Nearby cycles and characteristic classes of singular spaces
aa r X i v : . [ m a t h . AG ] M a y NEARBY CYCLES AND CHARACTERISTIC CLASSES OFSINGULAR SPACES
J ¨ORG SCH ¨URMANN
Abstract.
In this paper we give an introduction to our recent work oncharacteristic classes of complex hypersurfaces based on some talks givenat conferences in Strasbourg, Oberwolfach and Kagoshima. We explainthe relation between nearby cycles for constructible functions or sheavesas well as for (relative) Grothendieck groups of algebraic varieties andmixed Hodge modules, and the specialization of characteristic classes ofsingular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences betweenthe corresponding virtual and functorial characteristic classes of complexhypersurfaces in terms of vanishing cycles related to the singularities ofthe hypersurface.
A natural problem in complex geometry is the relation between invariantsof a singular complex hypersurface X (like Euler characteristic and Hodgenumbers) and the geometry of the singularities of the hypersurface (like thelocal Milnor fibrations). For the Euler characteristic this is for example aspecial case of the difference between the Fulton- and MacPherson- Chernclasses of X , whose differences are the now well studied Milnor classes of X ([1, 6, 7, 8, 27, 29, 34, 35, 45]). Their degrees are related to Donaldson-Thomas invariants of the singular locus ([3]).A very powerful approach to this type of questions is by the theory ofthe nearby and vanishing cycle functors. For example a classical result ofVerdier [44] says that the MacPherson Chern class transformation [25, 22]commutes with specialization, which for constructible functions means thecorresponding nearby cycles. Here we explain the corresponding result forour motivic Chern- and Hirzebruch class transformations as introduced inour joint work with J.-P. Brasselet and S. Yokura [5], i.e. they also com-mute with specialization defined in terms of nearby cycles. Here one canwork either in the motivic context with relative Grothendieck group of vari-eties [4, 19], or in the Hodge context with Grothendieck groups of M. Saito’smixed Hodge modules [30, 31]. The key underlying specialization result [36]is about the filtered de Rham complex of the underlying filtered D -modulein terms of the Malgrange-Kashiwara V -filtration. But here we focus on thegeometric motivations and applications as given in our joint work with S.E.Cappell, L. Maxim and J.L. Shaneson [13]. In this paper we work (for simplicity) in the complex algebraic context,since this allows us to switch easily between an algebraic geometric languageand an underlying topological picture. Many results are also true in the com-plex analytic or algebraic context over base field of characteristic zero. Firstwe introduce the virtual characteristic classes and numbers of hypersurfacesand local complete intersetions in smooth ambient manifolds. Next we recallsome of the theories of functorial characteristic classes for singular spaces[25, 2, 10, 5, 37]. Finally we explain the relation to nearby and vanishingcycles following our earlier results [34, 35] about different Chern classes forsingular spaces.
Acknowledgements:
This paper is an extended version of some talks givenat conferences in Strasbourg, Oberwolfach and Kagoshima. Here I wouldlike to thank the organizers for the invitation to these conferences. I alsowould like to thank Sylvain Cappell, Laurentiu Maxim and Shoji Yokura forthe discussions on our joint work related to the subject of this paper.
Contents
1. Virtual classes of local complete intersections 22. Functorial characteristic classes of singular spaces 73. Nearby and vanishing cycles 12References 221.
Virtual classes of local complete intersections
Recall that we are working in the complex algebraic context. A charac-teristic class cl ∗ of (complex algebraic) vector bundles over X is a map cl ∗ : V ect ( X ) → H ∗ ( X ) ⊗ R from the set V ect ( X ) of isomorphism classes of complex algebraic vectorbundles over X to some cohomology theory H ∗ ( X ) ⊗ R with a coefficientring R , which is compatible with pullbacks. Here we use as a cohomologytheory H ∗ ( X ) = H ∗ ( X, Z ) , the usual cohomology in even degrees. CH ∗ ( X ) , the operational Chow cohomology of [16]. K ( X ) , the Grothendieck group of vector bundles.We also assume that cl ∗ is multiplicative , i.e. cl ∗ ( V ) = cl ∗ ( V ′ ) ∪ cl ∗ ( V ′′ )for any short exact sequence0 → V ′ → V → V ′′ → EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 3 of vector bundles on X , with ∪ given by the cup- or tensor-product. Such acharacteristic class cl ∗ corresponds by the “splitting principle” to a uniqueformal power series f ( z ) ∈ R [[ z ]] with cl ∗ ( L ) = f ( c ( L )) for any line bundle L on X . Here c ( L ) ∈ H ( X ) is the nilpotent first Chern class of L , whichin the case H ∗ ( X ) = K ( X ) is given by c ( L ) := 1 − [ L ∨ ] ∈ K ( X ) (with( · ) ∨ the dual bundle). Finally cl ∗ should be stable in the sense that f (0) ∈ R is a unit so that cl ∗ induces a functorial group homomorphism cl ∗ : (cid:0) K ( X ) , ⊕ (cid:1) → ( H ∗ ( X ) ⊗ R, ∪ ) . Let us now switch to smooth manifolds, which will be an important in-termediate step on the way to characteristic classes of singular spaces. Fora complex algebraic manifold M its tangent bundle T M is available and acharacteristic class cl ∗ ( T M ) of the tangent bundle
T M is called a character-istic cohomology class cl ∗ ( M ) of the manifold M . We also use the notation cl ∗ ( M ) := cl ∗ ( T M ) ∩ [ M ] ∈ H ∗ ( M ) ⊗ R for the corresponding characteristic homology class of the manifold M , with[ M ] ∈ H ∗ ( M ) the fundamental class (or the class of the structure sheaf) in H ∗ ( M ) := H BM ∗ ( M ) , the Borel-Moore homology in even degrees. CH ∗ ( M ) , the Chow group. G ( M ) , the Grothendieck group of coherent sheaves.If M is moreover compact, i.e. the constant map k : M → { pt } is proper,one gets the corresponding characteristic number (1.1) ♯ ( M ) := k ∗ ( cl ∗ ( M )) =: deg ( cl ∗ ( M )) ∈ R .
