Global Euler obstruction, global Brasselet numbers and critical points.

UNIVERSIDADE DE SÃO PAULO

Instituto de Ciências Matemáticas e de Computação

ISSN 0103-2577

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GLOBAL EULER OBSTRUCTION, GLOBAL BRASSELET NUMBERS AND CRITICAL POINTS

NICOLAS DUTERTRE NIVALDO G. GRULHA JR.

N

438

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NOTAS DO ICMC

SÉRIE MATEMÁTICA

São Carlos – SP

Jan./2018

GLOBAL EULER OBSTRUCTION, GLOBAL BRASSELET NUMBERS AND CRITICAL POINTS

NICOLAS DUTERTRE AND NIVALDO G. GRULHA JR.

Abstract. LetX ⊂ Cⁿ be an equidimensional complex algebraic set and let f : X → C be a polynomial function. For eachc ∈ C, we define the global Brasselet number offatc, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity off atc. Then we establish several formulas relating these numbers to the topology ofX and the critical points off.

1. Introduction

The local Euler obstruction is an invariant defined by MacPherson in [27] as one of the main ingredients in his proof of the Deligne-Grothendieck conjecture on the existence of Chern classes for singular varieties. The local Euler obstruction at 0∈X, whereX is a sufficietly small representative of the equidimensional analytic germ (X,0), is denoted by Eu_X(0). After MacPherson’s pioneer work, the Euler obstruction was studied by many authors. Let us mention briefly some of the most important results on this subject. IfV ={V_i}^t_i=1 is a Whitney stratification ofX, then Brylinski, Dubson and Kashiwara [6] proved a famous formula that relates the Eu_V

i’s to the Euler characteristic of the normal links of the strata. In [25], Lˆe and Teissier showed that EuX(0) is equal to an alterned sum of multiplicities of generic polar varieties of X at 0. In [3], Brasselet, Lˆe and Seade proved a Lefschetz type formula for EuX(0), i.e. they relate EuX(0) to the topology of the real Milnor fibre onX of a generic linear function. There are also integral formulas for EuX(0) in [26] and [15].

In [4], Brasselet, Massey, Parameswaran and Seade defined a relative version of the local Euler obstruction, introducing information for a function f defined on the varietyX, called the Euler obstruction of a function and denoted by Eu_f,X(0).

They prove a Lefschetz type formula for this invariant. The Euler obstruction of a function can be seen as a generalization of the Milnor number ([4, 33, 21]). For instance, in [33], Seade, Tib˘ar and Verjovsky showed that Eu_f,X(0) is equal up to sign to the number of critical points of a Morsefication of f lying on the regular part ofX.

In [16], we study topological properties of functions defined on analytic complex varieties. In order to do it, we define an invariant called the Brasselet number, denoted by Bf,X(0). This number is well defined even when f has arbitrary sin- gularity. When f has isolated singularity we have Bf,X(0) = EuX(0)−Euf,X(0).

Mathematics Subject Classification (2010) : 14B05, 32C18, 58K45.

Keywords: Euler obstruction, Global Euler obstruction, Constructible functions, Morsefications.

1

We established several formulas for Bf,X, among them a relative version of the multiplicity formula of Lˆe and Teissier, a relative version of the Brylinski-Dubson- Kashiwara formula and an integral formula.

In a manner similar to the local case, in [34], working with an affine equidi- mensional singular varietyX ⊂C^N, Seade, Tib˘ar and Verjovsky defined the global Euler obstruction, denoted by Eu(X). WhenX is smooth the global Euler obstruc- tion ofX coincides with the Euler characteristic ofX. They prove a global version of the Lˆe-Teissier polar multiplicities formula. Later, this formula was generalized in [32] to an index formula for MacPherson cycles.

As the Euler obstruction of a function and the Brasselet number are useful to study the singularities of f in the local case, we introduce in this work the global Brasselet numbers and the Brasselet numbers at infinity, in order to investigate the topological behavior of the singularities, globally and at the infinity, of a given polynomial functionfdefined on an algebraic varietyX ⊂C^N. The main references we use in this paper about the study of singularities at infinity are [9, 10, 41].

In Section 2 we give prerequisities on the topology of complex algebraic sets:

stratified Morse functions, the complex link and the normal Morse datum, con- structible functions, the local Euler obstruction and the Brasselet number, the global Euler obstruction. In Section 3 we recall the notions of t-regularity at in- finity and ρ-regularity at infinity, some basic results and we adapt them to the stratified setting.

In Section 4 we define the global Brasselet numbers and the Brasselet numbers at infinity. We compare the global Brasselets number of f with the global Euler obstruction of the fibres off. The relation presented in Corollary 4.8 can be seen as a global relative version of the local index formula of Brylinsky, Dubson and Kashiwara [6].

In Section 5 we prove several formulas that relate the number of critical points of a Morsefication of a polynomial functionf on an algebraic setX, to the global Brasselet numbers and the Brasselet numbers at infinity of f. The main result in this section is Theorem 5.2. From this result we obtain many interesting corollaires.

Corollary 5.6, for instance, is a Brylinski–Dubson–Kashiwara type formula for the total Brasselet number at infinity. We also prove a relative version of the polar multiplicity formula of Seade, Tib˘ar and Verjovsky (Corollary 5.13).

We finish the paper at Section 6, relating the global Euler obstruction of an equidimensional algebraic setX ⊂Cⁿ to the Gauss-Bonnet curvature of its regular part and the Gauss-Bonnet curvature of the regular part of its link at infinity. The result is a global counterpart of the formula that the first author established for analytic germs in [15].

Acknowledgments. The authors thank the USP-Cofecub project “UcMa163/17 - Topologie et g´eom´etrie des espaces singuliers et applications”.

The fisrt author is supported by the ANR project LISA.

The second author is partially supported by Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo - FAPESP, Brazil, grant 2017/09620-2 and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq, Brazil, grant 303046/2016-3.

