On the Chern classes of singular complete intersections
aa r X i v : . [ m a t h . AG ] S e p ON THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS
ROBERTO CALLEJAS-BEDREGAL, MICHELLE F. Z. MORGADO, AND JOS´E SEADE
Abstract.
We consider two classical extensions for singular varieties of the usual Chern classes of complexmanifolds, namely the total Schwartz-MacPherson and Fulton-Johnson classes, c SM ( X ) and c F J ( X ). Theirdifference (up to sign) is the total Milnor class M ( X ), a generalization of the Milnor number for varietieswith arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes c SM ( X )and c F J ( X ), and use these to prove a surprisingly simple formula for the total Milnor class when X isdefined by a finite number of local complete intersection X , · . . . · , X r in a complex manifold, satisfyingcertain transversality conditions. As applications we obtain a Parusi´nski-Pragacz type formula and anAluffi type formula for the Milnor class, and a description of the Milnor classes of X in terms of the globalLˆe classes of the X i . Introduction
There are various different notions of Chern classes for singular varieties, each having its own interest andcharacteristics. Perhaps the most important of these are the total Schwartz-MacPherson class c SM ( X ) andthe total Fulton-Johnson class c F J ( X ). In the complex analytic context these are elements in the homologyring H ∗ ( X, Z ) and in the algebraic context these are elements in the Chow group A ∗ ( X ). Both of theseclasses c SM ( X ) and c F J ( X ) are defined by means of an embedding of X in some complex manifold M , butthey turn out to be independent of the choice of embedding; when X is non-singular these are the Poincar´eduals of the usual Chern classes. By definition the total Milnor class of X is:(1) M ( X ) := ( − dim X (cid:0) c F J ( X ) − c SM ( X ) (cid:1) . Milnor classes are a generalization of the classical Milnor number to varieties X with arbitrary singularset. These have support in the singular set Sing( X ). There is a Milnor class in each dimension from 0 tothat of Sing( X ). In particular, when the singularities of X are all isolated, then there is only a 0-degreeMilnor class which is an integer, and if X further is a local complete intersection, then this integer is the sumof the local Milnor numbers of X at its singular points (by [24, 25]). Milnor classes are important invariantsthat encode much information about the varieties in question, see for instance [1, 2, 3, 6, 7, 18, 19, 22]. Yet,most of the work on Milnor classes in the literature is for hypersurfaces, the complete intersection case beingmuch harder ( cf. [5, 9, 16]): that is the setting we envisage here. Our work is somehow inspired by theproduct formulas for the Milnor class of Ohmoto and Yokura in [17]. We prove: Theorem 1.
Let M be an n -dimensional compact complex analytic manifold and let { E , · . . . · , E r } , r ≥ ,be holomorphic vector bundles over M of ranks d i ≥ . For each i = 1 , · . . . · r , let X i be the ( n − d i ) -dimensional local complete intersection in M defined by the zeroes of a regular section s i of E i . Assumefurther that the X i are equipped with Whitney stratifications S i such that all the intersections amongst stratain the various X i are transversal. Set X = X ∩ · . . . · ∩ X r , a local complete intesection of dimension n − d − · . . . · − d r . Then: (i) c SM ( X ) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:16) c SM ( X ) · . . . · c SM ( X r ) (cid:17) ;(ii) c F J ( X ) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:16) c F J ( X ) · . . . · c F J ( X r ) (cid:17) ; and therefore (iii) M ( X ) = ( − dim X c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:16) c F J ( X ) · . . . · c F J ( X r ) − c SM ( X ) · . . . · c SM ( X r ) (cid:17) . Mathematics Subject Classification.
Primary ; 14C17, 55N45, 14M10 Secondary; 14B05, 32S20 .
Key words and phrases.
Complete intersections, Milnor classes, Whitney stratifications, Schwartz-MacPherson classes,Fulton-Johnson classes.Research partially supported by CAPES, CNPq and FAPESP, Brazil, and by FORDECYT-CONACYT and PAPIIT-UNAM,Mexico.
The transversality condition in this Theorem can be relaxed (see section 2). Similar transversality condi-tions were used in [22] to prove a refined intersection formula for the Chern-Schwartz-MacPherson classes.The proof of Theorem 1 takes most of this article. The first step is proving Verdier-Riemann-Roch typeformulae for the Schwartz-MacPherson, the Fulton-Johnson and, therefore, for the Milnor classes of localcomplete intersections. In the last section of this article we give various applications. The first is Theorem3.1 that describes the Milnor class of X in terms of the Milnor and the Schwartz-MacPherson classes of the X i and the Chern classes of M restricted to X . For instance, for r = 2 we get the beautiful formula: M ( X ) = c (( T M | X )) − ∩ (cid:16) ( − n M ( X ) · M ( X ) + ( − d c SM ( X ) · M ( X ) + ( − d M ( X ) · c SM ( X ) (cid:17) . For r = 3 we get: M ( X ) = c (cid:0) ( T M | X ) ⊕ (cid:1) − ∩ (cid:16) M ( X ) · M ( X ) · M ( X ) + ( − ( d + d ) c SM ( X ) · c SM ( X ) · M ( X )++( − ( d + d ) c SM ( X ) · M ( X ) · c SM ( X ) + ( − ( d + d ) M ( X ) · c SM ( X ) · c SM ( X )++( − ( n − d ) c SM ( X ) · M ( X ) · M ( X ) + ( − ( n − d ) M ( X ) · c SM ( X ) · M ( X )++( − ( n − d ) M ( X ) · M ( X ) · c SM ( X ) (cid:17) , and so on. This highlights why understanding the Milnor classes of complete intersections is a priori farmore difficult than in the hypersurface case, though the formula in Theorem 1 is surprisingly simple.We then restrict the discussion to the case where the bundles in question are all line bundles L i . We gettwo interesting applications of Theorem 3.1: i) A Parusi´nski-Pragacz type formula for local complete intersections as above (Corollary 3.5). This expressesthe Milnor classes using only Schwartz-MacPherson classes, and it answers positively the expected descriptiongiven by Ohmoto and Yokura in [17] for the total Milnor class of a local complete intersection. We noticethat a different generalization of the Parusi´nski-Pragacz formula for complete intersections has been givenrecently in [16]. ii)
A description of the total Milnor class of the local complete intersection X in the vein of Aluffi’s formulain [1] for hypersurfaces, using Aluffi’s µ -classes (Corollary 3.3).This work is a refinement of our unpublished article [8] (cf. also [9]). We are indebted to the referee andto J¨org Sch¨urmann for valuable suggestions. We are also grateful to Nivaldo Medeiros and Marcelo Saia forfruitful conversations. 1. Chern classes and the diagonal embedding
Derived categories.