Example 1.1 (Hirzebruch ’54) . The famous Hirzebruch χ y -genus is thecharacteristic number whose associated characteristic class can be given intwo versions (see [20] ): (1) The cohomological version, with R = Q [ y ] , is given by the Hirzebruchclass cl ∗ = T ∗ y corresponding to the normalized power series f ( z ) := Q y ( z ) := z (1 + y )1 − e − z (1+ y ) − zy ∈ Q [ y ][[ z ]] . (2) The K -theoretical version, with R = Z [ y ] , is given by the dual totalLambda-class cl ∗ = Λ ∨ y , with Λ ∨ y ( · ) := Λ y (cid:0) ( · ) ∨ (cid:1) = X i ≥ (cid:2) Λ i (cid:0) ( · ) ∨ (cid:1)(cid:3) · y i corresponding to the unnormalized power series f ( z ) = 1 + y − yz ∈ Z [ y ][[ z ]] . J. SCH ¨URMANN
So the χ y -genus of the compact complex algebraic manifold M is givenby χ y ( M ) := X p ≥ χ ( M, Λ p T ∗ M ) · y p = X p ≥ X i ≥ ( − i dim C H i ( M, Λ p T ∗ M ) · y p , with T ∗ M the algebraic cotangent bundle of M . The equality(gHRR) χ y ( M ) = deg (cid:0) T ∗ y ( T M ) ∩ [ M ] (cid:1) ∈ Q [ y ] , is (called) the generalized Hirzebruch Riemann-Roch theorem [20]. The cor-responding power series Q y ( z ) (as above) specializes to Q y ( z ) = z for y = − z − e − z for y = 0, z tanh z for y = 1.Therefore the Hirzebruch class T ∗ y ( T M ) unifies the following important(total) characteristic cohomology classes of
T M :(1.2) T ∗ y ( T M ) = c ∗ ( T M ) the
Chern class for y = − td ∗ ( T M ) the
Todd class for y = 0, L ∗ ( T M ) the
Thom-Hirzebruch L-class for y = 1.The gHHR-theorem specializes to the calculation of the following impor-tant invariants: χ − ( M ) = e ( M ) = deg ( c ∗ ( T M ) ∩ [ M ]) the Euler characteristic , χ ( M ) = χ ( M ) = deg ( td ∗ ( T M ) ∩ [ M ]) the arithmetic genus , χ ( M ) = sign ( M ) = deg ( L ∗ ( T M ) ∩ [ M ]) the signature ,(1.3)which are, respectively, the Poincar´e-Hopf or Gauss-Bonnet theorem , the
Hirzebruch Riemann-Roch theorem and the
Hirzebruch signature theorem .If X is a singular complex algebraic variety, then the algebraic tangentbundle of X doesn’t exist so that a characteristic (co)homology class of X can’t be defined as before. But if X can be realized as a local completeintersection inside a complex algebraic manifold M , then a substitute for T X is available. Indeed this just means that the closed inclusion i : X → M is a regular embedding into the smooth algebraic manifold M , so that the normal cone N X M → X is an algebraic vector bundle over X (compare[16]). Then the virtual tangent bundle of X (1.4) T vir X := [ i ∗ T M − N X M ] ∈ K ( X ) , is independent of the embedding in M (e.g., see [16][Ex.4.2.6]), so it is awell-defined element in the Grothendieck group of vector bundles on X . Of EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 5 course T vir X = [ T X ] ∈ K ( X )in case X is a smooth algebraic submanifold.If cl ∗ : K ( X ) → H ∗ ( X ) ⊗ R denotes a characteristic cohomology classas before, then one can associate to X an intrinsic homology class (i.e.,independent of the embedding X ֒ → M ) defined as:(1.5) cl vir ∗ ( X ) := cl ∗ ( T vir X ) ∩ [ X ] ∈ H ∗ ( X ) ⊗ R .
Here [ X ] ∈ H ∗ ( X ) is again the fundamental class (or the class of the struc-ture sheaf) of X in H ∗ ( X ) := H BM ∗ ( X ) , the Borel-Moore homology in even degrees. CH ∗ ( X ) , the Chow group. G ( X ) , the Grothendieck group of coherent sheaves.Here ∩ in the K -theoretical context comes from the tensor product with thecoherent locally free sheaf of sections of the vector bundle. Moreover, forthe class cl ∗ = Λ ∨ y one has to take R := Z [ y, (1 + y ) − ] to make it a stablecharacteristic class defined on K ( X ).Let i : X → M be a regular embedding of (locally constant) codimension r between possible singular complex algebraic varieties. Using the famous deformation to the normal cone , one gets functorial Gysin homomorphisms (compare [16, 43, 44])(1.6) i ! : H ∗ ( M ) → H ∗− r ( X )and(1.7) i ! : G ( M ) → G ( X ) . Note that i is of finite tor-dimension, so that the last i ! can also be describedas i ! = Li ∗ : G ( M ) ≃ K ( D bcoh ( M )) → K ( D bcoh ( X )) ≃ G ( X )coming from the derived pullback Li ∗ between the bounded derived cate-gories with coherent cohomology sheaves. If M is also smooth , then one getseasily the following important relation between the virtual characteristicclasses cl vir ∗ ( X ) of X and the Gysin homomorphisms:(1.8) i ! ( cl ∗ ( M )) = i ! ( cl ∗ ( T M ) ∩ [ M ]) = cl ∗ ( N X M ) ∩ cl vir ∗ ( X ) . From now on we assume that X = { f = 0 } = { f i = 0 | i = 1 , . . . , n } J. SCH ¨URMANN is a global complete intersection in the complex algebraic manifold M comingfrom a cartesian diagram(1.9) { f = 0 } X i −−−−→ M f y f = y ( f ,...,f n ) { } i −−−−→ C n . Then N X M ≃ f ∗ (cid:0) N { } C n (cid:1) = X × C n is a trivial vector bundle of rank n on X so that(1.10) cl ∗ ( N X M ) = ( cl ∗ = T ∗ y , c ∗ , td ∗ or L ∗ .(1 + y ) n for cl ∗ = Λ ∨ y .Assume now that f is proper so that X is compact. Since the Gysinhomomorphisms i ! commute with proper pushdown (compare [16, 43, 44]),one gets by the projection formula ♯ vir ( X ) := f ∗ (cid:0) cl vir ∗ ( X ) (cid:1) = f ∗ (cid:16) cl ∗ ( N X M ) − ∩ i ! cl ∗ ( M ) (cid:17) = cl ∗ ( N { } C n ) − ∩ i !0 ( f ∗ cl ∗ ( M )) . Taking a (small) regular value 0 = t ∈ C n , in the same way from thecartesian diagram(1.11) { f = 0 } X i −−−−→ M i ′ ←−−−− X t { f = t } f y f y y f t { } i −−−−→ C n i t ←−−−− { t } for the “nearby” smooth submanifold X t = { f = t } , one gets the equality ♯ ( X t ) := f t ∗ ( cl ∗ ( X t )) = cl ∗ ( N { t } C n ) − ∩ i ! t ( f ∗ cl ∗ ( M )) . Note that the set of critical values of f is a proper algebraic subset of C n ,as can be seen by “generic smoothness” or from an adapted stratificationof the proper algebraic map f . Now N { } C n ≃ C n ≃ N { t } C n and the smooth pullback π ∗ for the (vector bundle) projection π : C n → { pt } is anisomorphism π ∗ : R = H ∗ ( { pt } ) ⊗ R ≃ H ∗ + n ( C n ) ⊗ R with inverse i !0 and i ! t (see [16, 43, 44]), so that the “virtual characteristicnumber”(1.12) ♯ vir ( X ) := f ∗ (cid:0) cl vir ∗ ( X ) (cid:1) = ♯ ( X t ) ∈ R is the corresponding characteristic number of a “nearby” smooth fiber X t . EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 7 Functorial characteristic classes of singular spaces
For a more general singular complex algebraic variety X its “virtual tan-gent bundle” is not available any longer, so characteristic classes for singularvarieties have to be defined in a different way. For an introduction to thissubject compare with our survey paper [38] (and see also [37, 46]). Thetheory of characteristic classes of vector bundles is a natural transformationof contravariant functorial theories. This naturality is an important guidefor developing various theories of characteristic classes for singular varieties.Almost all known characteristic classes for singular spaces are formulatedas natural transformations cl ∗ : A ( X ) → H ∗ ( X ) ⊗ R of covariant functorial theories. Here A is a suitable theory (depending onthe choice of cl ∗ ), which is covariant functorial for proper algebraic mor-phisms.There is always a distinguished element I X ∈ A ( X ) such that the corre-sponding characteristic class of the singular space X is defined as cl ∗ ( X ) := cl ∗ (I X ) . Finally one has the normalization cl ∗ (I M ) = cl ∗ ( T M ) ∩ [ M ] ∈ H ∗ ( M ) ⊗ R for M a smooth manifold, with cl ∗ ( T M ) the corresponding characteristiccohomology class of M . This justifies the notation cl ∗ for this homologyclass transformation, which should be seen as a homology class version ofthe following characteristic number of the singular space X : ♯ ( X ) := cl ∗ ( k ∗ I X ) = deg ( cl ∗ (I X )) ∈ H ∗ ( { pt } ) ⊗ R ≃ R , with k : X → { pt } a constant