2. Prerequisites on the topology of complex algebraic sets In this section, we work with a reduced complex algebraic setX ⊂Cⁿ of dimen- siond. We assume thatX is equipped with a finite Whitney stratificationV whose

strata are connected. We denote byXreg the regular part ofX, i.e. the union of all the strata of dimensiond.

2.1. Stratified Morse functions. The main reference for this subject is [20].

Definition 2.1. Letxbe a point inXand letVbbe the stratum that contains it. A degenerate tangent plane of the stratificationV is an elementT (of an appropriate Grassmannian) such thatT = limxi→xTxiVa, whereVa is a stratum that contains V_b in its frontier and where thex_i’s belong toV_a.

Definition 2.2. A degenerate covector ofV at a point x∈X is a covector which vanishes on a degenerate tangent plane of V at x, i.e., an element η of T_x^∗Cⁿ such that there exists a degenerate tangent planeT of the stratification atxwith η(T) = 0.

Letf :X →Cbe an analytic function. We assume thatf is the restriction to X of an analytic functionF :Cⁿ →C, i.e. f =F_|X. A pointxin X is a critical point off if it is a critical point ofF_|V(x), whereV(x) is the stratum containingx.

Definition 2.3. Letxbe a critical point off. We say thatf is general atxwith respect to the stratificationV ifDF(q) is not a degenerate covector ofV at x.

We say that f is general with respect to V if it is general at all critical points with respect toV.

Definition 2.4. Letxbe a critical point off. We say thatxis a stratified Morse critical point of f if f is general at x and the functionf_|V(x) : V(x) → C has a non-degenerate critical point atxwhen dimV(x)>0.

We say that thatfis a stratified Morse function if it admits only stratified Morse critical points.

2.2. The complex link and the normal Morse datum. The complex link is an important object in the study of the topology of complex analytic sets. It is analogous to the Milnor fibre and was studied first in [23]. It plays a crucial role in complex stratified Morse theory (see [20]) and appears in general bouquet theorems for the Milnor fibre of a function with isolated singularity (see [24, 35, 39]).

LetV be a stratum of the stratification V of X and letxbe a point inV. Let g: (Cⁿ, x)→(C,0) be an analytic complex function-germ such that the differential form Dg(x) is not a degenerate covector of V atx. LetN_x,V^C be a normal slice to V at x, i.e. N_x,V^C is a closed complex submanifold of Cⁿ which is transversal toV atxandN_x,V^C ∩V ={x}.

Definition 2.5. The complex linkL^X_V ofV is defined by L^X_V =X∩N_x,V^C ∩B(x)∩ {g=δ},

where 0<|δ| 1. HereB(x) is the closed ball of radiuscentered atx. The normal Morse datum NMD(V) ofV is the pair of spaces

NMD(V) = X∩N_x,V^C ∩B(x), X∩N_x,V^C ∩B(x)∩ {g=δ}

.

The fact that these two notions are well-defined, i.e. independent of all the choices made to define them, is explained in [20].

2.3. Constructible functions. We start with a presentation of Viro’s method of integration with respect to the Euler characteristic with compact support [43]. We work in the semi-algebraic setting.

Definition 2.6. Let Y ⊂ Rⁿ be a semi-algebraic set. A constructible function α:Y →Zis a Z-valued function that can be written as a finite sum:

α=X

i∈I

mi1Y_i,

where Yi is a semi-algebraic subset of Y and 1Y_i is the characteristic function on Yi.

The sum and the product of two constructible functions on Y are again con- structible. The set of constructible functions on Y is thus a commutative ring, denoted byF(Y).

Definition 2.7. If α∈F(Y) andW ⊂Y is a semi-algebraic set then the Euler characteristicχ(W, α) is defined by

χ(W, α) =X

i∈I

m_iχ_c(W∩Y_i), whereα=P

i∈Imi1Y_i andχc is the Euler characterictic of Borel-Moore homology.

The Euler characteristicχ(W, α) is also called the Euler integral ofαand denoted byR

Wαdχc. Here we follow the terminology and notations used in [4, 16, 32].

Definition 2.8. Let f : Y → Z be a continuous semi-algebraic map and let α:Y →Z be a constructible function. The pushforwardf∗αofα alongf is the functionf_∗α:Z→Zdefined by:

f_∗α(z) =χ(f⁻¹(z), α).

Proposition 2.9. The pushforward of a constructible function is a constructible function.

Theorem 2.10 (Fubini’s theorem). Letf :Y →Z be a continuous semi-algebraic map and letαbe a constructible function onY. Then we have:

χ(Z, f∗α) =χ(Y, α).

Let us go back to the complex situation. Here we writeV ={V1, . . . , V_t} for the Whitney stratification ofX.

Definition 2.11. A constructible function with respect to the stratificationVofX is a functionα:X →Zwhich is constant on each stratumV of the stratification.

This means that there exist integersn_i, i∈ {1, . . . , t}, such thatα=Pt

i=1n_i·1_Vi. In most of the cases that we will consider, we can use the topological Euler charac- teristicχinstead ofχc. First since eachVi is an even-dimensional submanifold, by Poincar´e dualityχ_c(V_i) is equal toχ(V_i) and so χ(X, α) =Pt

i=1n_iχ(V_i). Now let B⊂Cⁿbe an euclidian closed ball that intersectsX transversally (in the stratified sense). We will give four equalities forχ(X∩B, α). By additivity ofχ_c, we have

χ(X∩B, α) =χ(X∩B, α) +˚ χ(X∩∂B, α).

ButX∩∂B is Whitney stratified by odd dimensional strata and soχ(X∩∂B) = 0 (see Lemma 5.0.3 in [31] or Proposition 1.6 in [28]). Therefore, we have

χ(X∩B, α) =χ(X∩B, α) =˚

t

X

i=1

niχc(Vi∩B),˚ and by Poincar´e duality,

χ(X∩B, α) =

t

X

i=1

niχ(Vi∩B˚).

