CHARACTERISTIC CLASSES OF HYPERSURFACES AND CHARACTERISTIC CYCLES. Adam Parusinski and Piotr Pragacz

Transcription

1 CHARACTERISTIC CLASSES OF HYPERSURFACES AND CHARACTERISTIC CYCLES Adam Parusinski and Piotr Pragacz Abstract. We give a new formula for the Chern-Schwartz-MacPherson class of a hypersurface with arbitrary singularities, generalizing the main result of [P-P], which was a formula for the Euler characteristic. Two dierent approaches are presented. The rst is based on the theory of characteristic cycle of a D-module (or a holonomic system) and the work of Sabbah [S], Briancon- Maisonobe-Merle [B-M-M], and L^e-Mebkhout [L-M]. In particular, this approach leads to a simple proof of a formula of Alu [A] for the above mentioned class. The second approach uses Verdier's [V] specialization property of the Chern-Schwartz-MacPherson classes. Some related new formulas for complexes of nearby cycles and vanishing cycles are also given. Introduction and statement of the main result Let X be a nonsingular compact complex analytic variety of pure dimension n and let L be a holomorphic line bundle on X. Take f 2 H 0 (X; L) a holomorphic section of L such that the variety Z of zeros of f is a (nowhere dense) hypersurface in X. Denoting by T X the tangent bundle of X, we will call (1) c F J (Z) := c(t Xj Z Lj Z ) \ [Z] ; the Fulton-Johnson class of Z. This terminology is justied by the fact that both canonical classes dened in [F-J] by c(t Xj Z ) \ s(n Z X), and in [F, Ex.4.2.6] by c(t Xj Z ) \ s(z; X), are equal in the present situation to the right-hand side of (1). Here, N Z X is the conormal sheaf to Z in X and s(z; X) is the Segre class of Z in X (cf. [F]). For more on this, consult [Su]; see also [B-L-S-S] and [Y3]. By c (Z) we denote the Chern-Schwartz-MacPherson class of Z, see [McP]. We recall its denition later in Section 1. Note that if Z is nonsingular then c F J (Z) = c (Z) = c(t Z) \ [Z]: The present research was supported by KBN grant No.2P03A Typeset by AMS-TEX

2 2 CHARACTERISTIC CLASSES OF HYPERSURFACES After [Y1,2,3] (see also [B-L-S-S]), we shall call (2) M(Z) := (1) n1 c F J (Z) c (Z) the Milnor class of Z. This class is supported on the singular locus of Z; it is convenient, however, to treat it as an element of H (Z). Example 0.1. Suppose that the singular set of Z is nite and equals x 1 ; : : : ; x k. Let xi denote the Milnor number of Z at x i (see [M]). Then M(Z) = kx i=1 xi [x i ] 2 H 0 (Z); see, for instance, Suwa [Su] where this result is generalized to complete intersections. Consider the function : Z! Zdened for x 2 Z by (x) := (F x ), where F x denotes the Milnor bre at x (see [M]) and (F x ) its Euler characteristic. Dene also the function : Z! Zby := (1) n1 ( 1 Z ). Fix now any stratication S = fsg of Z such that is constant on the strata of S. For instance, any Whitney stratication of Z satises this property, see [B-M-M] and [Pa]. Actually, it is not dicult to see that the topological type of the Milnor bres is constant along the strata of Whitney stratication of Z. Let us denote the value of on the stratum S by S. Let (3) (S) := S X S 0 6=S;SS 0 (S 0 ) be the numbers dened inductively on descending dimension of S. (These numbers appear as the coecients in the development of as a combination of the 1 S 's { see Lemma 4.1.) The main result of the present paper is Theorem 0.2. In the above notation, (4) M(Z) = X S2S (S)c(Lj Z ) 1 \ (i S;Z ) c (S); where i S;Z : S! Z denotes the inclusion. When X is projective, (4) was conjectured by Yokura in [Y2]. Under this last assumption, the equality (5) Z Z M(Z) = (Z) = X S2S (S) Z S c(lj S ) 1 \ c (S)

