²ⁿ

Ω²ⁿ₀ .

By integration of the form Π²ⁿ⁻¹ on∂ImΨ and using (4.7), one hase Z

ImΨe1

Π²ⁿ⁻¹−I(γ)· Z

Γe

Π²ⁿ⁻¹= Z

∂ImΨe

Π²ⁿ⁻¹= Z

ImΨe

dΠ²ⁿ⁻¹

=− Z

p(ImΨ)e

Ω²ⁿ₀ =− Z

βe²ⁿ

Ω²ⁿ₀ . Then, using (4.8), one has

Obs(v, Te ^∗F,βe²ⁿ) = Z

ImΨe1

Π²ⁿ⁻¹+ Z

βe²ⁿ

Ω²ⁿ₀ =I(γ).

Z

eΓ

Π²ⁿ⁻¹ and Proposition 4.5 follows withC=R

ΓeΠ²ⁿ⁻¹.

One the other hand, ifI(γ) = 0, the result is obvious.

4.4. Proof of Theorem 3.1. — The proof of Theorem 3.1 now goes as follows:

firstly, we showed in 4.2 that the obstruction Obs(ev, T^∗F, π⁻¹(Bε)) depends only on the index I(v,0) ofv at 0 as a section ofT Vi and not on the vector fieldv itself.

Moreover, ifI(v,0) = 0, then Obs(ev, T^∗F, π⁻¹(Bε)) = 0. Then we proved Proposition 4.5, which is the proportionality itself,i.e.we showed that there is a constantCsuch that Obs(ev, T^∗F, π⁻¹(Bε)) =C·I(v,0). Using 2.4 one obtains that if w is a radial vector field of index +1, thenC=χ(F). This proves the theorem.

4.5. Proof of Theorem 3.2. — The previous argument is also valid in the case ofr-frames. Since an important part of the proof is similar to the case of vector fields, we give only the main indications for the proof.

Let us consider a complex manifold M of (complex) dimension (n+ 1), and 06r6n+ 1. We recall that 2q = 2((n+ 1)−r+ 1) is the obstruction dimension to the construction of an r-frame tangent toM. This implies that we can construct a radial r-frame v^r with isolated singularities on the 2q-cells d^2q_α of a cellular de- composition (D) of M, with index I(v^r,σb_α) in the barycenter {bσ_α} = d^2q_α ∩σ_α. One can write the r-frame asv^r = (v^r−1, ur), the (r−1)-framev^r−1 being without singularities on the (2q)-skeleton of (D). The singularities ofv^rare zeroes of the last vectoru_r.

In the neighbourhood of 0, the (r−1)-frame v^r−1 generates a sub-bundle P^r−1 ofT M, of (complex) rank (r−1). Let us denote byQthe sub-bundle ofT M orthog- onal to P^r−1 relatively to an Riemannian metric induced by the one of Cⁿ⁺¹. The projection of the vector fieldu_r onQparallel toP^r−1, defines a section ofQoverd^2q_α with an isolated singularity at 0. The indexI(v^r,bσα) is equal to

(4.9) I(v^r,bσ_α) =I(v^r, T^rM, d^2q_α) =I(ur, Q^∗, d^2q_α).

Since the mapπhas constant rank on the strata, the liftingev^r−1defines an (r−1)- frame tangent to F over βe^2p = π⁻¹(d^2q_α ∩X). In the same way, whenever it is defined, the lifting eur is linearly independent of ev^r−1 and they define an r-frame e

v^r= (ev^r−1,eu_r).

At any pointy ofβe^2p, the (r−1)-frameev^r−1generates a (r−1)-subspacePe^r−1(y) ofT_yF. One obtains a trivial fiber sub-bundle ofT F of rank (r−1) with basis βe^2p. Let us denote byQ(y) the vector subspace orthogonal toe Pe^r−1(y) in TyF, with the Riemannian metric induced by the one of Cⁿ⁺¹. One obtains a fiber sub-bundle Qe of T F of rankp, with basis βe^2p. Let us denote byQe^∗ the associated bundle whose fiber is the previous one without the zero section.