But eachVi∩B is a manifold with boundary, soχ(Vi∩B) =˚ χ(Vi∩B) and χ(X∩B, α) =

t

X

i=1

niχ(Vi∩B).

Similarly, ifE=Cⁿ\B˚then χ(X∩E, α) =

t

X

i=1

n_iχ_c(V_i∩E) =

t

X

i=1

n_iχ_c(V_i∩E)˚

=

t

X

i=1

niχ(Vi∩E) =˚

t

X

i=1

niχ(Vi∩E).

If the radius ofB is sufficiently big, then X∩∂B is homeomorphic to the link at infinity ofX, denoted by Lk^∞(X), andX∩Bis a retract by deformation ofXwhich implies thatχ(X) =χ(X∩B). SinceX∩B is compact, χ(X∩B) =χc(X ∩B) and so, by additivity, χ_c(X) =χ(X) +χ_c(X ∩E). But˚ X∩E˚is homeomorphic to the product of Lk^∞(X) and an open interval in R. Sinceχ_c(Lk^∞(X)) = 0, by multiplicativity ofχcwe obtain thatχc(X∩E) = 0 and finally that˚ χ(X) =χc(X).

Definition 2.12. Letα:X →Z be a constructible function with respect to the stratificationV. Its normal Morse indexη(V, α) alongV is defined by

η(V, α) =χ(NMD(V), α) =χ(X∩N_x,V^C ∩B(x), α)−χ(L^X_V, α), wherexis a point inV.

IfZ⊂X is a closed union of strata, thenη(V,1Z) = 1−χ(LV ∩Z).

2.4. The local Euler obstruction and the Brasselet number. Here we assume that X is equidimensional. The Euler obstruction atx∈X, denoted by EuX(x), was defined by MacPherson, using 1-forms and the Nash blow-up (see [27] for the original definition). An equivalent definition of the Euler obstruction was given by Brasselet and Schwartz in the context of vector fields [5]. Roughly speaking, EuX(x) is the obstruction for extending a continuous stratified radial vector field aroundxinX to a non-zero section of the Nash bundle over the Nash blow-up of X.

The Euler obstruction is a constructible function and there are two distinguished bases for the free abelian group of constructible functions: the characteristic func- tions1_V and the Euler obstruction Eu_V of the closureV of all strataV. Moreover, the key role of the Euler obstruction comes from the following identities (see [32]

p.34 or [31] p.292 and p.323-324):

η(V⁰,Eu_V) = 1 ifV⁰=V,

and:

η(V⁰,Eu_V) = 0 ifV⁰6=V.

In [3], Brasselet, Lˆe and Seade study the Euler obstruction using hyperplane sections, following ideas of Dubson and Kato. Let us assume that 0 belongs toX.

Theorem 2.13 ([3]). For each generic linear forml, there is0 such that for any with 0< < 0 , the Euler obstruction of (X,0) is equal to:

Eu_X(0) =χ X∩B(0)∩l⁻¹(δ),Eu_X , where0<|δ| 1.

Letf :X → Cbe a holomorphic function. We assume that f has an isolated singularity (or an isolated critical point) at 0, i.e. thatf has no critical point in a punctured neighborhood of 0 inX.

In [4] Brasselet, Massey, Parameswaran and Seade introduced an invariant which measures, in a way, how far the equality given in Theorem 2.13 is from being true if we replace the generic linear formlwith some other function onX with at most an isolated stratified critical point at 0. This number is called the Euler obstruction of a function and denoted by Euf,X(0). The following result is the Brasselet, Massey, Parameswaran and Seade formula [4] that compares, in the same point, the local Euler obstruction with the Euler obstruction of a function.

Theorem 2.14. Letf :X→Cbe a function with an isolated singularity at0. For 0<|δ| ε1 we have:

EuX(0)−Euf,X(0) =χ X∩B(0)∩f⁻¹(δ),EuX

, where0<|δ| 1.

In [33], J. Seade, Tib˘ar and Verjovsky show that the Euler obstruction of f is closely related to the number of Morse points of a Morsefication off, as it is stated in the next proposition.

Proposition 2.15([33]). Let f :X →Cbe the an analytic function with isolated singularity at the origin. Then:

Euf,X(0) = (−1)^dnreg,

wherenreg is the number of Morse points onXreg in a stratified Morsefication off lying in a small neighborhood of 0.

Definition 2.16. A good stratification ofX relative tof is a stratificationV ofX which is adapted toX^f, (i.e. X^f is a union of strata) , whereX^f =X∩f⁻¹(0), such that{Vi ∈ V;V_i 6⊂X^f} is a Whitney stratification of X\X^f and such that for any pair of strata (V_a, V_b) such that V_a 6⊂ X^f and V_b ⊂ X^f, the (a_f)-Thom condition is satisfied.

Let us now recall the definition of the Brasselet number, defined in [16].

Definition 2.17. Let V be a good stratification of X relative to f. We define B_f,X(0) by:

Bf,X(0) =χ X∩B(0)∩f⁻¹(δ),EuX , where 0<|δ| 1.

Remark 2.18. Note that if f has a stratified isolated singularity at the origin then, by Theorem 2.14, we have that B_f,X(0) = Eu_X(0)−Eu_f,X(0).

2.5. Global Euler obstruction. Here we assume thatX is equidimensional and we writeV ={V1, . . . , Vt}. In [34], Seade, Tib˘ar and Verjovsky introduced a global analogous of the Euler obstruction called the global Euler obstruction and denoted by Eu(X). LetXe →^ν X denote the Nash modification of X, and let us consider a stratified real vector fieldvon a subsetV ⊂X: this means that the vector field is continuous and tangent to the strata. The restriction ofv toV has a well-defined canonical lifting ev to ν⁻¹(V) as a section of the real bundle underlying the Nash bundleTe→Xe.