We assume some basic knowledge on derived categories as described for instancein [10]. If X is a complex analytic space then D bc ( X ) denotes the derived category of bounded constructiblecomplexes of sheaves of C -vector spaces on X . We denote the objects of D bc ( X ) by something of the form F • . The shifted complex F • [ l ] is defined by ( F • [ l ]) k = F l + k and its differential is d k [ l ] = ( − l d k + l . Theconstant sheaf C X on X induces an object C • X ∈ D bc ( X ) by letting C X = C X and C kX = 0 for k = 0. If h : X → C is an analytic map and F • ∈ D bc ( X ), then we denote the sheaf of vanishing cycles of F • withrespect to h by φ h F • . For F • ∈ D bc ( X ) and p ∈ X , we denote by H ∗ ( F • ) p the stalk cohomology of F • at p ,and by χ ( F • ) p its Euler characteristic. That is, χ ( F • ) p = X k ( − k dim C H k ( F • ) p . We also denote by χ ( X, F • ) the Euler characteristic of X with coefficients in F • , i.e. , χ ( X, F • ) = X k ( − k dim C H k ( X, F • ) , where H ∗ ( X, F • ) denotes the hypercohomology groups of X with coefficients in F • . When F • ∈ D bc ( X ) is S -constructible, where S is a Whitney stratification of X , we denote it by F • ∈ D b S ( X ). Setting χ ( F • S ) = χ ( F • ) p for an arbitrary point p ∈ S , we have [10, Theorem 4.1.22]:(2) χ ( X, F • ) = X S ∈S χ ( F • S ) χ ( S ) . For a subvariety X in a complex manifold M we denote its conormal variety by T ∗ X M . That is, T ∗ X M := closure { ( x, θ ) ∈ T ∗ M | x ∈ X reg and θ | TxX reg ≡ } , N THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS where T ∗ M is the cotangent bundle and X reg is the regular part. One has (see [12]): Definition 1.1.
Let X be an analytic subvariety of a complex manifold M , { S α } a Whitney stratificationof M adapted to X and x ∈ S α a point in X . Consider g : ( M, x ) → ( C ,
0) a germ of holomorphic functionsuch that d x g is a non-degenerate covector at x with respect to the fixed stratification. That is, d x g ∈ T ∗ S α M and d x g T ∗ S ′ M for all stratum S ′ = S α . Let N be a germ of a closed complex submanifold of M which istransversal to S α , with N ∩ S α = { x } . Define the complex link l S α of S α by: l S α := X ∩ N ∩ B δ ( x ) ∩ { g = w } for 0 < | w | << δ << . The normal Morse datum and the normal Morse index of the stratum S α are, respectively: N M D ( S α ) := ( X ∩ N ∩ B δ ( x ) , l S α ) and η ( S α , F • ) := χ ( N M D ( S ) , F • ) , where the right-hand-side means the Euler characteristic of the relative hypercohomology.In fact, the slice N normal to the stratum S α at x is transversal to all other stratum that contain x in their closure, by Whitney regularity. Therefore the Whitney stratification on X induces a Whitneystratification on N M D ( S α ). Hence the sheaf F • restricted to N M D ( S α ) is constructible and therefore therelative hypercohomology is well-defined.By [12, Theorem 2.3] we get that η ( S α , F • ) does not depend on the choices of x ∈ S α , g and N . By [21,p. 283], the normal Morse index η ( S α , F • ) can be computed in terms of sheaves of vanishing cycles as(3) η ( S α , F • ) = − χ ( φ g | N ( F • | N )) . By [10, Remark 2.4.5(ii)] this can also be expressed as:(4) η ( S α , F • ) = χ ( X ∩ N ∩ B δ ( x ) , F • ) − χ ( l S α , F • ) . Chern classes for singular varieties.
From now on, let M be an n -dimensional compact complexanalytic manifold and let E be a holomorphic vector bundle over M of rank d . Let X be the zero schemeof a regular holomorphic section of E , which is an ( n − d )-dimensional local complete intersection. Consider the virtual bundle τ ( X ; M ) := T M | X − E | X , where T M denotes the tangent bundle of M and the differenceis in the K-theory of X . The element τ ( X ; M ) actually is independent of M (see [11, Appendix B.7.6.]) andis called the virtual tangent bundle of X . The Fulton-Johnson homology class of X is defined by the Chernclass of τ ( X ; M ) via the Poincar´e morphism, that is (cf. [11]): c F J ( X ; M ) = c ( τ ( X ; M )) ∩ [ X ] := c ( T M | X ) c ( E | X ) − ∩ [ X ] . For simplicity we denote the virtual bundle and the Fulton-Johnson classes simply by τ ( X ) and c F J ( X ).Consider now the Nash blow up ˜ X ν → X of X , its Nash bundle ˜ T π → ˜ X and the Chern classes of ˜ T , c j ( ˜ T ) ∈ H j ( ˜ X ), j = 1 , . . . , n . The Mather classes of X are: c Mak ( X ) := v ∗ ( c n − d − k ( ˜ T ) ∩ [ ˜ X ]) ∈ H k ( X ) , k = 0 , . . . , n . We equip X with a Whitney stratification X α . The MacPherson classes are obtained from the Mather classesby considering appropriate “weights” for each stratum, determined by the local Euler obstruction Eu X α ( x ).This is an integer associated in [15] to each point x ∈ X α . It is proved in [15] that there exists a unique setof integers b α , for which the equation P b α Eu ¯ X α ( x ) = 1 is satisfied for all points x ∈ X . Here, ¯ X α denotesthe closure of the stratum, which is itself analytic; the sum runs over all strata X α containing x in theirclosure. Then the MacPherson class of degree k is defined by c Mk ( X ) := X b α i ∗ ( c Mak ( ¯ X α )) , where i : ¯ X α ֒ → X is the inclusion map. We remark that by [4], the MacPherson classes coincide, up toAlexander duality, with the classes defined by M.-H. Schwartz in [23]. Thus, following the modern literature(see for instance [19, 5, 6]), these are called Schwartz-MacPherson classes of X and denoted by c SMk ( X ). Definition 1.3.