map. Note that the normalization impliesthat for M smooth: ♯ ( M ) = deg ( cl ∗ ( M )) = deg ( cl ∗ ( T M ) ∩ [ M ])so that this is consistent with the notion of characteristic number of thesmooth manifold M as used before.But only few characteristic numbers and classes have been extended inthis way to singular spaces. For example the three characteristic numbers(1.3) and classes (1.2) have been generalized to a singular complex algebraicvariety X in the following way (where the characteristic numbers are onlydefined for X compact):( y = − e ( X ) = deg ( c ∗ ( X )) , with c ∗ : F ( X ) → H ∗ ( X )the Chern class transformation of MacPherson [25, 22] from the abeliangroup F ( X ) of complex algebraically constructible functions to homology, J. SCH ¨URMANN where one can use the Chow group CH ∗ ( · ) or the Borel-Moore homologygroup H BM ∗ ( · , Z ) (in even degrees). Here e ( X ) is the (topological) Eulercharacteristic of X , and the distinguished element I X := 1 X ∈ F ( X ) issimply given by the characteristic function of X . Then c ∗ ( X ) := c ∗ (1 X )agrees by [9] via “Alexander duality” for compact X embeddable into acomplex manifold with the Schwartz class of X as introduced before byM.-H. Schwartz [39].( y = 0) χ ( X ) = deg ( td ∗ ( X )) , with td ∗ : G ( X ) → H ∗ ( X ) ⊗ Q the Todd transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson [2] (for Borel-Moore homology) or Fulton [16] (for Chowgroups). Here G ( X ) is the Grothendieck group of coherent sheaves, with χ ( X ) the arithmetic genus (or holomorphic Euler characteristic) of X . Then td ∗ ( X ) := td ∗ ([ O X ]), with the distinguished element I X := [ O X ] the class ofthe structure sheaf.Finally for compact X one also has( y = 1) sign ( X ) = deg ( L ∗ ( X )) , with L ∗ : Ω( X ) → H ∗ ( X, Q )the homology L -class transformation of Cappell-Shaneson [10] as formulatedin [5]. Here Ω( X ) is the abelian group of cobordism classes of selfdual con-structible complexes. Then L ∗ ( X ) := L ∗ ([ I C X ]) is the homology L-class ofGoresky-MacPherson [18], with the distinguished element I X := [ I C X ] theclass of their intersection cohomology complex. So sign ( X ) is the intersec-tion cohomology signature of X . For a rational PL-homology manifold X ,these L -classes are due to Thom [42].So all these theories have the same formalism, but they are defined oncompletely different theories. Nevertheless, it is natural to ask for anothertheory of characteristic homology classes of singular complex algebraic va-rieties, which unifies the above characteristic homology class transforma-tions. Of course in the smooth case, this is done by the Hirzebruch class T ∗ y ( T M ) ∩ [ M ] of the tangent bundle. An answer to this question was givenin [5] (together with some improvements in [37]). Using Saito’s deep theoryof algebraic mixed Hodge modules [30, 31], we introduced in [5] the motivicChern class transformations as natural transformations (commuting withproper push down) fitting into a commutative diagram: G ( X )[ y ] −−−−→ G ( X )[ y, y − ] G ( X )[ y, y − ] mC y x mC y x x MHC y K ( var/X ) −−−−→ M ( var/X ) χ Hdg −−−−→ K ( M HM ( X )) . Here K ( M HM ( X )) is the Grothendieck group of algebraic mixed Hodgemodules on X , and K ( var/X ) (resp. M ( var/X ) := K ( var/X )[ L − ]) is EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 9 the (localization of the) relative Grothendieck group of complex algebraicvarieties over X (with respect to the class of the affine line L , compare e.g.[4, 19]). The distinguished element is given by the constant Hodge module(complex) resp. by the class of the identity arrowI X := [ Q HX ] ∈ K ( M HM ( X )) resp. I X := [ id X ] ∈ K ( var/X ) , and the canonical “Hodge realization” homomorphism χ Hdg is given by(2.1) χ Hdg : K ( var/X ) → K ( M HM ( X )); [ f : Y → X ] [ f ! Q HY ] . The motivic Chern class transformations mC y , M HC y capture informa-tion about the filtered de Rham complex of the filtered D -module underlyinga mixed Hodge module. The corresponding characteristic class of the space X , mC y ( X ) = M HC y ( X ) ∈ G ( X )[ y ] , can also be defined with the help of the (filtered) Du Bois complex of X [15],and satisfies for M smooth the normalization condition(2.2) mC y ( M ) = M HC y ( M ) = Λ ∨ y ( T M ) ∩ [ M ] ∈ G ( M )[ y ] . The motivic Chern class transformations are a K -theoretical refinementof the Hirzebruch class transformations T y ∗ , M HT y ∗ , which can be definedby the (functorial) commutative diagram : M ( var/X ) χ Hdg −−−−→ K ( M HM ( X )) MHC y −−−−→ G ( X )[ y, y − ] T y ∗ y MHT y ∗ y y td ∗ H ∗ ( X ) ⊗ Q [ y, y − ] −−−−→ H ∗ ( X ) ⊗ Q loc (1+ y ) −∗ · ←−−−−−− H ∗ ( X ) ⊗ Q [ y, y − ] , with td ∗ : G ( X ) → H ∗ ( X ) ⊗ Q the Todd class transformation of Baum-Fulton-MacPherson [2, 16] and (1+ y ) −∗ · the renormalization given in degree i by the multiplication(1+ y ) − i · : H i ( − ) ⊗ Q [ y, y − ] → H i ( − ) ⊗ Q [ y, y − , (1+ y ) − ] =: H ∗ ( − ) ⊗ Q loc . This renormalization is needed to get for M smooth the normalization con-dition(2.3) T y ∗ ( M ) = M HT y ∗ ( M ) = T ∗ y ( T M ) ∩ [ M ] ∈ H ∗ ( M ) ⊗ Q [ y ] . It is the Hirzebruch class transformation T y ∗ , which unifies the (rational-ized) Chern class transformation c ∗ ⊗ Q , Todd class transformation td ∗ and L -class transformation L ∗ (compare [5]). The corresponding characteristicnumber χ y ( X ) := deg ( M HT y ∗ ( X )) ∈ Z [ y ]for a singular (compact) algebraic variety X captures information about the Hodge filtration of Deligne’s ([14]) mixed Hodge structure on the rationalcohomology (with compact support) H ∗ ( c ) ( X ; Q ) of X . In fact, by M. Saito’swork [31] one has an equivalence M HM ( { pt } ) ≃ mHs p between mixed Hodge modules on a point space, and rational (graded) polar-izable mixed Hodge structures. Moreover, the corresponding mixed Hodgestructure on rational cohomology with compact support H ∗ c ( X ; Q ) = H ∗ ( { pt } ; k ! Q HX )(with k : X → { pt } a constant map) agrees with Deligne’s one by anotherdeep theorem of M. Saito [32]. Therefore the transformations M HC y and M HT y ∗ can be seen as a characteristic class version of the ring homomor-phism χ y : K (mHs p ) → Z [ y, y − ]defined on the Grothendieck group of (graded) polarizable mixed Hodgestructures by(2.4) χ y ([ H ]) := X p dim Gr pF ( H ⊗ C ) · ( − y ) p , for F the Hodge filtration of H ∈ mHs p . Note that χ y ([ L ]) = − y .These characteristic class transformations are motivic refinements of the(rationalization of the) Chern class transformation c ∗ ⊗ Q of MacPherson. M HT y ∗ factorizes by [37] as M HT y ∗ : K ( M HM ( X )) → H ∗ ( X ) ⊗ Q [ y, y − ] ⊂ H ∗ ( X ) ⊗ Q loc , fitting into a (functorial) commutative diagram F ( X ) χ stalk ←−−−− K ( D bc ( X )) rat ←−−−− K ( M HM ( X )) c ∗ ⊗ Q y c ∗ ⊗ Q y y MHT y ∗ H ∗ ( X ) ⊗ Q H ∗ ( X ) ⊗ Q y = − ←−−−− H ∗ ( X ) ⊗ Q [ y, y − ] . Here D bc ( X ) is the derived category of algebraically constructible sheaveson X (viewed as a complex analytic space), with rat associating to a (com-plex of) mixed Hodge module(s) the underlying perverse (constructible)sheaf complex, and χ stalk is given by the Euler characteristic of the stalks.Let us go back to the case when X is a local complete intersection in someambient smooth algebraic manifold. Then it is natural to compare cl ∗ ( X )for a functorial homology characteristic class theory cl ∗ as above with thecorresponding virtual characteristic class cl vir ∗ ( X ). If M is smooth, thenclearly we have that cl vir ∗ ( M ) = cl ∗ ( T M ) ∩ [ M ] = cl ∗ ( M ) . However, if X is singular, the difference between the homology classes cl vir ∗ ( X ) and cl ∗ ( X ) depends in general on the singularities of X . Thismotivates the following Problem 2.1.