Definition 2.19. We say that the stratified vector fieldvonX is radial-at-infinity if it is defined on the restriction toX of the complement of a sufficiently large ball BM centered at the origin of C^N, and it is transversal to SR, pointing outwards, for anyR > M. In particular,v is without zeros onX\BM.

The “sufficiently large” radiusM is furnished by the following well-known result.

Lemma 2.20. There exists M ∈ R such that, for any R ≥ M, the sphere SR

centered at the origin of C^N and of radius R is stratified transversal to X, i.e.

transversal to all strata of the stratificationV.

Using this last lemma and inspired by [5] and [11], Seade, Tib˘ar and Verjovsky defined the global Euler obstruction in [34] as follows:

Definition 2.21. Let ˜vbe the lifting to a section of the Nash bundle ˜T of a radial- at-infinity stratified vector field v over X \BR. We call global Euler obstruction of X, and denote it by Eu(X), the obstruction for extending ˜v as a nowhere zero section ofTe withinν⁻¹(X∩BR).

To be precise, the obstruction to extend ˜vas a nowhere zero section ofTewithin ν⁻¹(X∩B_R) is in fact a relative cohomology class

o(˜v)∈H^2d(ν⁻¹(X∩BR), ν⁻¹(X∩SR))'H_c^2d(Xe).

The global Euler obstruction ofX is the evaluation ofo(˜v) on the fundamental class of the pair (ν⁻¹(X∩B_R), ν⁻¹(X∩S_R)). Thus Eu(X) is an integer and does not depend on the radius of the sphere defining the link at infinity ofX. Since two radial-at-infinity vector fields are homotopic as stratified vector fields, it does not depend on the choice ofv either.

Remark 2.22. The global Euler obstruction has the following properties (see [34]

p. 396):

(1) if X is non-singular, then Eu(X) =χ(X), (2) Eu(X) =χ(X,Eu_X).

3. Regularity conditions at infinity

A natural question is if the concepts of the Euler obstruction and the Brasselet number of a function could be extended to the global setting, as Seade, Tib˘ar and Verjovsky did for the local Euler obstruction, and what kind of information we could obtain with these possible new global invariants.

But, before extending the local notions of the Euler obstruction and Brasselet number of a function, we recall in this section some definitions and results about the study of singularities at infinity and we adapted some results to the stratified

setting. The main references for this section are [9, 10, 41] and we refer to these papers for details.

We considerX ⊂C^N a reduced algebraic set of dimensiond. We use coordinates (x₁, . . . , x_N) for the spaceC^N and coordinates [x₀:x₁:· · ·:x_N] for the projective space P^N. We consider the algebraic closure X of X in the complex projective spaceP^N and we denote by

H^∞=

[x0:x1:· · ·:xN]|x0= 0 , the hyperplane at infinity of the embeddingC^N ⊂P^N.

One may endow X with a semi-algebraic Whitney stratification W such that Xreg is a stratum and the part at infinityX∩H^∞ is a union of strata.

SinceX is projective and since the stratification ofX is locally finite, it follows that W has finitely many strata. We denote byXsing the set of singular points of X, i.e. Xsing=X\Xreg.

In order to recall the definition of thet-regularity, let us recall first the definition of the conormal spaces.

Definition 3.1. We denote byC(X) the conormal modification of X, defined as:

C(X) = closure

(x, H)∈Xreg×Pˇ^N−1| TxXreg⊂H ⊂X×Pˇ^N−1. Letπ:C(X)→X denote the projectionπ(x, H) =x.

Definition 3.2. Letg:X →Cbe an analytic function defined in some neighbour- hood ofX inC^N. LetX₀ denote the subset ofX_reg whereg is a submersion. The relative conormal space ofg is defined as follows:

Cg(X) = closuren

(x, H)∈X0×Pˇ^N−1 |Txg⁻¹((g(x))⊂H ⊂X×Pˇ^N−1, together with the projectionπ:C_g(X)→X,π(x, H) =x.

Let f : X → C be a function such that f = F_|X, where F : C^N → C is a polynomial function.

Let X = graphf be the closure of the graph of f in P^N ×C and let X^∞ = X∩(H^∞×C). One has the isomorphism graphf 'X.

We consider the affine chartsUj×CofP^N×C, where Uj={[x0:· · ·:xN]|xj 6= 0}, j= 0,1, . . . , N.

Identifying the chartU₀with the affine spaceC^N, we haveX∩(U0×C) =X\X^∞= graphf, and X^∞ is covered by the chartsU₁×C, ..., U_N×C.

Ifg denotes the projection to the variable x₀ in some affine chart U_j×C, then the relative conormalC_g(X\X^∞∩U_j×C)⊂Xס

P^N is well defined.

With the projectionπ(y, H) =y, let us then consider the spaceπ⁻¹(X^∞), which is well defined for every chartUj×Cas a subset ofCg(X\X^∞∩Uj×C).

Definition 3.3. We call space of characteristic covectors at infinity the setC^∞= π⁻¹(X^∞). For somep₀∈X^∞, we denoteC_p^∞

0 :=π⁻¹(p₀).

By Lemma 2.8 in [41], these notions are well-defined, i.e. they do not depend on the chartU_j.

Let τ : P^N ×C → C denote the second projection. One defines the relative conormal spaceC_τ(P^N ×C) as in Definition 3.2 where the function g is replaced by the mappingτ.

Definition 3.4. We say thatf ist-regular atp0∈X^∞ if Cτ(P^N ×C)∩C_p^∞₀ =∅.

We say thatf⁻¹(t₀) ist-regular iff ist-regular at all pointsp₀∈X^∞∩τ⁻¹(t₀).

Let us now recall the definition ofρ-regularity. Let K⊂C^N be some compact (eventually empty) set and letρ:C^N\K→R≥0be a proper analytic submersion.

Definition 3.5 (ρ-regularity at infinity). We say that f is ρ-regular atp0 ∈ X^∞ if there is an open neighbourhoodU ⊂P^N ×Cofp0 and an open neighbourhood D ⊂C ofτ(p0) such that, for all t∈D, the fibref⁻¹(t)∩Xreg∩U intersects all the levels of the restrictionρ_|U∩Xreg and this intersection is transversal.