One has

(4.10) Obs(ev^r, T^rF,βe^2p) = Obs(eur,Qe^∗,βe^2p)

Now, working withu_ras a section ofQ⊂Eoverd^2q_α and withue_ras a section ofQe overβe^2p, one can use the proof of Theorem 3.1 with the following modifications:

b^2s−→b^2e=d^2q_α ∩V_i^2s b²ⁿ⁺²=b^2s×D^2n+2−2s−→b^2q=b^2e×D^2n+2−2s

β²ⁿ=b²ⁿ⁺²∩X ∼=b^2s×β^2n−2s−→β^2p=b^2q∩X∼=b^2e×β^2n−2s βe²ⁿ=π⁻¹(β²ⁿ) ; ∂βe²ⁿ=π⁻¹(∂β²ⁿ)−→βe^2p=π⁻¹(β^2p) ; ∂βe^2p=π⁻¹(∂β^2p)

Π²ⁿ⁻¹; Ω²ⁿ⁻¹−→Π^2p−1; Ω^2p−1

Let us denote byV_ithe stratum containingσ_α. The celld^2q_α, dual ofσ_α, is transverse to X, i.e.to all strata of X, in particular to Vi. Recalling that we use Whitney stratifications, the intersectionY :=d^2q_α ∩X is homeomorphic to the conec(L_bσα) over the link of bσ_α and a distinguished neighbourhoodU_bσα of bσ_α in X is homeomorphic toB_i×c(L_bσα) whereB_i is an open ball inV_i whose dimension is the one ofV_i. One can consider two (local) Milnor fibres ofbσα. The first oneF_bσα =F_X,bσα is the Milnor fibre ofσbαconsidered as a singularity ofX, inM, the second oneF_Y,bσαis the Milnor fibre ofσb_αconsidered as a singularity ofY =d^2q_α ∩X, ind^2q_α.

Lemma 4.8. — The Milnor fibresF_X,bσα andF_Y,bσα satisfy the following relation:

F_X,bσα∼=Bi×F_Y,bσα

and one has

(4.11) χ(F_X,bσα) =χ(F_Y,bσα).

Let us return to the proof of Theorem 3.2. Theorem 3.1 implies (4.12) Obs(uer,Qe^∗,βe^2p) =χ(F_Y,bσα)·I(ur, Q^∗,bσα).

Combining the equalities (4.9) to (4.12), one obtains the result.

5. The Fulton-Johnson classes

Let us consider now a compact complex manifoldM of dimensionm=n+ 1 and a holomorphic functionf :M →D, whereDis an open disc around 0 inCandf has a critical value at 0∈C. We setX =f⁻¹(0) and denote by Σ the singular set ofX, which consists of the points in X where the differentialdf vanishes. We denote by Xreg=XrΣ the regular part ofX. One has an exact sequence of vector bundles:

0−→T Xreg−→T M|_Xreg−→L|_Xreg −→0,

whereL is a trivial line bundle, pull back byf of the tangent bundle ofC, T Xreg is the tangent bundle ofXreg, which is a sub-bundle of the tangent bundle ofM,T M. Thus, L|_Xreg is isomorphic to the normal bundle ofXreginM and, in particular, ifX is smooth then its tangent bundleT X is equivalent toT M|X−L|X in the K-theory groupKU(X). In general, whenX is singular, we set

τ(X) =T M|X−L|X,

and call it the virtual tangent bundle of X. This is not an actual bundle generally speaking, but it represents an element inKU(X), that we still denote byτ(X). Thus its total Chern class:

c(τ(X)) =c(T M|_X)·c(L|_X)⁻¹

is well defined. The image ofc(τ(X)) inH∗(X) under the Poincar´e homomorphism coincides with the Fulton-Johnson class of X, defined in [8, 9]. We denote it by

c^{F J}_∗ (X)∈H2∗(X) and we refer to [4] for background on these classes. IfX is smooth, these are the Poincar´e duals of the Chern classes of the tangent bundleT X.

Our aim now is to prove Theorem 1.1 stated in the introduction. For this, let us denote byX_t the fibersf⁻¹(t),t6= 0. This is a 1-parameter family ofn-dimensional complex submanifolds ofM that degenerate toX whent= 0.

Since for t6= 0 eachXt is a smooth complex manifold, its Chern classescⁱ(Xt)∈ H²ⁱ(Xt) are well defined, and since it is compact, by Poincar´e duality one can think of these as homology classes inH2n−2i(Xt), denoted byc_n−i(Xt). The class in degree 0, corresponding tocⁿ(Xt), is the Euler-Poincar´e characteristic ofX_t.