Describe the difference cl vir ∗ ( X ) − cl ∗ ( X ) in terms of thegeometry of singular locus of X . EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 11
The above problem is usually studied in order to understand the compli-cated homology classes cl ∗ ( X ) in terms of the simpler virtual classes cl vir ∗ ( X ),with the difference terms measuring the complexity of the singularities of X .This question was first studied for the Todd class transformation td ∗ ,where this difference term is vanishing. More precisely one has the Theorem 2.2 (Verdier ’76) . Assume that i : X → Y is regular embeddingof (locally constant) codimension n . Then the Todd class transformation td ∗ commutes with specialization (see [43] ), i.e. (2.5) i ! ◦ td ∗ = td ∗ ◦ i ! : G ( Y ) → H ∗− n ( X ) . Note that Y need not be smooth. Corollary 2.3.
Assume that X can be realized as a local complete intersec-tion in some ambient smooth algebraic manifold. Then td vir ∗ ( X ) = td ∗ ( X ) .Especially, if X is a global complete intersection given as the zero-fiber X = { f = 0 } of a proper morphism f : M → C n on the algebraic man-ifold M , then the arithmetic genus (2.6) χ ( X ) = χ vir ( X ) = χ ( X t ) of X agrees with that of a nearby smooth fiber X t for = t small and generic. The next case studied in the literature is the L -class transformation L ∗ for X a compact global complex hypersurface. Theorem 2.4 (Cappell-Shaneson ’91) . Assume X is a global compact hy-persurface X = { f = 0 } for a proper complex algebraic function f : M → C on a complex algebraic manifold M . Fix a complex Whitney stratificationof X and let V be the set of strata V with dim V < dim X . Assume forsimplicity, that all V ∈ V are simply-connected (otherwise one has to usesuitable twisted L -classes, see [11, 12] ). Then (2.7) L vir ∗ ( X ) − L ∗ ( X ) = X V ∈V σ (lk( V )) · L ∗ ( ¯ V ) , where σ (lk( V )) ∈ Z is a certain signature invariant associated to the linkpair of the stratum V in ( M, X ) . This result is in fact of topological nature, and holds more generally for asuitable compact stratified pseudomanifold X , which is PL-embedded intoa manifold M in real codimension two (see [11, 12] for details).If cl ∗ = c ∗ is the Chern class transformation, the problem amounts tocomparing the Fulton-Johnson class c F J ∗ ( X ) := c vir ∗ ( X ) (e.g., see [16, 17])with the homology Chern class c ∗ ( X ) of MacPherson. The difference be-tween these two classes is measured by the so-called Milnor class M ∗ ( X ) of X , which is studied in many references like [1, 6, 7, 8, 27, 29, 34, 35, 45].This is a homology class supported on the singular locus of X , and for a global hypersurface it was computed in [29] (see also [34, 35, 45, 27]) as aweighted sum in the Chern-MacPherson classes of closures of singular strataof X , the weights depending only on the normal information to the strata.For example, if X has only isolated singularities, the Milnor class equals(up to a sign) the sum of the local Milnor numbers attached to the singularpoints. In the following section we explain our approach [34, 35] through nearby and vanishing cycles (for constructible functions), which recently wasadapted to the motivic Hirzebruch and Chern class transformations [13, 36].3.
Nearby and vanishing cycles
Let us start to explain some basic constructions for constructible functionsin the complex algebraic context (compare [33, 34, 35]). Here we work in theclassical topology on the complex analytic space X associated to a separatedscheme of finite type over Spec ( C ). Definition 3.1.
A function α : X → Z is called (algebraically) constructible,if it satisfies one of the following two equivalent properties: (1) α is a finite sum α = P j n j · Z j , with n j ∈ Z and Z j the charac-teristic function of the closed complex algebraic subset Z j of X . (2) α is (locally) constant on the strata of a complex algebraic Whitneyb-regular stratification of X . This notion is closely related to the much more sophisticated notion of(algebraically) constructible (complexes of) sheaves on X . A sheaf F of (ra-tional) vector-spaces on X with finite dimensional stalks is (algebraically)constructible, if there exists a complex algebraic Whitney b-regular stratifi-cation as above such that the restriction of F to all strata is locally constant.Similarly, a bounded complex of sheaves is constructible, if all it cohomologysheaves have this property, and we denote by D bc ( X ) the corresponding de-rived category of bounded constructible complexes on X . The Grothendieckgroup of the triangulated category D bc ( X ) is denoted by K ( D bc ( X )).Since we assume that all stalks of a constructible complex are finite di-mensional, by taking stalkwise the Euler characteristic we get a naturalgroup homomorphism(3.1) χ stalk : K ( D bc ( X )) → F ( X ); [ F ] ( x χ ( F x )) . Here F ( X ) is the group of (algebraically) constructible functions on X . Itis easy to show that natural transformation χ stalk is surjective.As is well known (and explained in detail in [33]), all the usual functorsin sheaf theory, which respect the corresponding category of constructiblecomplexes of sheaves, induce by the epimorphism χ stalk well-defined grouphomomorphisms on the level of constructible functions. We just recall these,which are important for later applications or definitions. EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 13
Definition 3.2.