The total Milnor class of X is (see [6, 19]): M ( X ) := ( − n − d (cid:0) c F J ( X ) − c SM ( X ) (cid:1) . ROBERTO CALLEJAS-BEDREGAL, MICHELLE F. Z. MORGADO, AND JOS´E SEADE
Milnor classes and the diagonal embedding.
Given a manifold M as before, set M ( r ) := M × . . . × M , r times. We let E be a holomorphic vector bundle over M ( r ) of rank d . Consider ∆ : M → M ( r ) the diagonal morphism, which is a regular embedding of codimension nr − n . Let t be a regular holomorphicsection of E . The set of the zeros of t is a closed subvariety Z ( t ) of M ( r ) of dimension nr − d . Consider Z (∆ ∗ ( t )) the set of the zeros of the pull back section of t by ∆.Following [11, Chapter 6] we have that ∆ induces the refined Gysin homomorphism∆ ! : H k ( Z ( t )) → H k − nr + n ) ( Z (∆ ∗ ( t ))) . The refined intersection product is defined by α · . . . · α r := ∆ ! ( α × . . . × α r ). For the usual homology thisis defined by duality between homology and cohomology:∆ ! = ∆ ∗ : H k ( Z ( t ); Z ) ≃ H nr − k ) ( Z ( t ); Z ) → H nr − k ) ( Z (∆ ∗ ( t )); Z ) ≃ H k − nr + n ) ( Z (∆ ∗ ( t )); Z ) . Remark 1.4. (1) In [11, Proposition 14.1, (c) and (d)(ii)] it is proved that if f : X ′ → X is a localcomplete intersection morphism between purely dimensional schemes, E is a vector bundle on X, s is a regular section of E and s ′ = f ∗ s is the induced section on f ∗ E, then f ! [ Z ( s )] = [ Z ( s ′ )] , where f ! is the refined Gysin homomorphism induced by f. (2) In [11, Proposition 6.3] it is proved that if ι : X → Y is a regular embedding and F is a vector bundleon Y, then ι ! ( c m ( F ) ∩ α ) = c m ( ι ∗ F ) ∩ ι ! α, for all α ∈ H k ( Y, Z ) . Applying this result to the diagonalmorphism ∆ : M → M ( r ) , which is a regular embedding, we have that for any vector bundle F on M ( r ) holds that ∆ ! ( c m ( F ) ∩ α ) = c m (∆ ∗ F ) ∩ ∆ ! ( α ) for all α ∈ H k ( M ( r ) ; Z ) and m ≥ . These two remarks are used for following a Verdier-Riemann-Roch type theorem for the Fulton-Johnsonclasses:
Proposition 1.5.
The refined Gysin morphism satisfies: ∆ ! (cid:0) c F J ( Z ( t )) (cid:1) = c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) ∩ c F J ( Z (∆ ∗ t )) . Proof.
By definition of the Fulton-Johnson class we have∆ ! c F J ( Z ( t )) = ∆ ! (cid:16) c (cid:16) T M ( r ) | Z ( t ) (cid:17) c (cid:0) E | Z ( t ) (cid:1) − ∩ [ Z ( t )] (cid:17) . Applying Remark 1.4 (2) to the diagonal morphism ∆ : M → M ( r ) , which is a regular embedding, andusing the virtual bundle we have that∆ ! c F J ( Z ( t )) = c (cid:16) ∆ ∗ (cid:16) T M ( r ) | Z ( t ) (cid:17)(cid:17) c (cid:0) ∆ ∗ (cid:0) E | Z ( t ) (cid:1)(cid:1) − ∩ ∆ ! [ Z ( t )] . Note that ∆ ∗ (cid:0) E | Z ( t ) (cid:1) = ∆ ∗ E | Z (∆ ∗ t ) and, applying Remark 1.4 (1) to the diagonal morphism ∆ : M → M ( r ) , which is a local complete intersection morphism, and to the regular section t of E we obtain that∆ ! [ Z ( t )] = [ Z (∆ ∗ ( t ))] . Moreover, since ∆ ∗ T M ( r ) = T M ⊕ . . . ⊕ T M , we have c (cid:16) ∆ ∗ (cid:16) T M ( r ) | Z ( t ) (cid:17)(cid:17) = c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r (cid:17) and the result follows. (cid:3) Let F ( M ) be the free abelian group of constructible functions on M with respect to a Whitney stratifi-cation { S α } . It is proved in [15] that every element ξ in F ( M ) can be written uniquely in the form: ξ = X n W Eu W , for some appropriate subvarieties W and integers n W . Let L ( M ) be the free abelian group of all cyclesgenerated by the conormal spaces T ∗ W M , where W varies over all subvarieties of M . Given ξ ∈ F ( M ) definean element Ch ( ξ ) in L ( M ) by:(5) Ch ( ξ ) := X α ( − dim W n W · T ∗ W M .