3 ADAM PARUSI NSKI AND PIOTR PRAGACZ 3 was proved in [P-P]; hence the theorem gives, in particular, a generalization of the main result (5) of [P-P] to compact varieties. Perhaps, it is in order to note at this point that when Z is a curve on a complex surface X, (5) is nothing but a classical \adjunction formula" [Ko, (2.2)]. Our proof of the theorem is based on a formula due to Sabbah [S], which allows one to calculate the Chern-Schwartz-MacPherson class of a subvariety in terms of the associated characteristic cycle. In the case of hypersurface Z, this characteristic cycle was calculated in [B-M-M] and [L-M] in terms of the blow-up of the Jacobian ideal of a local equation of Z in X. So the proof of Theorem 0.2 is obtained by putting this local description and the global data together, and expressing the characteristic cycle of Z in terms of the global blow-up of the singular subscheme of Z. Here by the singular subscheme of Z we mean the one dened locally by the ideal ; : : : ; n, where (z 1 ; : : : ; z n ) are local coordinates on X. The approach used leads to a very simple proof of a formula for the Chern- Schwartz-MacPherson class of hypersurface in terms of some divisors associated with the above blow-up. This formula was originally obtained by Alu [A] by dierent methods. Some new formulas for the Chern-Schwartz-MacPherson classes of the constructible functions and are also given. In the last section, we show, using Verdier's specialization property of the Chern- Schwartz-MacPherson classes (see [V], and also [S] and [K2]), how to prove another conjecture of Yokura, which, combined with a result from [Y2,3], gives an alternative proof of Theorem 0.2. (More precisely, this comment concerns a variant of Theorem 0.2, where X is projective and the classes are pushed forwarded to the homology of the ambient space X. See the remark after Theorem 5.3.) We nd that this specialization argument somewhat better explains the essence of the theorem. Another expression for the Milnor class M(Z) was given by Alu in [A]. Finally, we note that one motivation for studying the Milnor classes comes from Riemann-Roch-type problems. Namely, it is pointed out by Yokura in [Y1] that the knowledge of the Milnor class is necessary to understand a generalized Verdier-type Riemann-Roch theorem for the Chern-Schwartz-MacPherson class. 1. Chern-Mather classes and Chern-Schwartz-MacPherson classes We start by recalling some results of Sabbah [S]. Let for X as in the introduction, T X denote the cotangent bundle of X. Let V be an (irreducible) subvariety of X. Denote by c M (V ) (resp. c _ M (V ) ) the Chern-Mather class of V (resp. the dual Chern-Mather class). Let us recall briey their denitions. Let : NB(V )! V be the Nash blow-up of V. By denition on NB(V ) there exists the \Nash tangent bundle" T V which extends T V 0, where V 0 is the regular part of V. Dene the following elements of H (V ) (6) c M (V ) := c(tv ) \ [NB(V )] c _ M(V ) := c(t V ) \ [NB(V )] ;

4 4 CHARACTERISTIC CLASSES OF HYPERSURFACES where T V is the dual bundle of T V. It is easy to see that (7) c _ M(V ) = (1) dim V c M (V ) _ ; where for a homology class a = a 0 + a 1 + a 2 + : : :, where a i 2 H 2i (V ), we denote a _ := a 0 a 1 + a 2 : : :. By T V X T X we denote the conormal space to V : T V X := Closure (x; ) 2 T X j x 2 V 0 ; j Tx V 0 0 ; and by C(V ) PT X its projectivization. Let : C(V )! V be the restriction of the projection PT X! X to C(V ). Then by [S], we have (8) c _ M (V ) = c(t Xj V ) \ c (O(1)) 1 \ [C(V )] c M (V ) = (1) n1dim V c (T Xj V ) \ c (O(1)) 1 \ [C(V )] where O(1) is the tautological line bundle on PT X restricted to C(V ). Let now ' be a constructible function on X, ' = X a j 1 Yj ; where Y j are (closed) subvarieties of X and a j 2 Z. By the characteristic cycle of ' we mean the Lagrangian conical cycle in T X dened by (9) Ch(') := Ch M j 1 iyj ;X C a j Y j A ; ; where C Yj is the constant sheaf on Y j and i Yj ;X : Y j! X denotes the inclusion. For a general denition of the characteristic cycle of a sheaf, we refer the reader to [B]. The characteristic cycle of a constructible function admits the following interpretation. Let F (X) and L(X) denote the groups of constructible functions on X and conical Lagrangian cycles in T X respectively. It is known that the assignment (10) T V X 7! (1)dim V Eu V ; where Eu V stands for the Euler obstruction (see [McP] and also [S], [K1]), denes a natural transformation of the functors of Lagrangian conical cycles and constructible functions, that is an isomorphism. In particular, we have an isomorphism between L(X) and F (X). The operation of taking the characteristic cycle is the inverse of this isomorphism; that is, it is given by (11) Ch(Eu V ) = (1) dim V T V X :

