NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES AND SOME PROBLEMS

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1 Journal of Singularities Volume 10 (2014), Proc. of 12th International Workshop on Singularities, São Carlos, 2012 DOI: /jsing b NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES AND SOME PROBLEMS VITTORIA BUSSI () AND SHOJI YOKURA () ABSTRACT. Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In [2] Behrend introduced a Donaldson-Thomas type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the Donaldson-Thomas type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations. 1. INTRODUCTION The Donaldson Thomas invariant χ (M) (abbr. invariant) is the virtual count of the moduli space M of stable coherent sheaves on a Calabi Yau threefold over k. Here k is an algebraically closed field of characteristic zero. Foundational materials for invariants can be found in [36], [2], [20], [23]. In [2] Behrend made the important observation that the Donaldson Thomas invariant χ (M) is described as the weighted Euler characteristic χ(m, ν M ) of the so-called Behrend (constructible) function ν M. For a scheme X of finite type, the Donaldson Thomas type invariant χ (X) is defined as χ(x, ν X ). The Euler characteristic χ defined by using the compactly-supported l-adic cohomology groups (see 2 for more details) satisfies the scissor formula χ(x) = χ(z) + χ(x \ Z) for a closed subvariety Z X. This scissor formula implies that χ can be considered as a homomorphism from the Grothendieck group of varieties χ : K 0 (V) Z, and furthermore it can be extended to the relative Grothendieck group, χ : Z for each scheme X. The Grothendieck Riemann Roch version of the homomorphism χ : Z is the motivic Chern class transformation T 1 : K 0(V/X) H BM (X) Q. Namely we have that When X is a point, T 1 : K 0(V/X) H BM (X) Q equals the homomorphism χ : K 0 (V) Z Q. The composite X T 1 = χ : K 0(V/X) Z Q. Here T 1 : K 0(V/X) H BM (X) Q is the specialization to y = 1 of the motivic Hirzebruch class transformation T y : H BM (X) Q[y] (see [5]). On the other hand the Donaldson Thomas type invariant χ (X) does not in general satisfy the scissor formula χ (X) χ (Z) + χ (X \ Z). Namely, χ ( ) cannot be captured as a homomorphism χ : K 0 (V) Z. Instead the following scissor formula holds: (1.1) χ (X id X X) = χ (Z i Z,X X) + χ (X \ Z i X\Z,X X). Here i Z,X and i X\Z,X are the inclusions. For this formula to make sense, we need a Donaldson Thomas type invariant χ (X f Y ) for a morphism f : X Y, which is also introduced in [2] and simply defined as χ(x, f ν Y ). Then χ can be considered as a homomorphism χ : Z. Note (*) Funded by EPSRC (**) Partially supported by JSPS KAKENHI Grant Number

2 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 27 that in the case when X is a point, χ : K 0 (V/pt) = K 0 (V) Z is the usual Euler characteristic homomorphism χ : K 0 (V) Z. In this paper we consider Grothendieck Riemann Roch type formulas for χ, using the motivic Hirzebruch class transformation T y ([5]). One of the key features on constructible functions and elements of when we state such Grothendieck Riemann Roch type formulas is that they are stable under morphisms. For example, δ assigning to each variety X a constructible function δ X is said to be stable under a morphism f : X Y if δ X = f δ Y. The 1 assigning to each variety X the characteristic function 1 X is stable under a (in fact, any) morphism and ν assigning to each variety X the signed Behrend function ν X := ( 1) dim X ν X is stable under a smooth morphism. We also propose to consider a bivariant-theoretic aspect for the categorification of the invariant. By this we mean a graded vector space encoding an appropriate cohomology theory whose Euler characteristic is equal to invariant. Naive reasons for the latter are the following. The categorification of the Euler characteristic is nothing but χ(x) := i ( 1) i dim R H i c(x; R). Note that the compact-support-cohomology Hc(X; i R) is isomorphic to the Borel Moore homology (X; R). The categorification of the Hirzebruch χ y -genus is H BM i χ y (X) = ( 1) i dim C Gr p F (Hi c(x; C))( y) p with F being the Hodge filtration of the mixed Hodge structure of H i c(x; C). Since the type invariant of a morphism satisfies the scissor formula (1.1) due to its definition, we propose to introduce some bivariant-theoretic homology theory Θ (X f Y ) categorifying χ (X f Y ), that is χ (X f Y ) = i ( 1)i dim Θ i (X f Y ). (Here we denote it symbolically ; as described in the case of χ y -genus, the above alternating sum of the dimensions might be complicated involving some other ingredients such as mixed Hodge structures.) 2. DONALDSON THOMAS TYPE INVARIANTS OF MORPHISMS Let K be an algebraically closed field of characteristic p, which is not necessarily zero. Let X be a K-scheme of finite type. For a prime number l such that l p and the field Q l of l-adic numbers, the following Euler characteristic χ(x) := i ( 1) i dim Ql H i c(x, Q l ) is independent on the choice of the prime number l. In fact the following properties hold (e.g., see [17, Theorem 3.10]): Theorem 2.1. Let K be an algebraically closed field and X, Y be separated K -schemes of finite type. Then (1) If Z is a closed subscheme of X, then χ(x) = χ(z) + χ(x \ Z). (2) χ(x Y ) = χ(x)χ(y ). (3) χ(x) is independent of the choice of l in the above definition (4) If K = C, χ(x) is the usual Euler characteristic with the analytic topology. (5) χ(k m ) = 1 and χ(kp m ) = m + 1 for m > 0 For a constructible function α : X Z on X the weighted Euler characteristic χ(x, α) is defined by χ(x, α) := m mχ(α 1 (m)).