This induces an isomorphism Ch : F ( M ) → L ( M ). Define the map cn : Z ( M ) → L ( M ) by cn ( X ) := T ∗ X M .Clearly, this is also an isomorphism. We know from [7, Section 3] that we have a commutative diagram: N THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS (6) Z ( M ) ˇ Eu / / cn (cid:15) (cid:15) F ( M )id (cid:15) (cid:15) L ( M ) Ch / / F ( M )The commutativity of this diagram amounts to saying: β = X α η ( S α , β ) · Eu S α , for any function β : X → Z which is constructible for the given Whitney stratification, where η ( S α , ξ ) = η ( S α , F • ), with F • being the complex of sheaves such that χ ( F • ) p = ξ ( p ). Substituting in equation (5) weget:(7) Ch ( ξ ) := X α ( − dim S α η ( S α , ξ ) · T ∗ S α M .
Now consider the projectivized cotangent bundles P ( T ∗ M ) and P ( T ∗ ( M ( r ) )); we denote by P (( T ∗ M ) ⊕ r )the bundle P ( T ∗ M ⊕ . . . ⊕ T ∗ M ). Notice that one has a fibre square diagram (see [11, pag. 428]):(8) P (( T ∗ M ) ⊕ r ) δ / / p (cid:15) (cid:15) P ( T ∗ ( M ( r ) )) π ( r ) (cid:15) (cid:15) M ∆ / / M ( r ) where π ( r ) is the natural proper map. Let i : P ( T ∗ M ) → P (( T ∗ M ) ⊕ r ) be the morphism induced by thediagonal embedding T ∗ M → T ∗ M ⊕ . . . ⊕ T ∗ M . Proposition 1.8.
Let β be a constructible function on M ( r ) with respect to a Whitney stratification {T γ } ,which we assume transversal to ∆( M ) . Then: δ ! [ P ( Ch ( β ))] = ( − nr − n i ∗ [ P ( Ch (∆ ∗ ( β )))] . Proof.
Since the stratification {T γ } is transversal to ∆( M ), we have that { ∆ − ( T γ ) } is a Whitney strati-fication of M with respect to which ∆ ∗ ( β ) is a constructible function. Moreover, if T is a normal slice of∆ − ( T γ ) at x then ∆( T ) is a normal slice of T γ at ( x, . . . , x ) . Set N = ∆( T ) . By equations (3) and (7) we have P ( Ch ( β )) = P m γ P (cid:16) T ∗T γ M ( r ) (cid:17) , where m γ := ( − nr − d − χ (cid:0) φ f | N F • | N (cid:1) ( x,...,x ) ,and F • is the bounded complex of sheaves such that χ ( F • ) p = β ( p ) and f : ( M ( r ) , ( x, . . . , x )) → ( C ,
0) is agerm such that d ( x,...,x ) f is a non-degenerate covector at ( x, . . . , x ) with respect to {T γ } .Analogously, P ( Ch (∆ ∗ ( β )) = X γ n γ P (cid:16) T ∗ ∆ − ( T γ ) M (cid:17) , where n γ := ( − n − d − χ (cid:0) φ g | T G • | T (cid:1) x , where G • = ∆ ∗ F • , which is the bounded complex of sheaves suchthat χ ( G • ) q = ∆ ∗ ( β )( q ) , and g : ( M, x ) → ( C ,
0) is a germ such that d x g is a non-degenerate covector at x with respect to { ∆ − ( T γ ) } . Notice that we can take g = ∆ ∗ f since these definitions do not depend on thechoices of g. Notice that ∆ | T : T → N is an isomorphism. Hence φ ∆ ∗ ( f | N ) ∆ ∗ ( F • | N ) ≃ ∆ ∗ (cid:0) φ f | N ( F • | N ) (cid:1) . But clearly φ ∆ ∗ ( f | N ) ∆ ∗ ( F • | N ) = φ g | T G • | T , thus χ (cid:0) φ g | T G • | T (cid:1) x = χ (cid:0) ∆ ∗ (cid:0) φ f | N ( F • | N ) (cid:1)(cid:1) x = χ (cid:0) φ f | N F • | N (cid:1) ( x,...,x ) . Therefore(9) m γ = ( − nr − n n γ . Proposition 1.8 is now an immediate consequence of the next lemma: (cid:3)
ROBERTO CALLEJAS-BEDREGAL, MICHELLE F. Z. MORGADO, AND JOS´E SEADE
Lemma 1.9.
One has: δ ! h P (cid:16) T ∗T γ M ( r ) (cid:17)i = i ∗ h P (cid:16) T ∗ ∆ − ( T γ ) M (cid:17)i . Proof.
Consider the projectivized cotangent bundles P ( T ∗ M ) and P ( T ∗ ( M ( r ) )); we denote by P (( T ∗ M ) ⊕ r )the bundle P ( T ∗ M ⊕ . . . ⊕ T ∗ M ). Notice that one has a fibre square diagram : P (cid:16) ∆ ∗ ( T ∗T γ M ( r ) ) (cid:17) δ ′ / / j ′ (cid:15) (cid:15) P (cid:16) T ∗T γ M ( r ) (cid:17) j (cid:15) (cid:15) P (∆ ∗ T ∗ ( M ( r ) )) δ / / p (cid:15) (cid:15) P ( T ∗ ( M ( r ) )) π ( r ) (cid:15) (cid:15) M ∆ / / M ( r ) where π ( r ) is the natural proper map.Notice that P (∆ ∗ T ∗ ( M ( r ) )) = P (( T ∗ M ) ⊕ r )and P (cid:16) ∆ ∗ ( T ∗T γ M ( r ) ) (cid:17) = P (cid:16) T ∗ ∆ − ( T γ ) M (cid:17) . Thus j ′ is induced by the diagonal embedding i. Notice that ∆ , δ and δ ′ are regular embeddings of codimension nr − n. Hence N P (cid:16) ∆ ∗ ( T ∗T γ M ( r ) ) (cid:17) P (cid:16) T ∗T γ M ( r ) (cid:17) = j ′∗ N P (∆ ∗ T ∗ ( M ( r ) )) P ( T ∗ ( M ( r ) )) . Therefore δ ! h P (cid:16) T ∗T γ M ( r ) (cid:17)i = j ′∗ h P (cid:16) ∆ ∗ ( T ∗T γ M ( r ) ) (cid:17)i = i ∗ h P (cid:16) T ∗ ∆ − ( T γ ) M (cid:17)i . Hence the result follows. (cid:3)
Corollary 1.10.