We say that the fibref⁻¹(t₀) isρ-regular at infinity iff isρ-regular at all points p0∈X^∞∩τ⁻¹(t0). We say thatt0 is an asymptoticρ-non-regular value iff⁻¹(t0) is notρ-regular at infinity.

The next proposition relatest-regularity toρ_E-regularity, whereρ_E denotes the Euclidian norm.

Proposition 3.6. If f ist-regular atp0∈X^∞, then f isρE-regular atp0. Proof. This is just an adaptation to our setting of the proof of Proposition 2.11 in

[41].

Corollary 3.7. The set of asymptotic non-ρ_E-regular values off is finite.

Proof. It is enough to prove that there are only finitely many values t0 such that f⁻¹(t0) is not t-regular. The proof of this fact is as in Corollary 2.12 in [41]. We can equip X with a Whitney stratification such thatX^∞ is a union of strata and such that any pair of strata (V, W), withV ⊂X\X^∞,W ⊂X^∞ andW ⊂V \V, satisfies the Thom (a_g)-regularity condition for some functiong definingX^∞ inX. Ifτ⁻¹(t₀) is transversal toX^∞in the stratified sense, thenf⁻¹(t₀) ist-regular. But the mappingτ_|X∞ :X^∞→Chas a finite number of critical values (in the stratified

sense).

Proposition 3.8. Let l : X → C be a generic linear projection then for all p₀∈X^∞,l ist-regular atp₀.

Proof. With Uj defined as before, let us work in the chart Uj ×C and with Cg(X\X^∞∩Uj×C) withπ(x, H) =x, as defined above.

Let us suppose thatl is nott-regular atp0= (q0, t0)∈X^∞. It means that there exists a sequencepn= (qn, l(qn))→p0, withqn∈Xregsuch that

T_pngraph(l)∩T_png⁻¹(g(p_n))⊂H_n,

and Hn →H and a sequence of hyperplanes {Ln} such thatC^N × {0} ⊂Ln and Ln→H, where (p0, H)∈ Cτ(P^N ×C)∩C^∞p0.

Since in fact each Ln =C^N× {0}, we conclude thatH =C^N× {0}. Note also that,

T_pngraph(l)∩T_png⁻¹(g(p_n))

= graph(l:Tq_nXreg→C)∩Tq_ng⁻¹(g(qn))×C. This implies that lim

n→+∞l(un) = 0 for any bounded sequence (un) of vectors such that u_n ∈T_qnX_reg∩T_qng⁻¹(g(q_n)). As in the previous corollary, we can equipX

with a Whitney stratification such thatX∩H^∞is a union of strata and such that any pair of strata (V, W), with V ⊂X, W ⊂X ∩H^∞ and W ⊂V \V, satisfies the Thom (ag)-regularity. Therefore we see that the axis of the pencil defined by l is not transversal to X∩H^∞. By Lemma 3.1 in [34], this is not possible if l is generic. So we conclude thatl generic ist-regular.

We assume now that X is equipped with a finite Whitney stratification V = {V_i}^t_i=1such thatV₁, . . . , V_t−1are connected,V₁, . . . , V_tare reduced andV_t=X_reg. Fori={1, . . . , t}, letf_i:V_i →Cbe the restriction toV_iof the polynomial function F. Note thatft=f.

Definition 3.9 (stratified t-regularity at infinity). We say that f is stratified t- regular atp₀∈X^∞ if fori= 1, . . . , t,f_i ist-regular atp₀.

We say thatf⁻¹(t₀) is stratifiedt-regular iff is stratifiedt-regular at all points p₀∈X^∞∩τ⁻¹(t₀).

Definition 3.10 (stratified ρ-regularity at infinity). We say that f is stratified ρ-regular atp0∈X^∞ if if fori= 1, . . . , t,fi isρ-regular atp0.

We say that the fibref⁻¹(t0) is stratified ρ-regular at infinity if f is stratified ρ-regular at all pointsp₀∈X^∞∩τ⁻¹(t₀). We say thatt₀is a stratified asymptotic ρ-non-regular value iff⁻¹(t₀) is not stratifiedρ-regular at infinity.

The following statements are easy consequences of the definitions of stratified t-regularity and stratified ρ-regularity.

Proposition 3.11. Stratifiedt-regularity implies stratifiedρE-regularity.

Corollary 3.12. The set of stratified asymptotic non-ρE-regular values of f is finite.

Corollary 3.13. Letl:X →Cbe a generic linear projection, then for allp0∈X^∞, l is stratified t-regular (and therefore stratified ρE-regular) at p0. Moreover the set of stratified asymptotic non-ρE-regular values off is empty.

4. Global Brasselet numbers and Brylinsky-Dubson-Kashiwara formulas

LetX ⊂ Cⁿ be a reduced algebraic set of dimension d, equipped with a finite Whitney stratification V = {Vi}^t_i=1. We assume that V1, . . . , V_t−1 are connected, V1, . . . , Vt are reduced and that Vt = Xreg, where Xreg has dimension d . Let f : X → C be a complex polynomial, restriction to X of a polynomial function F :Cⁿ→C, i.e.,f =F_|X. We assume thatf has a finite number of critical points q₁, . . . , q_s and we denote by {a1, . . . , a_r} the set of stratified asymptotic non-ρ_E- regular values off.

For simplicity, we will writeB_Rfor the ballB_R(0) andS_Rfor∂B_R.