3 28 VITTORIA BUSSI () AND SHOJI YOKURA () Let X be embeddable in a smooth scheme M and let C M X be the normal cone of X in M and let π : C M X X be the projection and C M X = m i C i, where m i Z are multiplicities and C i s are irreducible components of the cycle. Then the following cycle C X/M := ( 1) dim(π(ci)) m i π(c i ) Z(X) is in fact independent of the choice of the embedding of X into a smooth M ([1, Lemma 1.1] and [2, Proposition 1.1], also see [11, Example ]), thus simply denoted by C X without referring to the ambient smooth M and is called the distinguished cycle of the scheme. Then consider the isomorphism from the abelian groups Z(X) of cycles to the abelian group F(X) of constructible functions Eu : Z(X) = F(X) which is defined by Eu( i m i[z i ]) := i m i Eu Zi, where Eu Z denotes the local Euler obstruction supported on the subscheme Z i. Then the image of the distinguished cycle C X under the above isomorphism Eu defines a canonical integer valued constructible function ν X := Eu(C X ), which is called the Behrend function. The fundamental properties of the Behrend function are the following. Theorem 2.2. (1) For a smooth point x of a scheme X of dimension n, ν X (x) = ( 1) n. In particular, if X is smooth of dimension n, then ν X = ( 1) n 1 X. (2) ν X Y = ν X ν Y. (3) If f : X Y is smooth of relative dimension n, then ν X = ( 1) n f ν Y. (4) In particular, if f : X Y is étale, then ν X = f ν Y. (5) (see also [32]) If Y is the critical scheme of a regular function f on a smooth scheme M, i.e., Y = Z(df), then for y Y ν Y (y) = ( 1) dim M (1 χ(f y )) (= ( 1) dim X (χ(f y ) 1)), where X := f 1 (0) is the hypersurface, thus Y is the singularity subscheme of X defined by the partial derivatives of f, and F y is the Milnor fiber of X at the point y. Remark 2.3. In [1, 1 Weighted Chern Mather Classes] Paolo Aluffi introduces the weighted Chern Mather class of Y M, denoted by c wma (Y ), as follows: c wma (Y ) := i ( 1) dim Y dim π(ci) m i c Ma (π(c i )), where c Ma (π(c i )) is the Chern Mather class of π(c i ), i.e. c Ma (π(c i )) = c (Eu π(ci)). Therefore we get the following: c wma (Y ) := i ( 1) dim Y dim π(ci) m i c Ma (π(c i )) = ( 1) dim Y dim π(ci) m i c (Eu π(ci)) i ( = c ( 1) ) dim Y ( 1) dim π(ci) m i Eu π(ci) i ( = c ( 1) dim Y ) ν Y.

4 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 29 In other words, Aluffi introduces the distinguished constructible function, i.e. the signed Behrend function ( 1) dim Y ν Y =: ν Y. In [1, Theorem 1.2.] he proves that if X is defined as the zero-scheme of a nonzero section of a line bundle L over M, then (2.4) c ( ν Y ) = ( 1) dim X dim Y c(l) (c F J (X) c (X)), where Y is the singularity subscheme of the hypersurface X, i.e. the subscheme locally defined by the partial derivatives of an equation for X, and c F J (X) is Fulton Johnson class of X or the canonical class of X (see [11, Example ] and [12]). In this hypersurface case he furthermore shows the following [1, Theorem 1.5.]: As in (5) of the above Theorem 2.2, if µ Y is the constructible function defined by µ Y (y) := ( 1) dim X (χ(f y ) 1), then c ( ν Y ) = ( 1) dim Y c (µ Y ). It follows from (2.4) and ( 1) dim Y c ( ν Y ) = c (ν Y ) that we get c(l) 1 c (ν Y ) = ( 1) dim X (c F J (X) c (X)). The right-hand-sided invariant ( 1) dim X (c F J (X) c (X)) is the so-called Milnor class of X (supported on the singular locus Y ). Hence, in particular, in the case when the line bundle L is trivial, i.e., in the case of (5) of Theorem 2.2, we have that c (ν Y ) = c (µ Y ) is nothing but the Milnor class of X. The weighted Euler characteristic of the above Behrend function is called the Donaldson Thomas type invariant and denoted by χ (X): χ (X) := χ(x, ν X ). Remark 2.5. We would like to emphasize that using the Aluffi function ν X we have that χ (X) = χ(x, ν X ) = ( 1) dim X χ(x, ν X ). In [2, Definition 1.7] Kai Behrend defined the following. Definition 2.6. The -invariant or virtual count of a morphism f : X Y is defined by χ (X f Y ) := χ(x, f ν Y ), where ν Y is the Behrend function of the target scheme Y. Remark 2.7. Here we emphasize that χ (X f Y ) is defined by the constructible function f ν Y on the source scheme X. From the definition we can observe the following: (1) χ (X id X X) = χ(x, ν X ) = χ (X) is the -invariant of X. (2) χ (X π X pt) = χ(x, f ν pt ) = χ(x, 1 X ) = χ(x) is the topological Euler-Poincaré characteristic of X. (3) If Y is smooth, whatever the morphism f : X Y is, we have χ (X f Y ) = ( 1) dim Y χ(x). The very special case is that Y = pt, which is the above (2). The Euler characteristic χ( ) satisfies the additivity χ(x) = χ(z) + χ(x \ Z) for a closed subscheme Z X. Hence, χ is considered as a homomorphism from the Grothendieck group of varieties χ : K 0 (V) Z and furthermore as a homomorphism from the relative Grothendieck group of varieties over a fixed variety X ([28]) χ : Z,