Let Z ( t ) be as in Proposition 1.5. Assume that Z ( t ) admits a Whitney stratification {T γ } transversal to ∆( M ) . Then: δ ! [ P ( Ch ( Z ( t ) ))] = ( − nr − n i ∗ [ P ( Ch ( Z (∆ ∗ t ) ))] , where ( ) denotes the characteristic function. Remark 1.11. (1) In [11, Theorem 6.2. (a)] it is proved the following: Consider a fiber square diagram X ′ ι ′ / / q (cid:15) (cid:15) Y ′ p (cid:15) (cid:15) X ι / / Y , where ι is a regular embedding of codimension d and p is a proper morphism, then ι ! p ∗ ( α ) = q ∗ ( ι ! ( α )) , for all α ∈ H k ( Y ′ , Z ) . Also in [11, Theorem 6.2. (c)] it is proved that if ι ′ is also a regular embeddingof codimension d, then ι ! ( α ) = ι ′ ! ( α ) , for all α ∈ H k ( Y ′ , Z ) . (2) In [19, Equation (14)] Parusi´nski and Pragacz gave the following description of the Schwartz-MacPherson classes: Let Z be a smooth complex manifold, let V be a closed subvariety of Z and π : P ( T ∗ Z ) → Z be the projectivized cotangent bundle of Z, then the Schwartz-MacPherson classof V is given by c SM ( V ) = ( − dim Z − c ( T Z | V ) ∩ π ∗ (cid:0) c ( O (1)) − ∩ [ P ( Ch ( V ))] (cid:1) , where O (1) is the tautological line bundle of P ( T ∗ Z ) . Theorem 1.12.
With the assumptions of Corollary 1.10 we have: ∆ ! (cid:0) c SM ( Z ( t )) (cid:1) = c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) ∩ c SM ( Z (∆ ∗ t )) . N THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS Proof.
Applying Remark 1.11 (2) to the projectivized cotangent bundle π ( r ) : P ( T ∗ M ( r ) ) → M ( r ) we obtain c SM ( Z ( t )) = ( − nr − c (cid:16) T M ( r ) | Z ( t ) (cid:17) ∩ π ( r ) ∗ (cid:0) c ( O r (1)) − ∩ [ P ( Ch ( Z ( t ) ))] (cid:1) , where O r (1) denotes the tautological line bundle of P ( T ∗ M ( r ) ) . Applying Remark 1.4 (2) we have that(10) ∆ ! c SM ( Z ( t )) = ( − nr − c (cid:16) ∆ ∗ (cid:16) T M ( r ) | Z ( t ) (cid:17)(cid:17) ∩ ∆ ! π ( r ) ∗ (cid:0) c ( O r (1))) − ∩ [ P ( Ch ( Z ( t ) ))] (cid:1) . Applying Remark 1.11 (1) to the fiber square diagram (8) we get that(11) ∆ ! π ( r ) ∗ (cid:0) c ( O r (1))) − ∩ [ P ( Ch ( Z ( t ) ))] (cid:1) = p ∗ (cid:0) δ ! ( c ( O r (1)) − ∩ [ P ( Ch ( Z ( t ) ))]) (cid:1) . Applying again Remark 1.4 (2) we have,(12) ∆ ! π ( r ) ∗ (cid:0) c ( O r (1))) − ∩ [ P ( Ch ( Z ( t ) ))] (cid:1) = p ∗ (cid:0) c ( δ ∗ O r (1))) − ∩ δ ! [ P ( Ch ( Z ( t ) ))] (cid:1) . Since δ ∗ O r (1) = O P (( T ∗ M ) ⊕ r ) (1) is the tautological line bundle on the projectivization P (( T ∗ M ) ⊕ r ) → M ,by Corollary 1.10 and the equations (10), (11) and (12), we get:∆ ! (cid:0) c SM ( Z ( t )) (cid:1) = ( − n − c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r (cid:17) ∩∩ p ∗ (cid:0) c ( O P (( T ∗ M ) ⊕ r ) (1)) − ∩ i ∗ [ P ( Ch ( Z (∆ ∗ t ) ))] (cid:1) . Hence, by the projection formula for proper morphism (see [11, Theorem 3.2 (c)]) we have that∆ ! (cid:0) c SM ( Z ( t )) (cid:1) = ( − n − c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) c (cid:0)(cid:0) T M | Z (∆ ∗ t ) (cid:1)(cid:1) ∩∩ ( p ◦ i ) ∗ (cid:0) c ( i ∗ O P (( T ∗ M ) ⊕ r ) (1)) − ∩ [ P ( Ch ( Z (∆ ∗ t ) ))] (cid:1) . Now, using the fact that i ∗ O P (( T ∗ M ) ⊕ r ) (1) = O P (( T ∗ M )) (1) and that p ◦ i = q : P ( T ∗ M ) → M is theprojectivized cotangent morphism we have that∆ ! (cid:0) c SM ( Z ( t )) (cid:1) = ( − n − c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) c (cid:0)(cid:0) T M | Z (∆ ∗ t ) (cid:1)(cid:1) ∩∩ q ∗ (cid:0) c ( O P (( T ∗ M )) (1)) − ∩ [ P ( Ch ( Z (∆ ∗ t ) ))] (cid:1) . Applying Remark 1.11 (2) to the projectivize cotangent bundle q : P ( T ∗ M ) → M we obtain that c SM ( Z (∆ ∗ t )) = ( − n − c (cid:0)(cid:0) T M | Z (∆ ∗ t ) (cid:1)(cid:1) ∩ q ∗ (cid:0) c ( O P (( T ∗ M )) (1)) − ∩ [ P ( Ch ( Z (∆ ∗ t ) ))] (cid:1) Hence we have that ∆ ! (cid:0) c SM ( Z ( t )) (cid:1) = c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) ∩ c SM ( Z (∆ ∗ t )) . (cid:3) Theorem 1.12 is a Verdier-Riemann-Roch type formula for the Schwarz-MacPherson classes (cf. [20]).Analogously, the next result is a Verdier-Riemann-Roch type theorem for the Milnor classes. The proof is astraightforward application of Proposition 1.5 and Theorem 1.12.