Lemma 4.1. Let α: X → Z be a constructible function with respect to V. The function c7→χ(f⁻¹(c), α)is constant onC\

{f(q₁), . . . , f(q_s)} ∪ {a₁, . . . , a_r} . Proof. Let c ∈ C\

{f(q1), . . . , f(qs)} ∪ {a1, . . . , ar}

and let us choose Rc > 0 such thatf⁻¹(c)∩ {ρE≥Rc}does not contain any critical point ofρE|f⁻¹(c)(here f⁻¹(c) is equipped with the Whitney stratification tVi ∩f⁻¹(c)). This implies

that f⁻¹(c)∩BR is a retract by deformation of f⁻¹(c) and that χ(f⁻¹(c)) = χ(f⁻¹(c)∩BR) for any R ≥Rc. Since f⁻¹(c) is stratifiedρE-regular at infinity, there isR≥Rc and >0 such that the mapping

(ρE, f) :f⁻¹(D(c))∩ {ρ≥R} →R×C,

whereD(c) is the closed disc of radiuscentered atcinC, is a stratified submersion and so forc⁰ ∈D(c),f⁻¹(c⁰)∩B_R is also a retract by deformation off⁻¹(c⁰). But sincecis a regular value off,

χ(f⁻¹(c⁰)∩BR) =χ(f⁻¹(c)∩BR),

for c⁰ in a small neighborhood of c. Hence the result is proved for the function c7→χ(f⁻¹(c)), i.e., whenα=1_X (see the discussion after Definition 2.11).

LetV be a stratum ofX. By additivity, we have

χc( ¯V ∩f⁻¹(c)) =χc(V ∩f⁻¹(c)) +χc( ¯V \V ∩f⁻¹(c)), and by the arguments after Definition 2.11, we get that

χ( ¯V ∩f⁻¹(c)) =χ(V ∩f⁻¹(c)) +χ( ¯V \V ∩f⁻¹(c)).

But since ¯V and ¯V \V are algebraic subsets of X stratified by strata of V, c is a regular value and f⁻¹(c) is stratified ρE-regular at infinity for f : ¯V → C and f : ¯V \V →C. Therefore the functions

c7→χ( ¯V ∩f⁻¹(c)) andc7→χ(( ¯V \V)∩f⁻¹(c)), are constant on C\

{f(q1), . . . , f(qs)} ∪ {a1, . . . , ar}

and so is the function c7→χ(V∩f⁻¹(c)). Hence fori∈ {1, . . . , t}, the functionc7→χ(Vi∩f⁻¹(c)) is con- stant onC\

{f(q₁), . . . , f(q_s)} ∪ {a1, . . . , a_r}

. This implies thatc7→χ(f⁻¹(c), α) is also constant on C\

{f(q1), . . . , f(qs)} ∪ {a1, . . . , ar}

for any constructible

functionα.

Definition 4.2. WhenXis equidimensional, we define the global Brasselet number off atc by

B^X_f,c=χ(f⁻¹(c),EuX), and the global Euler obstruction off atc by

Eu^X_f,c= Eu(X)−B^X_f,c.

Leta∈Cand letR_a>0 be such thatf⁻¹(a)∩BRa does not contain any critical point ofρ_E|f−1(a). Then there existsδ₁>0 such that

f :f⁻¹

Dδ₁(a)\ {a}

→Dδ₁(a)\ {a},

is a locally trivial topological fibration (this is just a singular version of the Milnor- Lˆe fibration) and soχ(f⁻¹(c)∩BR_a) is constant forc∈Dδ₁(a)\ {a}.

Sinceχ(f⁻¹(c)) is constant onC\

{f(q1), . . . , f(qs)}∪{a1, . . . , ar}

, there exists δ₂>2 such thatχ(f⁻¹(c)) is constant forc∈D_δ2(a)\ {a}. Since

χ(f⁻¹(c)) =χ(f⁻¹(c)∩B_Ra) +χ(f⁻¹(c)∩ {ρ_E≥R_a})−χ(f⁻¹(c)∩S_Ra)

=χ(f⁻¹(c)∩B_Ra) +χ(f⁻¹(c)∩ {ρ_E≥R_a}),

we see that χ(f⁻¹(c)∩ {ρE ≥ R_a}) is constant for c in D_δ(a) \ {a}, where δ= min{δ1, δ₂}.

Let R⁰_a >0 be such that f⁻¹(a)∩BR⁰_a does not contain any critical point of ρE|f⁻¹(a). Then there existsδ⁰>0 such thatχ(f⁻¹(c)∩ {ρE≥R⁰_a}) is constant for cin D_δ⁰(a)\ {a}. We can suppose thatR_a⁰ > Ra. Since there are no critical points ofρ_E|f−1(a)on{Ra ≤ρ_E ≤R_a⁰}, ais a regular value off :{Ra ≤ρ_E≤R⁰_a} →C (note that a critical point off onf⁻¹(a) is also a critical point ofρ_E|f−1(a)). Hence there existsν >0 such that

f :f⁻¹(D_ν(a))∩ {R_a≤ρ_E ≤R⁰_a} →D_ν(a),

is a locally trivial topological fibration. Letν⁰= min{δ, δ⁰, ν}. ForcinDν⁰(a)\{a}, we have

χ(f⁻¹(c)∩ {ρ_E≥R_a}) =χ(f⁻¹(c)∩ {ρ_E≥R⁰_a}) +χ(f⁻¹(c)∩ {R_a ≤ρ_E ≤R⁰_a})

=χ(f⁻¹(c)∩ {ρE≥R⁰_a}) +χ(f⁻¹(a)∩ {Ra≤ρE≤R⁰_a}).

Butρ:f⁻¹(a)∩ {Ra≤ρE≤Ra⁰} →Cis a stratified submersion and so χ(f⁻¹(a)∩ {Ra≤ρE≤Ra⁰}) =χ(f⁻¹(a)∩ {ρE=Ra}) = 0.

We have proved that

χ(f⁻¹(c)∩ {ρE≥Ra}) =χ(f⁻¹(c)∩ {ρE≥R⁰_a}), ifcis sufficiently close toa.