Let f : X → Y be an algebraic map of complex spacesassociated to separated schemes of finite type over Spec ( C ) . Then one hasthe following transformations: (1) pullback: f ∗ : F ( Y ) → F ( X ); α α ◦ f , which corresponds to theusual pullback of sheaves f ∗ : D bc ( Y ) → D bc ( X ) . (2) exterior product: α ⊠ β ∈ F ( X × Y ) for α ∈ F ( X ) and β ∈ F ( Y ) ,given by α ⊠ β (( x, y )) := α ( x ) · β ( y ) . This corresponds on the sheaf level tothe exterior product ⊠ L : D bc ( X ) × D bc ( Y ) → D bc ( X × Y ) . (3) Euler characteristic: Suppose X is compact and Y = { pt } is a point.Then one has χ : F ( X ) → Z , corresponding to R Γ( X, · ) = k ∗ : D bc ( X ) → D bc ( { pt } ) on the level of constructible complexes of sheaves, with k : X → { pt } theconstant proper map. By linearity it is characterized by the convention thatfor a compact complex algebraic subspace Z ⊂ X (3.2) χ (1 Z ) := χ ( H ∗ ( Z ; Q )) is just the usual Euler characteristic of Z . (4) proper pushdown: Suppose f is proper. Then one has f ∗ = f ! : F ( X ) → F ( Y ) , corresponding to Rf ∗ = Rf ! : D bc ( X ) → D bc ( Y ) on the level of constructible complexes of sheaves. Explicitly it is given by (3.3) f ∗ ( α )( y ) := χ ( α | { f = y } ) , and in this form it goes back to the paper [25] of MacPherson. (5) nearby cycles: Assume Y = C and let X := { f = 0 } be the zero fiber.Then one has ψ f : F ( X ) → F ( X ) , corresponding to Deligne’s nearby cyclefunctor ψ f : D bc ( X ) → D bc ( X ) . This was first introduced in [44] by using resolution of singularities (comparewith [33] for another approach using stratification theory). By linearity, ψ f is uniquely defined by the convention that for a closed complex algebraicsubspace Z ⊂ X the value (3.4) ψ f (1 Z )( x ) := χ (cid:0) H ∗ ( F f | Z ,x ; Q ) (cid:1) is just the Euler-characteristic of a local Milnor fiber F f | Z ,x of f | Z at x .Here this local Milnor fiber at x is given by (3.5) F f | Z ,x := Z ∩ B ǫ ( x ) ∩ { f = y } , with < | y | << ǫ << and B ǫ ( x ) an open (or closed) ball of radius ǫ around x (in some local coordinates). Here we use the theory of a Milnorfibration of a function f on the singular space Z (compare [24, 33] ). (6) vanishing cycles: Assume Y = C and let i : X := { f = 0 } ֒ → X bethe inclusion of the zero-fiber. Then one has φ f : F ( X ) → F ( X ); φ f := ψ f − i ∗ , corresponding to Deligne’s vanishing cycle functor φ f : D bc ( X ) → D bc ( X ) . By linearity, φ f is uniquely defined by the convention that for a closed com-plex algebraic subspace Z ⊂ X the value (3.6) φ f (1 Z )( x ) := χ (cid:0) H ∗ ( F f | Z ,x ; Q ) (cid:1) − χ (cid:16) ˜ H ∗ ( F f | Z ,x ; Q ) (cid:17) is just the reduced Euler-characteristic of a local Milnor fiber F f | Z ,x of f | Z at x . Remark 3.3.
Let the global hypersurface X = { f = 0 } be the zero-fiberof an algebraic function f : M → C on the complex algebraic manifold M .Then the support of φ f (1 M ) is contained in the singular locus X sing of X : supp ( φ f (1 M )) ⊂ X sing . And φ f (1 M ) | X sing is (up to a sign) the Behrend function of X sing (see [3] ),an intrinsic constructible function of the singular locus appearing in relationto Donaldson-Thomas invariants. A beautiful result of Verdier [44, 23] shows that for a global hypersurfaceMacPherson’s Chern class transformation c ∗ commutes with specialization , ifone uses the nearby cycle functor ψ f on the level of constructible functions(as opposed to the pullback functor i ∗ for the corresponding inclusion i : X = { f = 0 } → Y ). Theorem 3.4 (Verdier ’81) . Assume that X = { f = 0 } is a global hypersur-face (of codimension one) in Y given by the zero-fiber of a complex algebraicfunction f : Y → C . Then the MacPherson Chern class transformation c ∗ commutes with specialization (see [44, 22] ), i.e. (3.7) i ! ◦ c ∗ = c ∗ ◦ ψ f : F ( Y ) → H ∗− ( X ) for the closed inclusion i : X = { f = 0 } → Y . Note that Y need not be smooth. As an immediate application one getsby (1.8) and (1.10) the following important result (compare [34, 35]): Corollary 3.5.
Assume that X = { f = 0 } is a global hypersurface (ofcodimension one) in some ambient smooth algebraic manifold M , given bythe zero-fiber of a complex algebraic function f : M → C . Then (3.8) c vir ∗ ( X ) − c ∗ ( X ) = c ∗ ( ψ f (1 M )) − c ∗ (1 X ) = c ∗ ( φ f (1 M )) ∈ H ∗ ( X sing ) , since supp ( φ f (1 M )) ⊂ X sing . Here we also use the naturality of c ∗ for theclosed inclusion X sing → X to view this difference term as a localized class in H ∗ ( X sing ) . In particular: (1) c viri ( X ) = c i ( X ) ∈ H i ( X ) for all i > dim X sing . EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 15 (2) If X has only isolated singularities (i.e. dim X sing = 0 ), then c vir ∗ ( X ) − c ∗ ( X ) = X x ∈ X sing χ (cid:16) ˜ H ∗ ( F x ; Q ) (cid:17) , where F x is the local Milnor fiber of the isolated hypersurface singu-larity ( X, x ) . (3) If f : M → C is proper, then deg ( c ∗ ( φ f (1 M ))) = deg (cid:0) c vir ∗ ( X ) − c ∗ ( X ) (cid:1) = χ ( X t ) − χ ( X ) is the difference between the Euler characteristic of a global nearbysmooth fiber X t = { f = t } (for = | t | small enough) and of thespecial fiber X = { f = 0 } . For a general local complete intersection X in some ambient smooth al-gebraic manifold (e.g. a local hypersurface of codimension one), one doesn’thave global equations so that the theory of nearby and vanishing cyclescan’t be applied directly. Instead one has to combine them with the de-formation to the normal cone leading to Verdier’s theory of specializationfunctors (compare [34, 35]). But even if X = { f = 0 } is a global completeintersection inside the ambient smooth algebraic manifold M , given by thezero-fiber of a complex algebraic map f : M → C n , one doesn’t have a the-ory of nearby and vanishing cycles, because a local theory of Milnor fibersfor f is missing (if n > ordering of the components of f (or of the coordinates on C n ), then a corresponding local Milnor fibration exists for any ordered tuple( f ) := ( f , . . . , f n ) : Z → C n of complex algebraic functions on the singular algebraic variety Z (as ob-served in [28]). Definition 3.6 (Nearby and vanishing cycles for an ordered tuple) . Let ( f ) := ( f , . . . , f n ) : Y → C n be an ordered n -tuple of complex algebraicfunctions on Y , with X := { f = 0 } = { f = 0 , . . . , f n = 0 } the zero-fiber of ( f ) . Then nearby cycles of ( f ) := ( f , . . . , f n ) are defined by iteration as (3.9) ψ ( f ) := ψ f ◦ · · · ◦ ψ f n : F ( Y ) → F ( X ) . By linearity, ψ ( f ) is uniquely defined by the convention that for a closedcomplex algebraic subspace Z ⊂ Y the value (3.10) ψ ( f ) (1 Z )( x ) := χ (cid:0) H ∗ ( F ( f ) | Z ,x ; Q ) (cid:1) is just the Euler-characteristic of a local Milnor fiber F ( f ) | Z ,x of ( f ) | Z at x .Here this local Milnor fiber of ( f ) at x is given by (3.11) F ( f ) | Z ,x := Z ∩ B ǫ ( x ) ∩ { f = y , . . . , f n = y n } , with < | y n | << · · · << | y | << ǫ << and B ǫ ( x ) an open (or closed) ballof radius ǫ around x (in some local coordinates, compare [28] ). The corresponding vanishing cycles of ( f ) are defined by (3.12) φ ( f ) := ψ ( f ) − i ∗ : F ( Y ) → F ( X ) , with i : X → Y the closed inclusion. By linearity, φ ( f ) is uniquely definedby the convention that for a closed complex algebraic subspace Z ⊂ X thevalue (3.13) φ ( f ) (1 Z )( x ) := χ (cid:0) H ∗ ( F ( f ) | Z ,x ; Q ) (cid:1) − χ (cid:16) ˜ H ∗ ( F ( f ) | Z ,x ; Q ) (cid:17) is just the reduced Euler-characteristic of a local Milnor fiber F ( f ) | Z ,x of ( f ) | Z at x . Note that again supp (cid:0) φ ( f ) (1 M ) (cid:1) ⊂ X sing in case the ambient space Y = M is a smooth algebraic manifold. Assume moreover that X is ofcodimension n so that the regular embedding i : X → Y factorizes into n regular embeddings of codimension one i = i n ◦ · · · ◦ i :(3.14) X = { f = 0 , . . . , f n = 0 } i −−−−→ { f = 0 , . . . , f n = 0 } i −−−−→ · · ·· · · { f n − = 0 , f n = 0 } i n − −−−−→ { f n = 0 } i n −−−−→ Y .