5 30 VITTORIA BUSSI () AND SHOJI YOKURA () which is defined by χ([v h X]) = χ(v ) = χ(v, 1 V ) = χ(v, h 1 X ) = χ(x, h 1 V ). Moreover, the following diagram commutes: (2.8) f K 0 (V/Y ) χ Z. χ On the other hand we have that χ (X) χ (Z) + χ (X \ Z). Thus χ ( ) cannot be captured as a homomorphism χ : K 0 (V) Z. However, we have that χ (X id X X) = χ (Z i Z,X X) + χ (X \ Z i X\Z,X X). Lemma 2.9. If we define χ ([V h X]) := χ(v, h ν X ), then we get the homomorphism χ : Z. Proof. The definition χ ([V h X]) := χ(v, h ν X ) is independent of the choice of the representative of the isomorphism class [V h X]. Indeed, let V h X be another representative of [V h X], i.e., we have the following commutative diagram, where ι : V = V is an isomorphism: V ι V h X. h Then we have that χ(v, h ν X ) = χ(v, ι (h ν X )) = χ(v, h ν X ). For a closed subvariety W V, we have χ ([V h X] = χ(v, h ν X ) = χ(w, h ν X ) + χ(v \ W, h ν X ) = χ(w, (h W ) ν X ) + χ(v \ W, (h V \W ) ν X ) = χ ([W h W X]) + χ ([V \ W h V \W X]). Thus we get the homomorphism χ : Z. Lemma If f : X Y satisfies the condition that ν X = f ν Y (such a morphism shall be called a Behrend morphism ), then the following diagram commutes: f K 0 (V/Y ) χ Z. χ

6 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 31 Proof. It is straightforward: χ f ([V h X]) = χ ([V f h X]) = χ(v, (f h) ν Y ) = χ(v, h f ν Y ) = χ(v, h ν X ) (since ν X = f ν Y ) = χ ([V h X]).. Remark An étale map is a typical example of a Behrend morphism. Remark For a general morphism f : X Y, we have that f ν Y = ( 1) reldim f ν X + Θ(X sing f 1 (Y sing )), where reldim f := dim X dim Y is the relative dimension of f and Θ(X sing f 1 (Y sing )) is some constructible functions supported on the singular locus X sing of X and the inverse image of the singular locus Y sing of Y. As then ν X = ( 1) dim X 1 X + some constructible function supported on X sing, f ν Y = ( 1) dim X f 1 Y + f (some constructible function supported on Y sing ). Hence in general we have (χ f )([V h X]) = ( 1) reldim f χ ([V h X]) + extra terms. Here the extra terms are supported on the singular locus X sing. To avoid taking care of the sign, we use the signed Behrend function, i.e., the Aluffi function ν X = ( 1) dim X ν X, which will be used later again. Note that if X is smooth, ν X = 1 X. Then we define the signed Donaldson Thomas type invariant χ (X) by χ (X f Y ) := χ(x, f ν Y ). (In other words, this invariant could be called an Aluffi Behrend Euler characteristic of a morphism f.) Then for a morphism f : X Y we have f ν Y = ν X + Θ(X sing f 1 (Y sing )). In particular the above lemma is modified as follows: Lemma If f : X Y satisfies the condition that ν X = f ν Y (such a morphism shall be called a signed Behrend morphism ; a smooth morphism is a typical example for ν X = f ν Y ), then the following diagram commutes: f K 0 (V/Y ) χ Z. χ