5 ADAM PARUSI NSKI AND PIOTR PRAGACZ 5 Since every constructible function is a combination of the Eu V 's (see [McP]), this allows \in theory" to compute Ch(') for a constructible function '. However, even for ' = 1 V, this would involve not only the Euler obstruction of V itself but also of some subvarieties of V. Now we associate with a constructible function ' on X its Chern-Schwartz- MacPherson class (abbreviation: CSM-class). Let : Supp PCh(')! Supp ' be the restriction of the projection PT X! X. Set (12) c (') := (1) n1 c (T Xj Supp ' ) \ c (O(1)) 1 \ [PCh'] { an element in H (Supp '). We note that, in particular, by (8), (11) and (12) one has (13) c (Eu V ) = c M (V ) : If V X is a (closed) subvariety, we will write c (V ) := c ( 1 V ) P as is customary. Note that (12) is in agreement with [McP] because for 1 V = i b ieu Yi, where b i 2 Zand Y i X are (closed) subvarieties, we have c ( 1 V ) = X i b i c (Eu Yi ) = X i b i c M (Y i ) = c (V ) : Thus, denoting by : Supp Ch( 1 V )! V the restriction of the projection PT X! X, we have (14) c (V ) = (1) n1 c (T Xj V ) \ c (O(1)) 1 \ [PCh( 1 V )] : 2. Characteristic cycle of a hypersurface (local case) Suppose that U C n is an open subset and Z U is a hypersurface of zeros of a holomorphic function f : U! C. Let J f denote the Jacobian ; : : : ; n of f, where (z 1 ; : : : ; z n ) are the standard coordinates of C n. Consider the blow-up : Bl Jf U! U of J f. Recall that we may interpret it as follows Bl Jf U = Closure (x; ) 2 U P _ n1 j x =2 Sing Z; 1 (x) : : : : : (x) n where Sing Z denotes the singular subscheme of Z, and P _ n1 stands for the dual projective (n 1)-space. Remark 2.1. Bl Jf U can be also interpreted as the projectivization of the relative conormal space Tf T U (see [B-M-M, x 2], where we put = X = U). Then by the Lagrangian specialization all bres of the restriction of ~ f : T U! U f! C

6 6 CHARACTERISTIC CLASSES OF HYPERSURFACES to Tf are conical Lagrangian subvarieties of T U. In particular, every irreducible component of f ~ 1 (0) \ Tf is conormal to its projection on U. For details, we refer to [B-M-M, x 2] and to references therein. Let Z be the total transform 1 (Z) of Z in Bl Jf U and Z S = i decomposition of Z into irreducible components. Set C i := (D i ) and denote by I Ci the ideal dening C i. Then dene n i := multiplicity of I Ci along D i m i := multiplicity of f along D i p i := multiplicity of J f along D i Let us now record the following result. Proposition 2.2. : m i = n i + p i. D i be the Proof. Observe that by Remark 2.1 we have D i = PT C i U. Let x be a generic point of C i and choose a system of coordinates (z 1 ; : : : ; z n ) at x such that C i = fz 1 = : : : = z k = 0g in a neighborhood of x. Then, over a neighborhood of x, (15) D i = C i P _ k1 ; where P _ k1 = f[ 1 : : : : : n ] 2 P _ n1 j k+1 = : : : = n = 0g: Let : E! U denote the blow-up of the product of J f and I Ci. So E = Closure x; [z 1 (x) : : : : : z k (x)]; in U P k1 P _ n1. Then factors through E 1 (x) : : : : :! Bl Jf U & j j # n (x) i jx =2 Sing Z and there exists at least one irreducible component, say B ij, of the exceptional divisor of which projects surjectively onto D i. Let (t) = z(t); v(t); (t) be an analytic curve in E such that z(0); v(0); (0) is a generic point of B ij, z k+1 (t) : : : z n (t) 0 and f z(t) 6= 0 for t 6= 0. Then we have for t 6= 0, v(t) = [z 1 (t) : : : : : z k (t)] 2 P k1 ; (t) 1 z(t) : : : : n z(t) 2 P _ n1