By the functoriality of the Gysin homomorphisms one gets i ! = i !1 ◦ · · · ◦ i ! n : H ∗ ( Y ) → H ∗− n ( X ) . Since in Verdier’s spezialisation theorem (3.4) the ambient space need not besmooth, we can apply it inductively to all embeddings i j (for j = n, . . . , Corollary 3.7.
Assume that X = { f = 0 } = { f = 0 , . . . , f n = 0 } isa global complete intersection (of codimension n ) in some ambient smoothalgebraic manifold M , given by the zero-fiber of an ordered n -tuple of complexalgebraic function ( f ) := ( f , . . . , f n ) : M → C n . Then (3.15) c vir ∗ ( X ) − c ∗ ( X ) = c ∗ ( φ ( f ) (1 M )) ∈ H ∗ ( X sing ) , since supp ( φ ( f ) (1 M )) ⊂ X sing . Here we also use the naturality of c ∗ for theclosed inclusion X sing → X to view this difference term as a localized class in H ∗ ( X sing ) . In particular: (1) c viri ( X ) = c i ( X ) ∈ H i ( X ) for all i > dim X sing . (2) If X has only isolated singularities (i.e. dim X sing = 0 ), then c vir ∗ ( X ) − c ∗ ( X ) = X x ∈ X sing χ (cid:16) ˜ H ∗ ( F x ; Q ) (cid:17) , where F x is the local Milnor fiber of the ordered n -tuple ( f ) at theisolated singularity x . (3) If ( f ) = ( f , . . . , f n ) : M → C n is proper, then deg (cid:0) c ∗ ( φ ( f ) (1 M )) (cid:1) = deg (cid:0) c vir ∗ ( X ) − c ∗ ( X ) (cid:1) = χ ( X t ) − χ ( X ) is the difference between the Euler characteristic of a global nearbysmooth fiber X t = { f = t , . . . , f n = t n } (for t = ( t , . . . , t n ) with EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 17 < | t n | << · · · << | t | small enough) and of the special fiber X = { f = 0 } . As explained in section 2, the motivic Hirzebruch and Chern class trans-formations T y ∗ , M HT y ∗ and mC y , M HC y can be seen as “motivic or Hodgetheoretical liftings” of the (rationalized) Chern class transformation c ∗ underthe comparison maps K ( var/Y ) χ Hdg −−−−→ K ( M HM ( Y )) rat −−−−→ K ( D bc ( Y )) χ stalk −−−−→ F ( Y ) . Here these Grothendieck groups have the same calculus as for constructiblefunctions in definition 3.2(1-4), respected by these comparison maps. So itis natural to try to extend known results about MacPherson’s Chern classtransformation c ∗ to these transformations. In the “motivic” (resp. “ Hodgetheoretical”) context this has been worked out in [5] (resp. [37]) for(1) the functorialty under push down for proper algebraic morphism.(2) the functorialty under exterior products .(3) the functorialty under smooth pullback given by a related Verdier-Riemann-Roch theorem .And recently we could also prove the “counterpart” of Verdier’s special-ization theorem (3.4). Let X = { f = 0 } be a global hypersurface in Y givenby the zero-fiber of a complex algebraic function f on Y : X := { f = 0 } i −−−−→ Y f −−−−→ C . First note that one can use the nearby and vanishing cycle functors ψ f and φ f either on the motivic level of localized relative Grothendieck groups M ( var/ − ) := K ( var/ − )[ L − ](see [4, 19]), or on the Hodge-theoretical level of algebraic mixed Hodgemodules ([30, 31]), “lifting” the corresponding functors on the level of al-gebraically constructible sheaves ([34]) and algebraically constructible func-tions as introduced before, so that the following diagram commutes:(3.16) M ( var/Y ) ψ mf ,φ mf −−−−→ M ( var/X ) χ Hdg y χ Hdg y K (MHM( Y )) ψ ′ Hf ,φ ′ Hf −−−−−→ K (MHM( X )) rat y rat y K ( D bc ( Y )) ψ f ,φ f −−−−→ K ( D bc ( X )) χ stalk y χ stalk y F ( Y ) ψ f ,φ f −−−−→ F ( X ) . We also use the notation ψ ′ Hf := ψ Hf [1] and φ ′ Hf := φ Hf [1] for the shiftedfunctors, with ψ Hf , φ Hf : M HM ( Y ) → M HM ( X ) and ψ f [ − , φ f [ −
1] :
P erv ( Y ) → P erv ( X ) preserving mixed Hodge modules and perverse sheaves,respectively. On the level of Grothendieck groups one simply has φ mf = ψ mf − i ∗ and φ ′ Hf = ψ ′ Hf − i ∗ . Remark 3.8.
The motivic nearby and vanishing cycles functors of [4, 19]take values in a refined equivariant localized Grothendieck group M ˆ µ ( var/X )of equivariant algebraic varieties over X with a “good” action of the pro-finite group ˆ µ = lim µ n of roots of unity. For mixed Hodge modules this cor-responds to an action of the semi-simple part of the monodromy. But in thefollowing applications we don’t need to take this action into account. Alsonote that for the commutativity of diagram (3.16) one has to use ψ ′ Hf , φ ′ Hf (as opposed to ψ Hf , φ Hf ).Now we are ready to formulate the main new result from [36]. Theorem 3.9 (Sch¨urmann ’09) . Assume that X = { f = 0 } is a global hy-persurface of codimension one given by the zero-fiber of a complex algebraicfunction f : Y → C . Then the motivic Hodge-Chern class transformation M HC y commutes with specialization in the following sense: (3.17) (1 + y ) · M HC y ( ψ ′ Hf ( − ) ) = i ! M HC y ( − ) as transformations K ( M HM ( Y )) → G ( X )[ y, y − ] . Again the smoothness of Y is not needed. The appearence of the factor(1 + y ) should not be a surprise, as it can already be seen in the case of asmooth hypersurface X inside a smooth ambient manifold Y ,(1 + y ) · M HC y ( X ) = i ! M HC y ( Y ) , if one recalls (1.8), (1.10) and the normalization condition (2.2), with Q HX = i ∗ Q HY ≃ ψ ′ Hf ( Q HY )in this special case. But the proof of this theorem given in [36] is far awayfrom the geometric applications described here. In fact it uses the algebraictheory of nearby and vanishing cycles in the context of D -modules given bythe V -filtration of Malgrange-Kashiwara , together with a specialization re-sult about the filtered de Rham complex of the filtered D -module underlyinga mixed Hodge module.Using Verdier’s result that the Todd class transformation td ∗ commuteswith specialization (see theorem 2.2), one gets ([36]): Corollary 3.10.
Assume that X = { f = 0 } is a global hypersurface ofcodimension one given by the zero-fiber of a complex algebraic function EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 19 f : Y → C . Then the motivic Hirzebruch class transformation M HT y ∗ commutes with specialization, that is: (3.18) M HT y ∗ ( ψ ′ Hf ( − )) = i ! M HT y ∗ ( − ) as transformations K (MHM( Y )) → H ∗ ( X ) ⊗ Q [ y, y − ] . Again the smoothness of Y is not needed here, but only the fact that X = { f = 0 } is a global hypersurface (of codimension one) is needed. Alsothe factor (1 + y ) in theorem 3.9 cancelled out by the renormalization factor(1 + y ) − i · on H i ( − ) used in the definition of M HT y ∗ , since the Gysin map i ! : H ∗ ( Y ) → H ∗− ( X ) shifts this degree by one.By the definition of ψ mf in [4, 19] one has that ψ mf ( K ( var/Y )) ⊂ im ( K ( var/X ) → M ( X )) , so M HT y ∗ ◦ ψ mf maps K ( var/Y ) into H ∗ ( X ) ⊗ Q [ y ] ⊂ H ∗ ( X ) ⊗ Q [ y, y − ].Together with [37][Prop.5.2.1] one therefore gets the following commutativediagram:(3.19) K ( var/Y ) T y ∗ ◦ ψ mf = −−−−−−→ i ! ◦ T y ∗ H ∗ ( X ) ⊗ Q [ y ] χ Hdg y y K (MHM( Y )) MHT y ∗ ◦ ψ ′ Hf = −−−−−−−−−→ i ! ◦ MHT y ∗ H ∗ ( X ) ⊗ Q [ y, y − ] χ stalk ◦ rat y y y = − F ( Y ) c ∗ ◦ ψ f = −−−−−→ i ! c ∗ H ∗ ( X ) ⊗ Q . As before one gets the following result from Theorem 3.9 and Corollary3.10 together with (1.8) and (1.10):
Lemma 3.11.