7 32 VITTORIA BUSSI () AND SHOJI YOKURA () 3. GENERALIZED DONALDSON THOMAS TYPE INVARIANTS OF MORPHISMS Mimicking the above definition of χ (X f Y ) and ignoring the geometric or topological interpretation, we define the following. Definition 3.1. For a morphism f : X Y and a constructible function δ Y F(Y ) we define χ δ Y (X f Y ) := χ(x, f δ Y ). Lemma 3.2. For a morphism f : X Y and a constructible function α F(X) we have χ(x, α) = χ(y, f α). Corollary 3.3. For a morphism f : X Y and a constructible function δ Y F(Y ) we have χ δ Y (X f Y ) = χ(y, f f δ Y ). Remark 3.4. For the constant map π X : X pt, the pushforward homomorphism π X : F(X) F(pt) = Z is nothing but the fact that π X (α) = χ(x, α) (by the definition of the pushforward). Hence, the above equality χ(x, α) = χ(y, f α) is rephrased as the commutativity of the following diagram: F(X) f F(Y ) π X F(pt) = Z. Namely, π X = (π Y f) = π Y f. This might suggest that F( ) is a covariant functor, but we need to be a bit careful. F( ) is a covariant functor provided that the ground field K is of characteristic zero. However, if it is not of characteristic zero, then it may happen that (g f) g f, for which see Schürmann s example in [17]. π Y Remark 3.5. If we define 1 : F(X) by 1 ([V f : X Y we have the following commutative diagrams: h X]) := h 1 V, then for a morphism f K 0 (V/Y ) 1 F(X) 1 f F(Y ) π X F(pt) = Z. π Y (π X 1 )([V h X]) = χ([v h X]) and the outer triangle is nothing but the commutative diagram (2.8) mentioned before. Here we emphasize that the above equality χ δ Y (X f Y ) = χ(y, f f δ Y ) have the following two aspects: The invariant on LHS for a morphism f : X Y is defined on the source space X. The invariant on RHS for a morphism f : X Y is defined on the target space Y.

8 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 33 So, in order to emphasize the distinction, we introduce the following notation: χ δ Y (X f Y ) := χ(y, f f δ Y ). Since we want to deal with higher class versions of the Donaldson Thomas type invariants and use the functoriality of the constructible function functor F( ), we assume that the ground field K is of characteristic zero. We consider MacPherson s Chern class transformation c : F(X) H BM (X), which is due to Kennedy [21]. For a morphism h : V X and for a constructible function δ X F(X) on the target space X, we have c (h δ X ) = χ(v, h δ X ) = χ δ h X (V X), V c (h h δ X ) = χ(x, h h δ X ) = χ δ X (V h X). X Here c (h δ X ) H BM (V ) on the side of the source space V and c (h h δ X ) H BM (X) on the side of the target space X. Hence when we want to deal with them as the homomorphism from to (X), we should consider the higher analogues c (h h δ X ), which we denote by H BM On the other hand we denote Note that c δ X (V h X) := c (h δ X ) H BM (V ). c δ X (V h X) := c (h h δ X ) H BM (X). c δ X (V h X) = h (c δ X (V h X)), for an isomorphism id X : X X, these two classes are identical and denoted simply by c δ X (X) := c (δ X ) = c δ X (X id X X) = c δ X (X id X X). In the following sections we treat these two objects c δ X (V h X) and c δ X (V h X) separately, since they have different natures. 4. MOTIVIC ALUFFI-TYPE CLASSES In [2] the Chern class c ν X (X) for the Behrend function ν X is called the Aluffi class, in which case (X) = χ (X). However, in this paper, for the signed Behrend function ν X the Chern class X cν X c ν X (X) shall be called the Aluffi class and denoted by c Al introduced in [1] as pointed out in [2, 1.4 The Aluffi class]. Note that X cal (X), since this is the class which Aluffi (X) = ( 1) dim X χ (X). In this sense, the Chern class c δ X (V h X) defined above shall be called a generalized Aluffi class of a morphism h : V X associated to a constructible function δ X F(X). So the original Aluffi class is c ν X (X id X X). Lemma 4.1. The following formulae hold: (1) If (V h X) = (V h X), i.e., there exists an isomorphism k : V = V such that h = h k, then we have c δ X (V h X) = c δ X (V h X). (2) For a closed subvariety W V, c δ X (V h X) = c δ X (3) For morphisms h i : V i X i (i = 1, 2), (W h W X) + c δ X (V \ W h V \W X). c δ X 1 δ X2 h (V 1 V 1 h 2 2 X1 X 2 ) = c δ X 1 h (V 1 1 X1 ) c δ X 2 h (V 2 2 X2 ).