We notice that, by the compactness ofX, given a regular neighbourhoodN ofX inM, we can findtsufficiently small so thatXt⊂ N. Thus, one has a homomorphism,

i∗:H∗(Xt)−→H∗(N), induced by the inclusion. One also has:

r∗:H∗(N)−→H∗(X), induced by a retractionrfromN intoX. The composition:

σ∗=r∗◦i∗:H∗(Xt)−→H∗(X)

is the Verdier specialization map. Notice that by construction, for each x ∈ X, σ∗ is induced by the degenerating map π of section 2 above, which is now globally defined on all of Xt. In other words, the Verdier specialization map is in this case the homomorphism in homology induced by the mapπ:Xt→X defined (locally) in section 2 above.

For eachX_t,t 6= 0, one has that [T X_t] = [T M|_Xt−L|_Xt] in K-theory. Thus the Chern classes of Xt are those of the virtual bundle [T M|Xt −L|Xt]. By [23], the homology specialization mapσ∗carries the Chern classes ofT M|Xt andL|Xt into the Chern classes ofT M|_X andL|_X, respectively. Thus, as noticed in [19], one has:

(5.13) c^{F J}_∗ (X) =σ∗c∗(Xt).

Let ev^r be, as before, a lifting toX_t via the degenerating map π, of a frame v^r on the 2p-skeleton ofX with isolated singularities. With the notations of 4.5, the Chern classc^p(Xt) is represented by the obstruction cocycleeγsatisfying

heγ,βe^2pi= Obs(ev^r, T^rXt,βe^2p) =X

I(ev^r, yλ)

where the pointsyλ are singular points ofev^r withinβe^2p=π⁻¹(d^2q_α ∩X).

For each point a∈X, the restriction off to a small neighbourhood of acan be regarded as a holomorphic function from an open set in Cⁿ⁺¹ into C. Hence there exists a (local) Milnor fiber Fa ofX ata. This can be identified withXt∩Bε(a) for t 6= 0 sufficiently near the origin inC and Bε(a) a small ball in M around a. We denote byχ(Fa) the Euler-Poincar´e characteristic ofF_a.

By Theorem 3.2 one has: heγ,βe^2pi=χ(F_bσα)·I(v^r,σb_α). The following lemma will prove Theorem 1.1:

Lemma 5.1. — Let eγ be a(D)-cocycle representing the Chern classe c^p(Xt)and let us denote kα=heγ,βe^2pi. Then the cycle

X

σ^2r−2α ⊂X

kασ^2r_α⁻²

represents the Fulton-Johnson class c^{F J}_r−1(X).

Proof. — Let us write the cycleγerepresentingc^p(Xt) as e

γ=X

µ_β(de^2p_β )^∗

where (de^2p_β )^∗ is the elementary (D)-cochain whose value is 1 on the celle de^2p_β and 0 on all others. In other words,µβ=heγ,de^2p_β i.

SinceX_tis smooth, the Chern classc_r−1(Xt) is the Poincar´e dual ofc^p(Xt). This means that if [Xt] denotes the fundamental class ofX_tand ifeσ_β^2r−2denotes the dual cell ofde^2p_β , then one has (see [2]):

e

γ∩[Xt] = X

e σ^2r−2_β ⊂Xt

µ_βσe_β^2r−2.

By (5.13), the Fulton-Johnson class is represented by the cycle π∗(eγ∩[Xt]). In the image ofγe∩[Xt] byπ∗, the only cellsσe^2r_β⁻² with non-zero contribution are the cells such thatπ(σeβ) =σα and dimπ(eσβ) = dim(σα). The images of other cells have dimension strictly less than 2r−2. Thus the cycleπ∗(eγ∩[Xt]) is homologous to

π∗

X

e σ_β^2r−2⊂Xt

µ_βeσ_β^2r−2

= X

σ^2rα⁻²⊂X

k_ασ^2r−2_α ,

wherekα=P

µβ =Pheγ,de^2p_β i,the sum being extended to all the indicesβ such that π(eσβ) =σα and dimπ(eσβ) = dim(σα). By Proposition 4.2 one has kα = heγ,βe^2pi, hence the lemma.

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J.-P. Brasselet, Institut de Math´ematiques de Luminy, UPR 9016 CNRS, Campus de Luminy - Case 907, 13288 Marseille Cedex 9, France • E-mail :[email protected]

J. Seade, Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Ciudad Universi- taria, Circuito Exterior, M´exico 04510 D.F., M´exico • E-mail :[email protected] T. Suwa, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

E-mail :[email protected]