Assume that X = { f = 0 } is a global hypersurface (of codi-mension one) in some ambient smooth algebraic manifold M , given by thezero-fiber of a complex algebraic function f : M → C . Then (3.20) mC vir y ( X ) = mC y (cid:0) ψ mf ([ id M ]) (cid:1) = M HC y ∗ (cid:0) ψ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) , and (3.21) T vir y ∗ ( X ) = T y ∗ (cid:0) ψ mf ([ id M ]) (cid:1) = M HT y ∗ (cid:0) ψ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) . If i : X = { f = 0 } → M is the closed inclusion, then one has i ∗ ([ id M ]) =[ id X ] and i ∗ ([ Q HM ]) = [ Q HX ]. So by φ mf = ψ mf − i ∗ and φ ′ Hf = ψ ′ Hf − i ∗ (onthe level of Grothendieck groups) one gets (compare [13]): Corollary 3.12.
Assume that X = { f = 0 } is a global hypersurface (ofcodimension one) in some ambient smooth algebraic manifold M , given bythe zero-fiber of a complex algebraic function f : M → C . Then mC viry ( X ) − mC y ( X ) = mC y (cid:0) φ mf ([ id M ]) (cid:1) = M HC y ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) ∈ G ( X sing )[ y ] , (3.22) and T vir y ∗ ( X ) − T y ∗ ( X ) = T y ∗ (cid:0) φ mf ([ id M ]) (cid:1) = M HT y ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) ∈ H ∗ ( X sing ) ⊗ Q [ y ] . (3.23) Here we use supp (cid:0) φ ′ Hf (cid:0) Q HM (cid:1)(cid:1) ⊂ X sing and the naturality of our characteristic class transformations for the closedinclusion X sing → X . In particular: (1) T viry,i ( X ) = T y,i ( X ) ∈ H i ( X ) ⊗ Q [ y ] for all i > dim X sing . (2) If X has only isolated singularities (i.e. dim X sing = 0 ), then mC viry ( X ) − mC y ( X ) = X x ∈ X sing χ y (cid:16) ˜ H ∗ ( F x ; Q ) (cid:17) = T vir y ∗ ( X ) − T y ∗ ( X ) , (3.24) where F x is the Milnor fiber of the isolated hypersurface singularity ( X, x ) . (3) If f : M → C is proper, then deg (cid:0) M HC y ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1)(cid:1) = χ y ( H ∗ ( X t ; Q )) − χ y ( H ∗ ( X ; Q ))= deg (cid:0) M HT y ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1)(cid:1) (3.25) is the difference between the χ y -characteristics of a global nearby smoothfiber X t = { f = t } (for = | t | small enough) and of the special fiber X = { f = 0 } . Remark 3.13. (Hodge polynomials vs. Hodge spectrum)
Let us explain theprecise relationship between the Hodge spectrum and the less-studied χ y -polynomial of the Milnor fiber of a hypersurface singularity. Here we follownotations and sign conventions similar to those in [19]. Denote by mHs mon the abelian category of mixed Hodge structures endowed with an automor-phism of finite order, and by K mon (mHs) the corresponding Grothendieckring. There is a natural linear map called the Hodge spectrum ,hsp : K mon0 (mHs) → Z [ Q ] ≃ [ n ≥ Z [ t /n , t − /n ] , such that(3.26) hsp([ H ]) := X α ∈ Q ∩ [0 , t α X p ∈ Z dim( Gr pF H C ,α ) t p EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 21 for any mixed Hodge structure H with an automorphism T of finite or-der, where H C is the underlying complex vector space of H , H C ,α is theeigenspace of T with eigenvalue exp(2 πiα ), and F is the Hodge filtration on H C . It is now easy to see that the χ y -polynomial χ y ([ H ]) of H is obtainedfrom hsp([ H ]) by substituting t = 1 in t α for α ∈ Q ∩ [0 ,
1) and t = − y in t p for p ∈ Z .As already explained before, Corollary 3.12 reduces for the value y = − X = { f = 0 } = { f = 0 , . . . , f n = 0 } (of codimension n ) in someambient smooth algebraic manifold M , given by the zero-fiber of an ordered n -tuple of complex algebraic function ( f ) := ( f , . . . , f n ) : M → C n . Herewe leave the details to the reader.It is also very interesting to look at the other specializations of Corollary3.12 for y = 0 and y = 1. Let us first consider the case when y = 0. Notethat in general T ∗ ( X ) = td ∗ ( X ) for a singular complex algebraic variety (see[5]). But if X has only Du Bois singularities (e.g., rational singularities, cf.[32]), then by [5] we have T ∗ ( X ) = td ∗ ( X ). So if a global hypersurface X = { f = 0 } has only Du Bois singularities , then by Corollaries 2.3 and3.12 we get:
M HT ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) = 0 ∈ H ∗ ( X ) ⊗ Q . This vanishing (which is in fact a class version of Steenbrink’s cohomo-logical insignificance of X [40]) imposes interesting geometric identities onthe corresponding Todd-type invariants of the singular locus. For example,we obtain the following Corollary 3.14.
If the global hypersurface X has only isolated Du Boissingularities , then (3.27) dim C Gr F H n ( F x ; C ) = 0 for all x ∈ X sing , with n = dim X . It should be pointed out that in this setting a result of Ishii [21] impliesthat (3.27) is in fact equivalent to x ∈ X sing being an isolated Du Boishypersurface singularity. Also note that in the arbitrary singularity case,the Milnor-Todd class T ∗ (cid:0) φ mf ([ id M ]) (cid:1) = M HT ∗ (cid:0) φ ′ Hf ( (cid:2) Q HM (cid:3) ) (cid:1) ∈ H ∗ ( X sing ) ⊗ Q carries interesting non-trivial information about the singularities of the hy-persurface X .Finally, if y = 1, the formula (3.23) should be compared to the Cappell-Shaneson topological result of (2.7). While it can be shown (compare with[26]) that the normal contribution σ (lk( V )) in (2.7) for a singular stratum V ∈ V is in fact the signature σ ( F v ) ( v ∈ V ) of the Milnor fiber (as amanifold with boundary) of the singularity in a transversal slice to V in v ,the precise relation between σ ( F v ) and χ ( F v ) is in general very difficult tounderstand. For X a rational homology manifold , one would like to have a“local Hodge index formula” σ ( F v ) ? = χ ( F v ) , which is presently not available. But if the hypersurface X is a rational ho-mology manifold with only isolated singularities , then this expected equalityfollows from [41][Thm.11]. One therefore gets in this case (by a comparisonof the different specialization results for L ∗ and T ∗ ) the following conjecturalinterpretation of L -classes from [5] (see [13] for more details): Theorem 3.15.