9 34 VITTORIA BUSSI () AND SHOJI YOKURA () (4) c δpt (pt pt) = δ pt (pt) Z. Corollary 4.2. Let δ X F(X) be a constructible function. Then the following hold: (1) The map c δ X : H BM (X) defined by c δ X ([V h X]) := c δ X (V h X) = c (h h δ X ) and linearly extended is a well-defined homomorphism. (2) c δ X commutes with the exterior product, i.e. for constructible functions δ Xi F(X i ) and for α i K 0 (V/X i ), c δ X 1 δ X2 (α 1 α 2 ) = c δ X 1 (α 1 ) c δ X 2 (α 2 ). Remark 4.3. If δ X is some function defined on X, such as the characteristic function 1 X, the Behrend function ν X, the signed Behrend function ν X, and if it is multiplicative, i.e. δ X Y = δ X δ Y, then the above Corollary 4.2 (2) can be simply rewritten as c δ X 1 X 2 (α 1 α 2 ) = c δ X 1 (α 1 ) c δ X 2 (α 2 ). Remark 4.4. If X is smooth and h : V X is proper (here properness is required since we use the pushforward h of the Borel Moore homology groups), then we have c Al ([V h X]) = c (h h ν X ) = h c (h 1 X ) = h c ( 1 V ) = h c SM (V ) is the pushforward of the Chern Schwartz MacPherson class of V, thus it depends on the morphism h : V X, although the degree zero part of it, i.e. the signed Donaldson Thomas type invariant is nothing but the Euler characteristic of V, thus it does not depend on the morphism at all. Therefore the higher class version is more subtle. The part h h δ X can be formulated as follows. Given a constructible function δ X F(X), we define by [δ X ]([V [δ X ] : F(X) h X]) := h h δ X and extend it linearly, i.e., ( ) [δ X ] m h [V h X] := h h m h (h h δ X ). If (V h X) = (V h X), i.e., there exists an isomorphism k : V = V such that h = h k, then we have (h ) (h ) δ X = h k k h δ X = h h δ X because k k = id F(X). For a morphism h : V X and for a closed subvariety W V, we have h h δ X = (h W ) (h W ) δ X + (h V \W ) (h V \W ) δ X, ( ) that is, we have that [δ X ] [V h X] [W h W X] [V \ W h V \W X] = 0. Therefore the homomorphism [δ X ] : F(X) is well-defined. Note that 1 : F(X) is nothing but [ 1 X ] : F(X). It is straightforward to see the following. Lemma 4.5. For any morphism g : X Y and any constructible function δ Y F(Y ), the following diagrams commute: [g δ Y ] F(X) g g, K 0 (V/Y ) [δ Y ] F(Y ). [δ Y ] K 0 (V/Y ) F(Y ) g g [g δ Y ] F(X).

10 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 35 The following corollary follows from MacPherson s theorem [29] and our previous results [34, 38], and here we need the properness of the morphism g : X Y, since we deal with the pushforward homomorphism for the Borel Moore homology. c δ X : H BM (X) is the composite of and MacPherson s Chern class c, in particular c Al we have the following corollary: [δ X ] : F(X) : H BM (X) is c Al = c [ ν X ]. Hence Corollary 4.6. (1) For a proper morphism g : X Y and any constructible function δ Y F(Y ), the following diagram commutes: g c g δ Y H BM (X) g K 0 (V/Y ) c δ Y H BM (Y ). (2) For a smooth morphism g : X Y with c(t g ) being the total Chern cohomology class of the relative tangent bundle T g of the smooth morphism and g : H BM (Y ) H BM (X) the Gysin homomorphism ([11, Example ]), the following diagram commutes: K 0 (V/Y ) g c δ Y H BM (Y ) c(t g) g c g δ Y H BM (X). Therefore, if δ assigning to each variety X a constructible function δ X F(X) is stable under a proper morphism g : X Y, then we have the following commutative diagrams: g c δ X H BM (X) g K 0 (V/Y ) g c δ Y H BM (Y ) c(t g) g K 0 (V/Y ) c δ Y H BM (Y ), c δ X In particular we get the following theorem for the Aluffi class c Al H BM (X). : K 0 (V/ ) H BM ( ): Theorem 4.7. For a smooth proper morphism g : X Y the following diagrams commute: g c Al H BM (X) g K 0 (V/Y ) g c Al H BM (Y ) c(t g) g K 0 (V/Y ) c Al H BM (Y ), c Al H BM (X). They are respectively Grothendieck Riemann Roch type and a Verdier Riemann Roch type formulas. Remark 4.8. In the above theorem the smoothness of the morphism g : X Y is crucial and the Aluffi class homomorphism c Al : H BM (X) cannot be captured as a natural transformation in a full generality, i.e. natural for any morphism. Indeed, if it were the case, then c Al : K 0 (V/ ) H BM ( ) H BM ( ) Q

11 36 VITTORIA BUSSI () AND SHOJI YOKURA () becomes a natural transformation such that for any smooth variety X we have c Al ([X id X X]) = c(t X ) [X]. Let T y : K 0 (V/ ) H BM ( ) Q[y] be the motivic Hirzebruch class transformation [5], which is the unique natural transformation satisfying the normalization condition that for a smooth X, T y ([X id X X]) = td y (T X) [X], where [X] is the fundamental class and td y (T X) is Hirzebruch characteristic cohomology class of the tangent bundle T X. Here the Hirzebruch class td y (E) of the complex or algebraic vector bundle E over X is defined to be (see [15, 16]): td y (E) := rank E i=1 Here α i s are the Chern roots of E, i.e., c(e) = ( α(1 + y) αy 1 e α(1+y) rank(e) i=1 following three well-known characteristic cohomology classes: td 1 (E) = td 0 (E) = td 1 (E) = rank(e) i=1 rank(e) i=1 rank(e) i=1 (1 + α) = c(e), the total Chern class, α = td(e), the total Todd class, 1 e α ). α = L(E), the total Thom Hirzebruch L-class. tanh α is equal to T 1 : K 0(V/ ) H BM Then c Al is the unique natural transformation satisfying the normalization condition that (1 + α i ). Then td y (E) is a unification of the ( ) Q, since T 1 : K 0(V/ ) H BM ( ) Q T 1 ([X id X X]) = c(t X ) [X] for a smooth X. Thus for any variety X, singular or non-singular, we have c Al ([X id X X]) = c SM (X) = c ( 1 X ) In particular X c ( 1 X ) = χ(x) the topological Euler Poincaré characteristic, which is a contradiction to the fact that c Al ([X id X X]) = ( 1) dim X χ (X). Remark 4.9. In fact c 1 X X is equal to the motivic Chern class transformation T 1 : K 0(V/X) H BM (X) H BM (X) Q. is a ring with the following fiber product [V h X] [W k X] := [V X W h X k X]. Proposition The operation h h δ X of pullback followed by pushforward of a constructible function makes F(X) a -module with the product [V h X] δ X := h h δ X. Namely, the following properties hold: [V h X] (δ X + δ X ) = [V h X] δ X + [V h X] δ X. ([V h X] + [W k X]) δ X = [V h X] δ X + [W k X] δ X.