7 ADAM PARUSI NSKI AND PIOTR PRAGACZ 7 and (0) = 1 (0) : : : : : k (0) : 0 : : : : : 0 by (15). Since z(0); (0) is a generic point of D i, the following equality would imply the proposition ord 0 (f ) (t) = ord 0 f z(t) (16) = ord 0 z1 (t); : : : ; z k (t) + ord 1 z(t) ; : : : n z(t) : We show (16). First we note that we may suppose that (z 1 ; : : : ; z k ) is generated by z i0 at (0) and 1 J f is generated j0 at (0), where j 0 2 f1; : : : ; kg by (15). We have (17) d dt f z(t) = kx i=1 z(t) z i i X k j0 z(t) z i0 (t) i j0 z(t) z i (t) z i0 (t) where z i stands for dz i dt. Note that the quotients make sense since =@z j 0 generates 1 J f, and z i (t)= z i0 (t) are analytic (because z i0 generates 1 (z 1 ; : : : ; z k )). We may suppose that j0 = 1 and v i0 = 1, which corresponds to choosing ane coordinates on P k1 P _ n1. Since lim [ z 1 (t) : : : : : z k (t)] = lim [z 1 (t) : : : : : z k (t)] ; t!0 t!0 1 A ; we get lim t!0 X k i j0 z(t) z i (t) z i0 (t) 1 A = lim t!0 kx i=1! i (t) j0 (t) v i(t) = v i0 (t) kx i=1 i (0)v i (0): This last sum is nonzero by the transversality of relative polar varieties, see, for instance, [H-M, 8.7, Lemme de transversalite]. Consequently, (17) implies ord 0 f z(t) 1 = ord 0 which gives (16), as required. z(t) + ord0 z i0 (t) j0 In the following theorem, the equality (i) and the second equality in (ii) were established in [B-M-M] (see also [L-M]).

8 8 CHARACTERISTIC CLASSES OF HYPERSURFACES Theorem 2.3. (i) Ch( 1 Z ) = (1) n1 X i n i T C i U ; (ii) (iii) Ch() = Ch(R f C U ) = (1) n1 X i Ch() = (1) n1 Ch R f C U = X i m i T C i U ; p i T C i U : (For a denition of the complexes of nearby cycles R f and vanishing cycles R f, we refer the reader to [D-K]. The rst equalities in (ii) and (iii) are well-known and follow from the local index theorem, see for instance [B-D-K] and [S, (1.3) and (4.4)].) Assertion (iii) follows from the equation combined with Proposition 2.2. Ch() = (1) n1 Ch() Ch( 1 Z ) ; Let Y denotes the exceptional divisor in Bl Jf U. Since D i = PT C i U, we can rewrite the assertions of the theorem as the following equalities. Corollary 2.4. (i) [PCh( 1 Z )] = (1) n1 ([Z] [Y]) ; (ii) [PCh()] = (1) n1 [Z] ; (iii) [PCh()] = [Y] : Observe that these equalities already take place on the level of cycles. Remark 2.5. Since f belongs to theintegral closure of J f (see [LJ-T]) the normalizations of the blow-ups of J f and 1 ; : : : n are equal. Hence Corollary 2.4 holds true if we replace the blow-up of the former ideal by the blow-up of the latter one. 3. Characteristic cycle of a hypersurface (global case) Let X; L; f and Z be as in the introduction. Let B = Bl Y X! X be the blow-up of X along the singular subscheme Y of Z. Let Z and Y denote the total transform of Z and the exceptional divisor in B, respectively. The following description of the CSM-class of Z was established by Alu [A] by dierent methods. Theorem 3.1. ([A]) Let : Z! Z be the restriction of the blow-up to Z. Then c (Z) = c (T Xj Z ) \ [Z] [Y] 1 + Z Y ;