Let X be a compact complex algebraic variety with onlyisolated singularities, which is moreover a rational homology manifold andcan be realized as a global hypersurface (of codimension one) in a complexalgebraic manifold. Then (3.28) L ∗ ( X ) = T ∗ ( X ) ∈ H ∗ ( X ; Q ) . References [1] Aluffi, P.,
Chern classes for singular hypersurfaces , Trans. Amer. Math. Soc. (1999), no. 10, 3989–4026.[2] Baum, P., Fulton, W., MacPherson, R.,
Riemann-Roch for singular varieties ,Publ. Math. I.H.E.S. (1975), 101–145.[3] Behrend, K., Donaldson-Thomas invariants via microlocal geometry , Annals ofMath. , (2009), no. 3, 1307–1338.[4] Bittner, F.,
On motivic zeta functions and motivic nearby cyles , Math. Z. (2005), 63–83.[5] Brasselet, J.-P., Sch¨urmann, J., Yokura, S.,
Hirzebruch classes and motivic Chernclasses of singular spaces , Journal of Topology and Analysis , No.1 (2010), 1–55.[6] Brasselet, J.-P., Lehmann, D., Seade, J., Suwa, T., Milnor numbers and classes oflocal complete intersections , Proc. Japan Acad. Ser. A Math. Sci. (1999), no.10, 179–183.[7] Brasselet, J.-P., Lehmann, D., Seade, J., Suwa, T., Milnor classes of local completeintersections , Trans. Amer. Math. Soc. (2001), 1351–1371.[8] Brasselet, J.-P., Seade, J., Suwa, T.,
An explicit cycle representing the Fulton-Johnson class, I. , S´eminaire & Congr`es (2005), 21–38.[9] Brasselet, J.P., Schwartz, M.H., Sur les classes de Chern d’une ensemble analy-tique complexe , in:
Caract´eristice d’Euler-Poincar´e, S´eminaire E.N.S. 1978-1979 , Ast´erisque (1981), 93–148.[10] Cappell, S. E., Shaneson, J. L.,
Stratifiable maps and topological invariants , J.Amer. Math. Soc. (1991), 521–551.[11] Cappell, S. E., Shaneson, J. L., Characteristic classes, singular embeddings, andintersection homology , Proc. Natl. Acad. Sci. USA, Vol (1991), 3954–3956.[12] Cappell, S. E., Shaneson, J. L., Singular spaces, characteristic classes, and inter-section homology , Ann. of Math. (1991), 325–374.[13] Cappell, S. E., Maxim, L., Shaneson, J. L., Sch¨urmann, J.,
Characteristic classesof complex hypersurfaces , arXiv:0908.3240 . To appear in Adv. in Math.
EARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 23 [14] Deligne, P.,
Th´eorie de Hodge, II, III , Inst. Hautes ´Etudes Sci. Publ. Math. No. (1971), 5–57; Inst. Hautes ´Etudes Sci. Publ. Math. No. (1974), 5–77.[15] Du Bois, Ph., Complexe de De Rham filtr´e d’une vari´et´e singuli`ere , Bull. Soc.Math. France (1981), 41–81.[16] Fulton, W.,
Intersection Theory , Second edition. Springer-Verlag, Berlin, 1998.[17] Fulton, W., Johnson, K.,
Canonical classes on singular varieties , ManuscriptaMath (1980), 381–389.[18] Goresky, M., MacPherson, R., Intersection homology theory , Topology (1980),no. 2, 135–162.[19] Guibert, G., Loeser, F., Merle, M., Iterated vanishing cycles, convolution, and amotivic analogue of a conjecture of Steenbrink , Duke Math. J. (2006), no. 3,409–457.[20] Hirzebruch, F.,
Topological methods in algebraic geometry , Springer-Verlag NewYork, Inc., New York 1966.[21] Ishii, S.,
On isolated Gorenstein singularities , Math. Ann. (1985), 541–554.[22] Kennedy, G.,
MacPherson’s Chern classes of singular varieties , Com. Algebra. (1990), 2821–2839.[23] Kennedy, G., Specialization of MacPherson’s Chern classes , Math. Scand. (1990), 12–16.[24] Lˆe, D.T., Some remarks on relative monodromy , In:
Real and complex singu-larities. Nordic Summer School (Oslo 1976) , Sijthoff-Noordhoff, Groningen(1977), 397–403.[25] MacPherson, R.,
Chern classes for singular algebraic varieties , Ann. of Math. (2) (1974), 423–432.[26] Maxim, L.,
Intersection homology and Alexander modules of hypersurface comple-ments , Comment. Math. Helv. (2006), no. 1, 123–155.[27] Maxim, L., On the Milnor classes of complex hypersurfaces . To appear in “Topol-ogy of Stratified Spaces”, MSRI Publications , Cambridge Univ. Press (2010)[28] McCrory, C., Parusi´nski, A., Complex monodromy and the topology of real alge-braic sets , Comp. Math. (1997), 211–233.[29] Parusi´nski, A., Pragacz, P.,
Characteristic classes of hypersurfaces and character-istic cycles , J. Algebraic Geom. (2001), no. 1, 63–79.[30] Saito, M., Modules de Hodge polarisables , Publ. RIMS (1988), 849–995.[31] Saito, M., Mixed Hodge Modules , Publ. RIMS (1990), 221–333.[32] Saito, M., Mixed Hodge complexes on algebraic varieties , Math. Ann. (2000),283–331.[33] Sch¨urmann, J.,
Topology of singular spaces and constructible sheaves , MonografieMatematyczne (New Series), Birkh¨auser, Basel, 2003.[34] Sch¨urmann, J., Lectures on characteristic classes of constructible functions ,in
Trends Math., Topics in cohomological studies of algebraic varieties ,Birkh¨auser, Basel (2005), 175–201,[35] Sch¨urmann, J.,
A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes , arXiv:math/0202175[36] Sch¨urmann, J.,
Specialization of motivic Hodge-Chern classes , arXiv:0909.3478[37] Sch¨urmann, J.,
Characteristic classes of mixed Hodge modules , arXiv:0907.0584 .To appear in “Topology of Stratified Spaces”, MSRI Publications , CambridgeUniv. Press (2010)[38] Sch¨urmann, J., Yokura, S., A survey of characteristic classes of singular spaces ,in
Singularity Theory (ed. by D. Ch´eniot et al), Dedicated to Jean PaulBrasselet on his th birthday, Proceedings of the 2005 Marseille SingularitySchool and Conference World Scientific, (2007), 865–952. [39] Schwartz, M.H.,
Classes caract´eristiques d´efinies par une stratification d’unevari´et´e analytique complexe , C. R. Acad. Sci. Paris (1965), 3262–3264 and3535–3537.[40] Steenbrink, J.,
Cohomologically insignificant degenerations , Compositio Math. (1980/81), no. 3, 315–320.[41] Steenbrink, J., Monodromy and weight filtration for smoothings of isolated singu-larities , Comp. Math. (1995), 285–293.[42] Thom, R., Les classes caract´eristiques de Pontrjagin des vari´et´es triangul´ees , in
Symp. Intern. de Topologia Algebraica. Unesco
Le th´eor`eme de Riemann-Roch pour les intersections compl`etes ,Ast´erisque (1976), 189–228.[44] Verdier, J.-L.,
Sp´ecialisation des classes de Chern , Ast´erisque , 149-159(1981).[45] Yokura, S.,
On characteristic classes of complete intersections , in
Algebraic ge-ometry: Hirzebruch 70 (Warsaw, 1998) , Contemp. Math., , Amer. Math.Soc., Providence, RI (1999), 349–369,[46] Yokura, S.,
Motivic characteristic classes , arXiv:0812.4584 . To appear in “Topol-ogy of Stratified Spaces”, MSRI Publications , Cambridge Univ. Press (2010)[47] Yokura, S., Motivic Milnor classes , Journal of Singularities (2010), 39–59. J. Sch¨urmann : Mathematische Institut, Universit¨at M¨unster, Einsteinstr.62, 48149 M¨unster, Germany.
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