12 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 37 ([V h X] [W k X]) δ X = [V h X] ([W k X] δ X ). [X id X X] δ X = δ X. Then the operation h h δ X gives rise to a map Φ : F(X) F(X) and the composition Φc := c Φ : F(X) H BM (X) of Φ and MacPherson s Chern class transformation c is a kind of extension of c. Lemma For any morphism g : X Y the following diagram commutes: K 0 (V/Y ) F(Y ) F(Y ) g g g Φ F(X) Φ F(X). Corollary For a smooth morphism g : X Y the following diagram commutes: Φc K 0 (V/Y ) F(Y ) H BM (Y ) g g c(t g) g F(X) Φc H BM (X). Remark Fix δ Y F(Y ), the composite of the inclusion homomorphism i δy : K 0 (V/Y ) K 0 (V/Y ) F(Y ) defined by i δy (α) := α δ Y and the map Φ : K 0 (V/Y ) F(Y ) F(Y ) is the homomorphism [δ Y ]; Φ i δy = [δ Y ] : K 0 (V/F ) F(Y ). The right-hand-sided commutative diagram in Lemma 4.5 is the outer square of the following commutative diagrams: i δy Φ K 0 (V/Y ) K0 (V/Y ) F(Y ) F(Y ) g g g g i g δy F(X) Φ F(X). Furthermore, if g : X Y is smooth, we get the following commutative diagrams: K 0 (V/Y ) i δy K0 (V/Y ) F(Y ) Φ F(Y ) c H BM (Y ) g g g g c(t g) g i g δy F(X) Φ F(X) c the outer square of which is the commutative diagram in Corollary 4.6 (2). H BM (X), Remark As to the pushforward we do knot know if there exists a reasonable pushforward? : F(X) K 0 (V/Y ) F(Y ) such that the following diagram commutes: F(X) F(X)? g K 0 (V/Y ) F(Y ) Φ F(Y ). Φ

13 38 VITTORIA BUSSI () AND SHOJI YOKURA () Indeed, for [V h X] δ X F(X) we have that g Φ([V h X] δ X ) = g h h δ X. But we do not know how to define? : F(X) K 0 (V/Y ) F(Y ) such that One possibility would be Φ(? ([V h X] δ X )) = g h h δ X.? = (g? )([V h X] δ X ) = [V gh Y ]? (δ X ) = (gh) (gh) (? (δ X )) = g h h g (? (δ X )), but here we do not know how to define? : F(X) F(Y ) so that g (? (δ X )) = δ X. At the moment we can see only that the following diagrams commute: i g δy F(X) Φ F(X) c H BM (X) g K 0 (V/Y ) i δy K 0 (V/Y ) F(Y ) Φ g g F(Y ) c H BM (Y ) Indeed, in the left long square, we do have that ) (g Φ i g δ Y ) ([V h X]) = g (Φ([V h X] g δ Y ) = g (h h (g δ Y )) = (gh) (gh) δ Y, ( (Φ i δy g ) ([V h X]) = Φ i δy ([V Thus the left long square is commutative. ) gh Y ]) = Φ([V gh Y ] δ Y ) = (gh) (gh) δ Y. 5. NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES In this section we give a further generalization of the above generalized Aluffi class c δ (X), using the motivic Hirzebruch class transformation T y : K 0 (V/ ) H BM ( ) Q[y]. In the above argument, a key part is the operation of pullback-followed-by-pushforward h h for a morphism h : V X on a fixed or chosen constructible function δ X of the target space X. It is quite natural to do the same operation on itself. For that purpose we need to define a motivic element δx mot K 0(V/X) corresponding to the constructible function δ X ; in particular we need to define a reasonable motivic element νx mot K 0(V/X) corresponding to the Behrend function ν X F(X). By considering the isomorphism 1 : Z(X) = F(X), 1 ( V n V [V ]) := V n V 1 V, we define another distinguished integral cycle: D X := 1 1 (ν X ) ( = 1 1 Eu(C X ) ). Then we set ν mot X := [D X X]. This can be put in as follows. Let s : F(X) be the section of 1 : F(X) defined by s( 1 S ) := [S X]. Then νx mot = s(ν X). Another way is νx mot := n n[ν 1 X (n) X] (see [10]). Remark 5.1. Obviously the homomorphism [ 1 X ] = 1 : F(X) is not injective and its kernel is infinite. In the case when X is the critical set of a regular function f : M C, then there is a notion of motivic element (which is called the motivic Donaldson Thomas invariant ) corresponding to the Behrend function (which is in this case described via the Milnor fiber), using the motivic Milnor fiber, due to Denef Loeser. In our general case, we do not have such a sophisticated machinery available, thus it seems to be natural to define a motivic element νx mot naively as above. Let Ψ : be the fiber product mentioned before: ( ) Ψ [V h X] [W k X] := [V h X] [W k X] = [V X W h X k X].