Corollary 1.13.
With the assumptions of Corollary 1.10 we have: ∆ ! M ( Z ( t )) = ( − nr − n c (cid:16)(cid:0) T M | Z (∆ ∗ t ) (cid:1) ⊕ r − (cid:17) ∩ M ( Z (∆ ∗ t )) . Intersection product formulas
As before, let M be an n -dimensional compact complex analytic manifold. Let { E i } be a finite collectionof holomorphic vector bundles over M of rank d i , 1 ≤ i ≤ r . For each of these bundles, let s i be a regularholomorphic section and X i the ( n − d i )-dimensional local complete intersections defined by the zeroes of s i .In this section we assume that we can equip the product X × . . . × X r with a Whitney stratificationsuch that the diagonal embedding ∆ is transversal to all strata. This transversality condition is necessaryfor using Proposition 1.8 and this is precisely the transversality condition that we need in Theorem 1.Let p i : M ( r ) → M be the i th -projection, then we have the holomorphic exterior product section s = s ⊕ . . . ⊕ s r : M ( r ) → p ∗ E ⊕ . . . ⊕ p ∗ r E r , given by s ( x , . . . , x r ) = ( s ( x ) , . . . , s r ( x r )) . Then Z ( s ) = X × . . . × X r and Z (∆ ∗ ( s )) = X ∩ . . . ∩ X r . Set X = Z (∆ ∗ ( s )) . The next result describes the total Schwartz-MacPherson class of X in terms of the totalSchwartz-MacPherson classes of the X i . ROBERTO CALLEJAS-BEDREGAL, MICHELLE F. Z. MORGADO, AND JOS´E SEADE
Proposition 2.1. c SM ( X ) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:16) c SM ( X ) · . . . · c SM ( X r ) (cid:17) . Proof.
By Theorem 1.12 we have that∆ ! (cid:0) c SM ( Z ( s )) (cid:1) = c (cid:16)(cid:0) T M | Z (∆ ∗ s ) (cid:1) ⊕ r − (cid:17) ∩ c SM ( Z (∆ ∗ s )) . Hence c SM ( X )) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ ∆ ! (cid:0) c SM ( X × . . . × X r ) (cid:1) . Now, M. Kwieci´nski proved in [13] that Schwartz-MacPherson classes behave well with respect to the exteriorproducts, that is c SM ( X × . . . × X r ) = c SM ( X ) × . . . × c SM ( X r ) . Hence ∆ ! (cid:0) c SM ( X × . . . × X r ) (cid:1) = c SM ( X ) · . . . · c SM ( X r )and the result follows. (cid:3) Remark 2.2.
In [11, Example 3.2.8.] it is proved the following: Let Y and Z be schemes, p and q theprojections from Y × Z to Y and Z, E and F vector bundles on Y and Z, α ∈ H ∗ ( Y, Z ) and β ∈ H ∗ ( Z, Z ) . Then ( c ( E ) ∩ α ) × β = c ( p ∗ E ) ∩ ( α × β )and ( c ( E ) ∩ α ) × ( c ( F ) ∩ β ) = c ( p ∗ E ⊕ q ∗ F ) ∩ ( α × β ) . Since c ( p ∗ E ) ∩ (( c ( E ) − ∩ α ) × β ) = α × β we have that( c ( E ) − ∩ α ) × β = c ( p ∗ E ) − ∩ ( α × β ) . Analogously, (cid:0) c ( E ) − ∩ α (cid:1) × (cid:0) c ( F ) − ∩ β (cid:1) = c ( p ∗ E ⊕ q ∗ F ) − ∩ ( α × β ) . In [17] was stated without proof that Fulton-Johnson classes behave well with respect to the exteriorproducts. For completeness we include it proof here
Lemma 2.3. c F J ( X × . . . × X r ) = c F J ( X ) × . . . × c F J ( X r ) . Proof. c F J ( X ) = c (cid:0) T M ( r ) | X × ... × X r (cid:1) c ( p ∗ E ⊕ . . . ⊕ p ∗ r E r ) − ∩ [ X × . . . × X r ]= c ( p ∗ T M | X ⊕ . . . ⊕ p ∗ r T M | X r ) c ( p ∗ E ⊕ . . . ⊕ p ∗ r E r ) − ∩ ([ X ] × . . . × [ X r ])= (cid:0) c ( T M | X ) c ( E ) − ∩ [ X ] (cid:1) × . . . × (cid:0) c ( T M | X r ) c ( E r ) − ∩ [ X r ] (cid:1) = c F J ( X ) × . . . × c F J ( X r ) . where the third equality follows by Remark 2.2. (cid:3) Proposition 2.4. c F J ( X ) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:16) c F J ( X ) · . . . · c F J ( X r ) (cid:17) . Proof.