Definition 4.3. Letα:X →Zbe a constructible function with respect toV. For anya∈C, we set

B^X,∞_f,a (α) = lim

c→aχ(f⁻¹(c)∩ {ρ_E ≥R_a}, α) andλ^X,∞_f,a = B^X,∞_f,a (1_X),

whereRa>0 is such thatf⁻¹(a)∩ {ρE≥Ra} does not contain any critical point ofρE|f⁻¹(a).

Note that B^X,∞_f,a (α) is well-defined since, by the previous considerations,

c→alimχ(V_i∩f⁻¹(c)∩ {ρE ≥R_a}), is well-defined and so is

c→alimχ(Vi∩f⁻¹(c)∩ {ρE ≥Ra}).

Lemma 4.4. Letα:X →Zbe a constructible function with respect toV. Ifc∈C is such that f⁻¹(c)is stratifiedρ_E-regular at infinity thenB^X,∞_f,c (α) = 0.

Proof. It is enough to prove thatλ^X,∞_f,c = 0. Since f⁻¹(c) is stratified ρE-regular at infinity, there isRc>0 and >0 such that the mapping

(ρ_E, f) :f⁻¹(D(c))∩ {ρ_E≥R_c} →R×C, is a stratified submersion. Letc⁰∈f⁻¹(D(c)). Then the mapping

ρE:{ρE≥Rc} ∩f⁻¹(c⁰)→R, is a proper stratified submersion and so

χ(f⁻¹(c⁰)∩ {ρE≥Rc}) =χ(f⁻¹(c⁰)∩ {ρE =Rc}) = 0.

Definition 4.5. Letα:X→Zbe a constructible function with respect toV. We set

B^X,∞_f (α) =X

c∈C

B^X,∞_f,c (α) andλ^X,∞_f = B^X,∞_f (1X).

Definition 4.6. WhenX is equidimensional, we define the Brasselet numbers at infinity off by:

B^X,∞_f,c = B^X,∞_f,c (EuX),

forc∈C, and the total Brasselet number at infinity off by:

B^X,∞_f = B^X,∞_f (EuX).

We start comparing the global Brasselet numbers off and the Euler obstructions of the fibres off.

Proposition 4.7. Let a∈C, we have B^X_f,a= Eu(f⁻¹(a)) + X

j |f(q_j)=a

Eu_X(q_j)−Eu_f−1(a)(q_j).

Proof. By definition,

B^X_f,a=

t

X

i=1

χ(Vi∩f⁻¹(a))EuX(Vi).

For eachi∈ {1, . . . , t}, let us denote by Γi the set consisting of theqj’s such that qj∈Vi∩f⁻¹(a). The partition

f⁻¹(a) = ti|Γ_i=∅Vi∩f⁻¹(a) [

ti|Γ_i6=∅V_i∩f⁻¹(a)\Γ_i [

ti |Γ_i6=∅Γ_i , gives a Whitney stratification off⁻¹(a), and

Eu(f⁻¹(a)) = X

i |Γ_i=∅

χ(Vi∩f⁻¹(a))Eu_f⁻¹_(a)(Vi∩f⁻¹(a))

+ X

i|Γi6=∅

χ(Vi∩f⁻¹(a)\Γi)Eu_f−1(a)(Vi∩f⁻¹(a)\Γi)

+ X

i|Γ_i6=∅

X

q∈Γi

Eu_f−1(a)(q).

If Γiis empty then the intersectionVi∩f⁻¹(a) is transverse (necessarily dimVi>0) and by [12], Proposition IV. 4.1.1

Eu_X(V_i) = Eu_f−1(a)(V_i∩f⁻¹(a)).

If Γi is not empty and dimVi>0, then

χ(Vi∩f⁻¹(a)) =χ(Vi∩f⁻¹(a)\Γi) + #Γi,

and EuX(Vi) = Eu_X∩f−1(a)(Vi∩f⁻¹(a)\Γi) because outside Γi,f⁻¹(a) intersects Vi transversally. If Γi is not empty and dimVi= 0, then

χ(V_i∩f⁻¹(a)) = 1 andχ(V_i∩f⁻¹(a)\Γ_i) = 0.

Therefore we get B^X_f,a= X

i|Γ_i=∅

χ(Vi∩f⁻¹(a))Eu_f⁻¹_(a)(Vi∩f⁻¹(a))

+ X

i|Γ_i6=∅

χ(V_i∩f⁻¹(a)\Γ_i)Eu_f−1(a)(V_i∩f⁻¹(a)\Γ_i)

+ X

i|Γ_i6=∅

X

q∈Γi

EuX(q)

= Eu(f⁻¹(a)) + X

i|Γi6=∅

EuX(q)−Eu_f−1(a)(q).

Note that for a regular valuecoff, B^X_f,c= Eu(f⁻¹(c)). Furthermore if X=Cⁿ then EuX(qj) = 1 and Eu_f⁻¹_(a)= 1 + (−1)ⁿ⁻²µ⁰(f, qj), whereµ⁰(f, qj) is the first Milnor-Teissier number off atqj, so

B^X_f,a=χ(f⁻¹(a)) = Eu(f⁻¹(a)) + (−1)ⁿ⁻¹ X

j|f(q_j)=a

µ⁰(f, q_j), and we recover Equality (3.3) in [42].

A direct corollary of the previous proposition is a global relative version of the local index formula of Brylinski, Dubson and Kashiwara.

Corollary 4.8. Let α:X →Zbe a constructible function with respect to V. For any a∈C, we have

χ(f⁻¹(a), α) =

t

X

i=1

B^V_f,aⁱ η(Vi, α).