9 ADAM PARUSI NSKI AND PIOTR PRAGACZ 9 where on the RHS, Z and Y mean the rst Chern classes of the line bundles associated with Z and Y i.e. those of (Lj Z ) and O B (1), the latter being the canonical line bundle on B. Proof. To get a convenient description of B, we use (after [A]) the bundle P 1 X L of principal parts of L over X (see e.g. [At]). Consider the section X! PX 1 L determined by f 2 H 0 (X; L). Recall that PX 1 L ts in an exact sequence 0! T X L! P 1 X L! L! 0 and the section in question is written locally as (df; f) 1 ; : : : n ; f, where (z 1 ; : : : ; z n ) are local coordinates on X. It follows that the closure of the image of the meromorphic map X > PPX 1 L induced by (df; f) is the blow-up B! X. Thus we may treat B as a subvariety of PPX 1 L. Clearly, the total transform Z of Z equals B \ P(T X L). The canonical line bundle O B (1) = O(Y) on B is the restriction of the tautological line bundle O(1) on PPX 1 L. Observe that the bundle O(1) restricted to Z is contained in (T X L)j Z (because f 0 over Z). Hence O B (1)j Z is the restriction of the tautological line bundle O e P (1) on ep = P(T X L). Using the natural identication P(T X L) = P(T X) the line bundle O e P (1) corresponds to the line bundle O P(1) L on P = P(T X). Thus O P (1) on P corresponds to O e P (1) L on P. e Hence using the characteristic cycle formula (14), we get c (Z) = (1) n1 c (T Xj Z ) \ c O B (1) 1 Lj Z \ PCh( 1 Z ) = c (T Xj Z ) \ [Z] [Y] 1 + Z Y because by (the global analogue of) Corollary 2.4, we have the equality [PCh( 1 Z )] = (1) n1 [Z] [Y]. By Corollary 2.4, we have [PCh()] = (1) n1 [Z] and [PCh()] = [Y]. Therefore, using similar arguments, we get the following result. Theorem 3.2. [Z] (i) c () = c (T Xj Z ) \ ; 1 + Z Y (ii) c () = (1) n1 c (T Xj Z ) \ [Y] 1 + Z Y (The constructible function is supported on Y but for later use we consider its CSM-class in H (Z).) Remark 3.3. One can add to the above formulas also [Z c M (Z) = c (Eu Z ) = c (T Xj Z ) \ 0 ] 1 + Z Y ; :

10 10 CHARACTERISTIC CLASSES OF HYPERSURFACES where Z 0 is the proper transform of Z. This equality for the Chern-Mather class was established originally by Alu [A] by dierent methods. Using the technique of characteristic cycles, it is a consequence of the equality P Ch(Eu Z ) = (1) n1 [Z 0 ] (see (11)). 4. Proof of Theorem 0.2 We start this section with the following fact about the constructible functions and dened in the introduction. P Lemma 4.1. : = (S) 1 S. S2S Proof. Fix an arbitrary stratum S 0 and a point x 2 S 0. We have X S = (S) 1 S (x) = X S6=S 0 ;SS 0 (S) + X S0 S6=S 0 ;SS 0 (S) + (S 0 ) X S6=S 0 ;SS 0 (S) 1 A = (x) : Now we pass to the proof of Theorem 0.2. Let : Z! Z be the restriction of the blow-up B = Bl Y X! X. We have, rewriting (1) as in [A], by using the projection formula, c F J (Z) = c (T Xj Z ) \ [Z] 1 + Z (Alternatively, one can use the expression from [F, Ex.4.2.6] and the birational invariance of Segre classes [F, Chap.4]: c(t Xj Z ) \ s(z; X) = c(t Xj Z ) \ s(z; B) Invoking (2) and using Theorem 3.1, we get (18) M(Z) = (1) n1 c F J (Z) c (Z) = c(t Xj Z ) \ [Z] 1 + Z [Z] = (1) n1 c (T Xj Z ) \ 1 + Z = (1) n1 c (T Xj Z ) \ : :) [Z] [Y] 1 + Z Y [Y] (1 + Z)(1 + Z Y) because Y \ [Z] = Z \ [Y] (see [F, Theorem 2.4). If we pass to the characteristic cycle approach, the equality (18) is rewritten, by Corollary 2.4, in the form (19) M(Z) = (1) n1 c (T Xj Z ) \ [PCh()] (1 + Z)(1 + Z Y) :