By Proposition 1.5 we have that∆ ! (cid:0) c F J ( Z ( s )) (cid:1) = c (cid:16)(cid:0) T M | Z (∆ ∗ s ) (cid:1) ⊕ r − (cid:17) ∩ c F J ( Z (∆ ∗ s )) . Hence c F J ( X )) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ ∆ ! (cid:0) c F J ( X × . . . × X r ) (cid:1) . By Lemma 2.3 we have that c F J ( X )) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ ∆ ! (cid:0) c F J ( X ) × . . . × c F J ( X r ) (cid:1) . N THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS Since ∆ ! (cid:0) c F J ( X ) × . . . × c F J ( X r ) (cid:1) = c F J ( X ) · . . . · c F J ( X r )the result follows. (cid:3) Proof of Theorem 1:
This follows immediately from Proposition 2.1 and Proposition 2.4. (cid:3)
Example 2.5.
Let Z and Z be the hypersurfaces of P defined by H ( x , . . . , x ) = x x and G ( x , . . . , x ) = x . The line bundle of Z is O (2 H ), where H = c ( O (1)), so the class of the virtual tangent bundle of Z is:(1 + H ) H/ (1 + 2 H ) = 2 H + 6 H + 8 H + 4 H , while the Schwartz-MacPherson class is, by the inclusion-exclusion formula in [2]:2 c ( T P ) − c ( T P ) = 2((1 + H ) H ) − (1 + H ) H = 2 H + 7 H + 9 H + 5 H Therefore the Milnor class of Z is H + H + H . On the other hand, since Z is smooth, the Schwartz-MacPherson class and the Fulton-Johnson class of Z are (1 + H ) H = H + 4 H + 6 H + 4 H . Therefore,by Theorem 1, the Milnor class of Z ∩ Z is given by M ( Z ∩ Z ) = − c ( T P ) − ∩ c SM ( Z ) M ( Z ) = − H . Remark 2.6.
Take the complete intersection X = X ∩ X , where X is a smooth quadric surface in P and X is a tangent plane get two distinct lines meeting at a point. The Milnor class of X is simply the class ofa point, but the Milnor classes of X and X are both zero because they both are smooth. This shows thata transversality condition is necessary for our formula in 3.1.3. Applications to line bundles
Theorem 3.1.
With the conditions of Theorem 1 we have: M ( X ) = ( − nr − n c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ X ( − ( n − d ) ǫ + ... +( n − d r ) ǫ r P · . . . · P r ∈ H ∗ ( X ) , where the sum runs over all choices of P i ∈ (cid:8) M ( X i ) , c SM ( X i ) (cid:9) , i = 1 , . . . , r, except ( P , . . . , P r ) =( c SM ( X ) , . . . , c SM ( X r )) and where ǫ i = (cid:26) , if P i = c SM ( X i )0 , if P i = M ( X i ) . Proof.
By Corollary 1.13,∆ ! M ( Z ( s )) = ( − nr − n c (cid:16)(cid:0) T M | Z (∆ ∗ s ) (cid:1) ⊕ r − (cid:17) ∩ M ( Z (∆ ∗ s )) . Thus, M ( X ) = ( − nr − n c (cid:16)(cid:0) T M | Z (∆ ∗ s ) (cid:1) ⊕ r − (cid:17) − ∩ ∆ ! M ( X × . . . × X r ) , and using the description of the Milnor classes of a product due to [17, Corollary 3.1], we have: M ( X ) = ( − nr − n c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ X ( − ( n − d ) ǫ + ... +( n − d r ) ǫ r ∆ ! ( P × . . . × P r ) , where the sum runs over all choices of P i ∈ (cid:8) M ( X i ) , c SM ( X i ) (cid:9) , i = 1 , . . . , r, except ( P , . . . , P r ) =( c SM ( X ) , . . . , c SM ( X r )) and where ǫ i = (cid:26) , if P i = c SM ( X i )0 , if P i = M ( X i ) . The result follows because ∆ ! ( P × . . . × P r ) = P · . . . · P r ∈ H ∗ ( X ) . (cid:3) Corollary 3.2. M ( X ) = c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ r X i =1 ( − D − d i a ,i · ... · a r − ,i · M ( X i ) , where D = r X j =1 d j and a j,i = (cid:26) c SM ( X j +1 ) if i ≤ jc F J ( X j ) if i > j . From now on we replace the bundles E i by line bundles L i . Aluffi type formula.
Let X = X ∩ . . . ∩ X r be as above. The µ -classes were introduced by P. Aluffiin [1]. For each X i , the Aluffi’s µ -class of the singular locus is defined by the formula µ L i (Sing( X i )) = c ( T ∗ M ⊗ L i ) ∩ s (Sing( X i ) , M ) , where s (Sing( X i ) , M ) is the Segre class of Sing( X i ) in M (see [11, Chapter 4]). Given a cycle α ∈ H ∗ ( X i , Z )and α = P j ≥ α j , where α j is the codimension j component of α, then Aluffi introduced the following cycles α ∨ := X j ≥ ( − j α j and α ⊗ L i := X j ≥ α j c ( L i ) j . Then Aluffi proved in [1] that the total Milnor class M ( X i ) can be described as follows:(13) M ( X i ) = ( − n − c ( L i ) n − ∩ ( µ L i (Sing( X i )) ∨ ⊗ L i ) . Again using Corollary 3.2, the above equation yields:
Corollary 3.3.
The Total Milnor class of X := X ∩ . . . ∩ X r is: M ( X ) = ( − n − c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ r X i =1 ( − r − a ,i · . . . · a r − ,i · c ( L i ) n − ∩ ( µ L i (Sing( X i ) ) ∨ ⊗ L i ) ! , where a j,i = (cid:26) c SM ( X j +1 ) if i ≤ jc F J ( X j ) if i > j . Parusi´nski-Pragacz-type formula.