14 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 39 Since [δ X ] = Φ i δx : i δ X K0 (V/X) F(X) Φ F(X) with δ X F(X), we consider its motivic analogue, which means the following homomorphism [γ X ] : iγ X Ψ, where γ X and i γx : is defined by i γx (α) := α γ X. Proposition 5.2. Let γ X. Then the following diagram commutes: [γ X ] [1 (γ X )] F(X). 1 Proof. Let γ X := [S h S X]. Then it suffices to show the following ( [ ]) [ ( )] 1 [S h S X] ([V h X]) = 1 [S h S X] ([V h X]). This can be proved using the fiber square V X S h S h S h S V X. h ( [ ]) ([ ] 1 [S h S X] ([V h X]) = 1 [S h S X] ([V ) h X]) = 1 ([V X S h h S X]) = (h h S ) 1 V X S (by the definition of 1 ) = h hs 1 V X S = h hs h 1 S = h h (h S ) 1 S (since h S h = h ( (h S ) ) ) = h h 1 ([S h S X]) [ ( )] = 1 [S h S X] ([V h X]). Corollary 5.3. we have (1) Let δ X F(X) and let δ mot X K 0(V/X) be such that 1 (δx mot) = δ X. Then [δ mot X ] [δ X ] F(X). 1 The motivic element δ mot X is called a naive motivic lift of δ X.

15 40 VITTORIA BUSSI () AND SHOJI YOKURA () (2) In particular, we have [ν mot X ] [ν X ] F(X). 1 Remark 5.4. Here we emphasize that the following diagrams commutes: [ν mot X ] T 1 [ν X ] F(X) 1 c Q H BM (X) Q. Thus, modulo the torsion and the choices of motivic elements νx mot, the composite T 1 [νmot X ] is a higher class analogue of the Donaldson Thomas type invariant. Thus it would be natural to generalize the Donaldson Thomas type invariant using the motivic Hirzebruch class T y. Let γ X, γ Y commute: K 0 (V/Y ). Then for any morphism g : X Y the following diagrams [γ X ] i γx Ψ K0 (V/X) g g or g g g g K 0 (V/Y ) [g γ X ] [g γ Y ] K 0 (V/Y ),, K 0 (V/Y ) i gγx K 0 (V/Y ) K 0 (V/Y ) Ψ K 0 (V/Y ) [γ Y ] i γy Ψ K 0 (V/Y ) K 0 (V/Y ) K 0 (V/Y ) K0 (V/Y ) K 0 (V/Y ) K 0 (V/Y ) g g or g g g g i g γy Ψ g [g γ Y ] g K 0 (V/Y ) K 0 (V/Y ). [γ Y ] The last commutative diagram is a bit more precisely the following i g γy Ψ g K 0 (V/Y ) i γy K 0 (V/Y ) K 0 (V/Y ) Ψ g K 0 (V/Y ) Here we do not know how to define a homomorphism in the middle so that the diagrams commute, just like in the case discussed in Remark 4.14.

16 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 41 Corollary 5.5. (1) Let γ X, γ Y K 0 (V/Y ). For a proper morphism g : X Y the following diagrams commute: g T y [γ X ] H BM (X) Q[y] g g T y [g γ Y ] H BM (X) Q[y] g K 0 (V/Y ) T y [g γ X ] H BM (Y ) Q[y], K 0 (V/Y ) T y [γ Y ] H BM (Y ) Q[y], (2) For a proper smooth morphism g : X Y and for γ Y K 0 (V/Y ) the following diagrams are commutative: T y [γ Y ] (Y ) Q[y] K 0 (V/Y ) g H BM td y(t g) g T y [g γ Y ] H BM (X) Q[y]. (3) Let ν X mot := ( 1)dim X νx mot, the signed one. Let T y := T y [ ν X mot ]. For a proper smooth morphism g : X Y the following diagrams are commutative: g T y H BM (X) Q[y] g K 0 (V/Y ) g T y H BM (Y ) Q[y] td y(t g) g K 0 (V/Y ) T y H BM (Y ) Q[y], T y H BM (X) Q[y]. Remark 5.6. The commutative diagram in Proposition 5.2 can be described in more details as follows: i γx Ψ id i 1X F(X) Ψ id i 1X F(X) id Φ F(X) Φ Φ F(X) If we denote Φ(α δ X ) simply by α δ X, then the bottom square on the right-hand-side commutative diagrams means that (α β) δ X = α (β δ X ), i.e. the associativity. Remark 5.7. We remark that the following diagrams commute: (1) for a proper marphism g : X Y n {}}{ Ψ n 1 T y H BM (X) Q[y] g g g g K 0 (V/Y ) K 0 (V/Y ) }{{} n Ψ n 1 K 0 (V/Y ) T y H BM (Y ) Q[y],