11 ADAM PARUSI NSKI AND PIOTR PRAGACZ 11 Since = P S2S (S) 1 S by Lemma 4.1, we have and hence (20) [PCh()] (1 + Z)(1 + Z Y) = = X S2S Ch() = X S2S (S) Ch( 1 S ) (S)c(Lj Z ) 1 \ c Lj Z O B (1) 1 \ [PCh( 1 S )] By (14) and the proof of Theorem 3.1, we get (21) is;z c (S) = (1) n1 c T Xj Z \ c Lj Z O B (1) 1 \ [PCh( 1 S )] for each stratum S 2 S. Finally, using (20) and (21), we rewrite (19) in the form M(Z) = X S2S which is the required expression. (S) c Lj Z 1 \ i S;Z c (S) 5. Another approach via specialization In this section, the setup is as in the Introduction. Additionally, let us assume that there exists a section g 2 H 0 (X; L) such that Z 0 = g 1 (0) is smooth and transverse to the strata of a (xed) Whitney stratication S = fsg of Z. For t 2 C, denote f t = f tg. In this section, by Z we will denote the following correspondence in X C : Z := (x; t) 2 X C j f t (x) = 0 : Denoting by p : Z! C the restriction to Z of the projection onto the second factor of X C, we have p 1 (t) = fx 2 X j f t (x) = 0g =: Z t for t 2 C. Let F (Z) (resp. F (Z) ) denote the group of constructible functions on Z (resp. on Z). Denote by F : F (Z)! F (Z 0 = Z) the specialization map of constructible functions (see [V], [S] and [K2], where a dierent notation is used). Recall briey its denition. If Y Z is a (closed) subvariety, one sets for the generator 1 Y, F 1 Y (x) := lim t!0 B(x; ") \ Y t for any suciently small " > 0, where B(x; ") is the closed ball of radius " about x and Y t = Y \Z t. In our situation, we are aiming to compute F 1 Z. More explicitly, for x 2 Z we want to calculate This is the content of the following ( F 1 Z )(x) = lim t!0 B(x; ") \ Z t : :

12 12 CHARACTERISTIC CLASSES OF HYPERSURFACES Proposition 5.1. One has F 1 Z (x) = (x) = 1 + (1) n1 (x) for x 62 Z \ Z 0 1 for x 2 Z \ Z 0 : Proof. If x 62 Z \ Z 0 i.e. g(x) 6= 0, then Z t = z j f(z) tg(z) = 0 = z j f(z)=g(z) = t after restriction to a small ball is the Milnor bre of f=g at x, and f=g also denes Z in a neighborhood of x. The assertion follows. Let now x 2 Z \ Z 0. We will use similar arguments to those used in Step 1 of the proof of Proposition 7 in [P-P]. Proceeding locally we can assume that x is the origin in C n, that in our local coordinates g(z) z n and that fz n = 0g is transverse to a xed Whitney stratication S = fsg of Z = ff = 0g. Our goal is to show that for suciently small " > 0 and 0 < << ", if t 2 C satises 0 < jtj <, then Z t \ B " = (z 1 ; : : : ; z n ) 2 C n jzj < "; f tz n = 0 is contractible, where B " = B(0; "). Set V = ff = z n = 0g. If " is suciently small then V \ B " is contractible. So it suces to retract Z t \ B " onto V \ B ". In what follows we shall proceed on Z t r V for t suciently small. First note that since the stratication is Whitney and hence satises the a f condition, we have by the assumption on transversality ; : : : n1 n for some universal c > 0. Therefore the linear forms df(p) and dz n (p) are linearly independent for p 62 ff = 0g. So are clearly the forms d(f tz n ) and dz n. Consequently the orthogonal projection of grad jz n j onto Z t = ff tz n = 0g r V is nonzero, and we may normalize it so that the normalized vector eld ~v satises (i) n j = tz n ) = We want, as well, the trajectories of this vector eld do not leave B ". For this we modify ~v near S " = z jzj = "g. Let p 2 V \ S " and let S be the stratum which contains p. Let p(s)! p as s! 0 be an analytic curve such that f p(s) 6= 0 for s 6= 0. Then the limit of df p(s) in P _ n1 as s! 0, exists. The forms and dz n are linearly independent by the assumption on transversality, and both vanish on