Proof. We keep the notations of the previous proposition and apply Equality (0.2) of [42] to get

χ(f⁻¹(a)) = X

i |Γi=∅

Eu(V_i∩f⁻¹(a))

1−χ(L^f_V^−1(a)

i∩f⁻¹(a))

+ X

i|Γi6=∅

Eu(Vi∩f⁻¹(a)\Γi)

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ X

i|Γi6=∅

X

q∈Γi

1−χ(L^f_{q}^−1(a))

= X

i |Γi6=Vi

Eu(Vi∩f⁻¹(a))

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ X

i |Γi6=∅

X

q∈Γi

1−χ(L^f_{q}^−1(a)),

because Γi =Vi if and only if Vi is just a 0-dimensional stratum and in this case, Vi∩f⁻¹(a)\Γi=∅. By the previous proposition, we obtain the equality

χ(f⁻¹(a)) = X

i |Γi6=Vi



B^V_f,aⁱ − X

j |f(qj)=a

−Eu_V

i(q_j) + Eu_V

i∩f⁻¹(a)(q_j)





1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ X

i|Γi6=∅

X

q∈Γi

1−χ(L^f_{q}^−1(a)), that we rewrite

χ(f⁻¹(a)) = X

i |Γi6=Vi



B^V_f,aⁱ − X

j |f(qj)=a

−Eu_V

i(q_j) + Eu_V

i∩f⁻¹(a)(q_j)





1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ X

j|f(q_j)=a

1−χ(L^f_{q^−1(a)

j} ).

If Γi6=Vi thenf⁻¹(a) intersectsVi\Γi transversally and so χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi) =χ(L^X_V

i).

Hence we have χ(f⁻¹(a)) = X

i |Γ_i6=V_i

B^V_f,aⁱ

1−χ(L^X_V

i)

+ X

j |f(q_j)=a

h1−χ(L^f_{q^−1(a)

j} )

+ X

i|Γi6=Vi

−Eu_V

i(qj)

1−χ(L^X_Vi) + Eu_V

i∩f⁻¹(a)(qj)

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi) i . Let us evaluate the second part of this sum and fixqa critical point off such that f(q) =a. Two cases are possible.

Ifqbelongs to a stratumVk with Γk 6=Vk then we add the stratumV0={q}to the Whitney stratification of X. By the Brylinski, Dubson and Kashiwara index formula ([6] or [31], p294), we know that

1 =

t

X

i=0

Eu_V

i(q)

1−χ(L^X_V

i) , and so

1 = X

i |Γi6=Vi

Eu_V

i(q)

1−χ(L^X_V

i)

+ Eu_V0(q)

1−χ(L^X_V

0) , because q /∈ V_i if Γ_i = V_i (i ≥ 1). But Eu_V0(q) = 1 andχ(L^X_V

0) = 1 because a generic linear form is a stratified submersion at q (see [17], p90 for details). The same index formula applied tof⁻¹(a) gives

1 =

t

X

i=1

Eu_V

i∩f⁻¹(a)\Γi(q)

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ 1−χ(L^f_{q}^−1(a)).

Therefore we get that

χ(L^f_{q}^−1(a)) = X

i|Γi6=Vi

Eu_V

i∩f⁻¹(a)(q)

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi) , and so the contribution ofqin the second summand of the above sum is zero.

Ifqbelongs to a stratumV_k with Γ_k=V_kthen, actuallyV_k={q}. By the index formula and the same arguments, we find that

1 = X

i|Γ_i6=Vi

Eu_V

i(q)

1−χ(L^X_Vi)

+ 1−χ(L^X_{q}),

and

1 = X

i|Γ_i6=Vi

Eu_V

i∩f⁻¹(a)(q)

1−χ(L^f_V^−1(a)

i∩f⁻¹(a)\Γi)

+ 1−χ(L^f_{q}^−1(a)).

Hence the contribution ofqin the above second summand is 1−χ(L^X_{q}), which we can write B^V_f,a^k

1−χ(L^X_Vk)

. Finally we have proved that χ(f⁻¹(a)) =

t

X

i=1

B^V_f,aⁱ

1−χ(L^X_Vi) ,

and the theorem follows because both sides of the equality are linear inα.

5. Global Brasselet numbers and critical points

In this section, we prove several formulas that relate the number of critical points of a Morsefication of a polynomial functionf on an algebraic setX, to the global Brasselet numbers and the Brasselet numbers at infinity off. We note that when X =Cⁿ, similar formulas have already appeared in the literature ([8, 22, 38, 40, 41, 36, 30, 1]).

The setting is the same as in the previous section: X ⊂Cⁿis a reduced algebraic set of dimensiond, equipped with a finite Whitney stratificationV ={Vi}^t_i=1 such thatV1, . . . , Vt−1are connected,V1, . . . , Vtare reduced andVt=Xreg ;f :X →C is a complex polynomial, restriction to X of a polynomial function F : Cⁿ → C. We assume thatf has a finite number of critical pointsq₁, . . . , q_sand we denote by {a1, . . . , a_r}the set of stratified asymptotic non-ρ_E-regular values of f.

Definition 5.1. We say that ˜f : X → C is a Morsefication of f if ˜f is a small deformation off which is a local (stratified) Morsefication at all isolated critical points off.

Let ˜f be a Morsefication off. As in the local, we can take ˜f of the form f+tl where t is a sufficiently small complex number and l is the restriction to X of a generic linear form (see Theorem 2.2 in [24]). Note that ˜f has two kinds of critical points: those appearing in a small neighborhood of a critical point off and those appearing at infinity, i.e., outside a ball of sufficiently big radius. We will only consider the first ones.

Letni, i = 1, . . . , t, be the number of critical points of ˜f appearing in a small neighborhood of a critical point off on the stratumVi. Note that

ni≥µ^T(f_|Vi) = X

j |q_j∈Vi

µ(f_|Vi, qj),

whereµ(f_|Vi, qj) is the Milnor number off_|Vi atqj, since we do not assume thatf is general with respect toV.

The following theorem relates the number of stratified critical points of ˜f ap- pearing on the stratumVi to the topology of X and a generic fibre of f.

Theorem 5.2. Letc∈Cbe a regular value off, which is not a stratified asymptotic non-ρE-regular value. We have

χ(X)−χ(f⁻¹(c)) =

t

X

i=1

(−1)^din_i

1−χ(L^X_V

i)

−λ^X,∞_f .