We now assume each X i has a Whitney stratification S i . Onehas in [19] the following characterization of the Milnor classes of hypersurfaces in compact manifolds:(14) M ( X i ) := X S ∈S i γ S (cid:16) c ( L i | Xi ) − ∩ c SM ( S ) (cid:17) ∈ H ∗ ( X i ) , where γ S is the function defined on each stratum S as follows: for each x ∈ S ⊂ X i , let F x be a local Milnorfibre , and let χ ( F x ) be its Euler characteristic. We set: µ ( x ; X i ) := ( − n ( χ ( F x ) − , and call it the local Milnor number . This number is constant on each Whitney stratum, so we denote it µ S .Then γ S is defined inductively by: γ S = µ S − X S ′ = S, S ′ ⊃ S γ S ′ . Lemma 3.4.
Let Y and Z be subschemes of M , W = Y ∩ Z, E a vector bundle on
Y, α ∈ H ∗ ( Y, Z ) and β ∈ H ∗ ( Z, Z ) . Then ( c ( E ) ∩ α ) · β = c ( E | W ) ∩ ( α · β ) . Proof.
Let p be the projections from Y × Z to Y and d : W → Y × Z be the diagonal embedding.( c ( E ) ∩ α ) · β = d ! (( c ( E ) ∩ α ) × β )= d ! ( c ( p ∗ E ) ∩ ( α × β ))= c ( d ∗ p ∗ E ) ∩ d ! ( α × β )= c ( E | W ) ∩ ( α · β ) . where the second and third equalities follows by Remark 2.2 and Remark 1.4 (2) respectively. (cid:3) Corollary 3.5 (Parusi´nski-Pragacz formula for local complete intersections) . We have: M ( X ) = ( − nr − n c (cid:16) ( T M | X ) ⊕ r − (cid:17) − ∩ (cid:18) X α ǫ ,...,ǫ r S ,...,S r c ( L ) ǫ · . . . · c ( L r ) ǫ r c ( L ⊕ . . . ⊕ L r ) ∩ c SM ( S ) · . . . · c SM ( S r ) (cid:19) , where the sum runs over all possible choices of the strata provided ( S , . . . , S r ) = (( X ) reg , . . . , ( X r ) reg ) , α ǫ ,...,ǫ r S ,...,S r = ( − ( n − ǫ + ... + ǫ r ) γ − ǫ S · . . . · γ − ǫ r S r , and ǫ i = (cid:26) , if S i ⊆ ( X i ) reg , if dim( S i ) < n − . N THE CHERN CLASSES OF SINGULAR COMPLETE INTERSECTIONS Proof.
The proof will be by induction on r. For r = 1 this is Parusi´nski-Pragacz formula given in equation(14) for X . Let Y = X ∩ . . . ∩ X r − . Then dim Y = n − ( r −
1) and X = Y ∩ X r . Hence M ( X ) = M ( Y ∩ X r ) . By Theorem 3.1 we have that M ( Y ∩ X r ) = ( − n c (( T M | X )) − ∩ (cid:16) M ( Y ) ·M ( X r )+( − dim Y c SM ( Y ) ·M ( X r )+( − n − M ( Y ) · c SM ( X r ) (cid:17) . By induction hypotheses, equation (14) for X r , Proposition 2.1 and Lemma 3.4 we have that M ( X ) = ( − n c (cid:16) ( TM | X ) ⊕ r − (cid:17) − ∩ h ( − nr (cid:18) X S = X α ǫ ,...,ǫr − S ,...,Sr − c ( L ) ǫ · . . . · c ( L r − ) ǫr − c ( L ⊕ . . . ⊕ L r − ) ∩ r − Y i =1 c SM ( S i ) (cid:19) · (cid:18) X Sr γ Sr (cid:16) c ( L r | Xr ) − ∩ c SM ( S r ) (cid:17) (cid:19) +( − n − r +1 (cid:16)Q r − i =1 c SM ( S i ) (cid:17) · (cid:18) P Sr γ Sr (cid:16) c ( L r | Xr ) − ∩ c SM ( S r ) (cid:17) (cid:19) +( − n − nr (cid:18) X S = X α ǫ ,...,ǫr − S ,...,Sr − c ( L ) ǫ · . . . · c ( L r − ) ǫr − c ( L ⊕ . . . ⊕ L r − ) ∩ r − Y i =1 c SM ( S i ) (cid:19) · c SM ( X r ) i where S = X means that ( S , . . . , S r − ) = (( X ) reg , . . . , ( X r − ) reg ) . Notice that γ S r = 0 if S r ⊆ ( X r ) reg and that α ǫ ,...,ǫ r S ,...,S r = ( − n − α ǫ ,...,ǫ r − S ,...,S r − , if S r ⊆ ( X r ) reg α ǫ ,...,ǫ r − S ,...,S r − · γ S r , if dim( S r ) < n − . Hence M ( X ) = ( − nr − n c (cid:16) ( TM | X ) ⊕ r − (cid:17) − ∩ h X S = X and Sr = Xr α ǫ ,...,ǫrS ,...,Sr c ( L ) ǫ · . . . · c ( L r ) ǫr c ( L ⊕ . . . ⊕ L r ) ∩ r Y i =1 c SM ( S i )+ X S = X and Sr = Xr α ǫ ,...,ǫrS ,...,Sr c ( L ) ǫ · . . . · c ( L r ) ǫr c ( L ⊕ . . . ⊕ L r ) ∩ r Y i =1 c SM ( S i )+ X S = X and Sr = Xr α ǫ ,...,ǫrS ,...,Sr c ( L ) ǫ · . . . · c ( L r ) ǫr c ( L ⊕ . . . ⊕ L r ) ∩ r Y i =1 c SM ( S i ) i . Now the result follows straightforwardly. (cid:3)
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Centro de Ciˆencias Exatas e da Natureza, Universidade Federal da Para´ıba-UFPb, Jo˜ao Pessoa, PB - Brasil.
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