13 ADAM PARUSI NSKI AND PIOTR PRAGACZ 13 the tangent space to S \ fz n = 0g. Therefore, by the Whitney condition (b) for the closure of S \ P n fz n = 0g, we get the linear independence of, dz n and i=1 z idz i at p. Consequently, the orthogonal projection of grad jz n j onto S " \(Z t rv ) is nonzero in a neighborhood of p. Since S " \ V is compact, there exist a neighborhood U of S " \ V and a vector eld ~w on U r fz n = 0g [ ff = 0g such that for t small n j ~w = 1 tz n ) (ii) = 0 ~w = 0, where (z) =k z k2 : Using partition of unity we \glue" ~w and ~v in order to get a vector eld ~u dened on Z t r V such n j = 1 ; (ii) tz n ) = @~u = 0 on S ": The ow of ~u allows us to retract Z t \ B " onto Z t;c = Z t \ B " \ fjz n j cg for c as small as we want. On the other hand, for c small enough, Z t;c retracts onto V \ B " = Z t \ B " \ fz n = 0g, as required. Now we want to pass to the specialization map of homology classes H : H (Z t )! H (Z 0 = Z) (see [V], [S] and [K2], where a dierent notation is used). Recall briey its denition. Let D C be a disk of a suciently small radius such that the inclusion Z = Z 0 p 1 (D) is a homotopy equivalence. Thus for small nonzero t 2 D one denes the above H as the composition H (Z t ) i! H (p 1 D) = H (Z 0 = Z) ; where i : Z t! p 1 D is the inclusion. Recall now that Verdier's specialization property of CSM-classes asserts the following. For ' 2 F (Z) and t suciently small, one has (22) H c 'j Zt = c ( F ') : (see [V] and also [S] and [K2]). Let us evaluate the both sides of (22) for ' = 1 Z. H c (Z t ). As for the RHS, we have by Proposition 5.1 (23) F 1 Z = 1 Z + (1) n1 1 ZrZ\Z 0 = 1 Z + (1) n1 1 Z 1 Z\Z 0 : The LHS reads simply

14 14 CHARACTERISTIC CLASSES OF HYPERSURFACES Invoking the equality = P S (S) 1 S (see Lemma 4.1), Equation (23) is rewritten as (24) F 1 Z = 1 Z + (1) n1 X S (S) 1 S X S (S) 1 S\Z 0 and applying c to (24) we get that the RHS of (22) is evaluated as c F 1 Z =! ; = c (Z) + (1) n1 ( X S (S) (i S;Z ) c (S) (i S\Z 0;Z ) c (S \ Z 0 ) ) ; where i S\Z0 ;Z denotes the inclusion S \ Z0! Z. Suming up, by virtue of the specialization property (22), we have proved Proposition 5.2. For the specialization map H : H (Z t )! H (Z), where t 6= 0 is small enough, one has H c (Z t ) = = c (Z) + (1) n1 ( X S2S (S) (i S;Z ) c (S) (i S\Z0 ;Z ) c (S \ Z 0 ) ) : We now state the folowing result which appeared as a conjecture in [Y2]. Theorem 5.3. In the above notation, one has M(Z) = X S2S (S) (i S;Z ) c (S) (i S\Z0 ;Z ) c (S \ Z 0 ) : Proof. Observe that for t like in Proposition 5.2, we have c (Z t ) = c F J (Z t ) because Z t is smooth. Moreover, since the Fulton-Johnson class is expressed in terms of the Chern classes of vector bundles, one has H c F J (Z t ) = c F J (Z). We thus have M(Z) = (1) n1 c F J (Z) c (Z) = (1) n1 H c F J (Z t ) c (Z) = (1) n1 H c (Z t ) c (Z) (S) (i S;Z ) c (S) (i S\Z0 ;Z ) c (S \ Z 0 ) = X S by Proposition 5.2.

15 ADAM PARUSI NSKI AND PIOTR PRAGACZ 15 Finally, arguing as in [Y2,3 x 2] one shows that Theorem 5.3 implies, for X projective, (i Z;X ) M(Z) = X S2S (S)c(L) 1 \ (i S;X ) c (S); where i Z;X : Z! X and i S;X : S! X denote the inclusions. Acknowledgements. We thank P. Alu and S. Yokura for sending us their preprints [A] and [Y2] in spring 1996 and summer 1997, respectively. These two preprints were very inspiring for us during the preparation of the present paper. Especially inspiring was the letter of Alu (dated March 4, 1996) attached to [A] asking us about the relationship between the approach taken in [A] and that of [P-P] in the context of Chern-Schwartz-MacPherson classes. We hope that the present paper answers Alu's question. During the preparation of this paper the second named author proted the hospitality of the Technion in Haifa (Israel) and Universite d'angers (France). Note. Milnor classes of singular varieties were also studied by Brasselet, Lehmann, Seade and Suwa in [B-L-S-S], as we have been informed by J.-P. Brasselet and S. Yokura. For a survey about Milnor classes, see [Y3]. References [A] [At] P. Alu, Chern classes for singular hypersurfaces, preprint February (1996); a revised version will appear in Trans. Amer. Math. Soc.. M. F. Atiyah, Complex analytic connections in bre bundles, Trans. Amer. Math. Soc. 85 (1957), no. 1, 181{207. [B-M-M] J. Briancon, P. Maisonobe, M. Merle, Localisation de systemes dierentiels, stratications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531{550. [B-L-S-S] J.-P. Brasselet, D. Lehmann, J. Seade, T. Suwa, On Milnor classes of local complete intersections, Hokkaido University Preprint Series in Mathematics No. 413, May (1998). [B] [B-D-K] [D-K] J.-L. Brylinski, (Co)-Homologie d'intersection et faisceaux pervers, Seminaire Bourbaki 585 ( ). J.-L. Brylinski, A. Dubson, M. Kashiwara, Formule de l'indice pour les modules holonomes et obstruction d'euler locale, C. R. Acad. Sci. Paris (Serie I) 293 (1981), 129{132. P. Deligne, N. Katz, Groupes de monodromie en Geometrie Algebrique, (S.G.A. 7 II), Springer Lecture Notes in Math. 340 (1973). [F-J] W. Fulton, K. Johnson, Canonical classes on singular varieties, Manuscripta Math. 32 (1980), 381{389. [F] W. Fulton, Intersection Theory, Springer-Verlag, [H-M] J.-P. Henry, M. Merle, Conditions de regularite et eclatements, Ann. Inst. Fourier 37(3) (1987), 159{190. [K1] G. Kennedy, MacPherson's Chern classes of singular algebraic varieties, Comm. Alg. 18(9) (1990), 2821{2839. [K2] G. Kennedy, Specialization of MacPherson's Chern classes, Math. Scand. 66 (1990), 12{16.

16 16 CHARACTERISTIC CLASSES OF HYPERSURFACES [Ko] K. Kodaira, On compact complex analytic surfaces I, Ann. of Math. 71 (1960), 111{152. [L-M] [LJ-T] L^e D~ung Trang, Z. Mebkhout, Varietes caracteristiques et varietes polaires, C. R. Acad. Sci. Paris 296 (1983), 129{132. M. Lejeune-Jalabert, B. Teissier, Cl^oture integrale des ideaux et equisingularite, Seminaire Ecole Polytechnique , Available at: Institut de Maths. Pures, Universite de Grenoble, F Saint-Martin-d'Heres, France. [McP] R. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974), 423{432. [M] J. Milnor, Singular points of complex hypersurfaces, vol. 61, Ann. of Math. Studies, Princeton University Press, [Pa] A. Parusinski, Limits of tangent spaces to bres and the w f condition, Duke Math. J. 72 (1993), 99{108. [P-P] [S] [Su] A. Parusinski, P. Pragacz, A formula for the Euler characteristic of singular hypersurfaces, J. Alg. Geom. 4 (1995), 337{351. C. Sabbah, Quelques remarques sur la geometrie des espaces conormaux, Asterisque 130 (1985), T. Suwa, Classes de Chern des intersections completes locales, C. R. Acad. Sci. Paris 324 (1996), 67{70. [V] J.-L. Verdier, Specialization des classes de Chern, Asterisque 82{83 (1981), 149{159. [Y1] S. Yokura, On a Verdier-type Riemann-Roch for Chern-Schwartz-MacPherson class, in press, Topology and Its Applications 95 (1999). [Y2] S. Yokura, On a Milnor class, preprint, June (1997). [Y3] S. Yokura, On characteristic classes of complete intersections, to appear in "Algebraic Geometry Conference - Hirzebruch 70", Contemporary Mathematics A.M.S.. A.P. : Departement de Mathematiques, Universite d'angers, 2 Bd. Lavoisier, Angers Cedex 01, France parus@tonton.univ-angers.fr P.P. : Mathematical Institute of Polish Academy of Sciences, Chopina 12, Torun, Poland pragacz@mat.uni.torun.pl