17 42 VITTORIA BUSSI () AND SHOJI YOKURA () (2) for a proper smooth morphism g : X Y n {}}{ Ψ K 0 (V/Y ) K 0 (V/Y ) n 1 T y K 0 (V/Y ) H BM (Y ) Q[y] g g g c(t g) g }{{} n Ψ n 1 T y H BM (X) Q[y], Here Ψ n 1 ([V X]) := [V X] [V X] is the fiber product of n copies of [V X]. When n = 1, Ψ 0 := id K0(V/X) is understood to be the identity. Let P (t) := a i t i Q[t] be a polynomial. Then we define the polynomial transformation Ψ P (t) : by Ψ P (t) ([V h X]) := a i Ψ i 1 ([V X]). Then we have the following commutative diagrams. (1) for a proper morphism g : X Y Ψ P (t) T y H BM (X) Q[y] g g g K 0 (V/Y ) Ψ P (t) K 0 (V/Y ) T y H BM (Y ) Q[y], (2) for a proper smooth morphism g : X Y Ψ P (t) T y K 0 (V/Y ) K 0 (V/Y ) H BM (Y ) Q[y] g g c(t g) g Ψ P (t) T y H BM (X) Q[y], These are a motivic analogue of the corresponding case of constructible functions: (1) for a proper morphism g : X Y F P (t) c F(X) F(X) H BM (X) g g g F(Y ) F P (t) F(Y ) c H BM (Y ) (2) for a proper smooth morphism g : X Y F(Y ) F P (t) F(Y ) c H BM (Y ) g g c(t g) g F(X) F P (t) F(X) c H BM (X)

18 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 43 Here F P (t) (β) := a i β i. Note also that the following diagram commutes 1 Ψ P (t) 1 Definition 5.8. F(X) F P (t) (1) We refer to the following class T y (X) := ( T y F(X). ) ([X id X X]) = Ty ([ ν mot X ]) as the naive motivic Donaldson Thomas type Hirzebruch class of X. (2) The degree zero of the naive motivic Donaldson Thomas type Hirzebruch class is called the naive motivic Donaldson Thomas type χ y -genus of X: χ y (X) := X T y (X). Remark 5.9. The cases of the three special values y = 1, 0, 1 are the following. (1) For y = 1, T 1 (X) = T 1 ([ νmot X ]) = cal (X). (2) For y = 0, T 0 (X) = T 0 ([ ν X mot ]) =: tdal (X), which we call an Aluffi-type Todd class of X. (3) For y = 1, T 1 (X) = T 1 ([ ν X mot ]) =: LAl (X), which we call an Aluffi-type Cappell Shaneson L-homology class of X. The degree zero part of these three motivic classes are respectively: (1) for y = 1, χ 1 (X) = ( 1) dim X χ (X), the original Donaldson Thomas type invariant (i.e. Euler characteristic) of X with the sign; (2) for y = 0, χ 0 (X) =: χ a (X), which we call a naive Donaldson Thomas type arithmetic genus of X and (3) for y = 1, χ 1 (X) = σ (X), which we call a naive Donaldson Thomas type signature of X. Remark Since ν X (x) = 1 for a smooth point x X, we have that ν X = 1 X + α Xsing for some constructiblee functions α Xsing supported on the singular locus X sing. For example, consider the simplest case that X has one isolated singularity x 0, say ν X = 1 X + a 0 1 x0. Then Here x 0 ν mot X i x0 X is the inclusion. Hence we have = [X id i X x0 X] + a 0 [x 0 X] K0 (V/X). T y (X) = T y ( ν X mot ) = T y ([X id i X x0 X] + a 0 [x 0 X]) = T y (X) + a 0 (i x0 ) T y (x 0 ) = T y (X) + a 0. Thus the difference between the motivic type Hirzebruch class T y (X) and the motivic Hirzebruch class T y (X) is just a 0, independent of the parameter y. Of course, if dim X sing 1, then the difference does depend on the parameter y. For example, for the sake of simplicity, assume that ν X = 1 X +a 1 Xsing. Then the difference is T y (X) T y (X) = a(i Xsing ) T y (X sing ), which certainly depends on the parameter y, at least for the degree zero part χ y (X sing ).

19 44 VITTORIA BUSSI () AND SHOJI YOKURA () If we take a different motivic element ν mot X = [X id X X] + [V h X] such that 1 ([V h X]) = a 0 1 x0 and dim V 1, then the difference T y (X) T y (X) = h (T y (V )), thus it does depend on the parameter y, at least for the degree zero part, again. In the case when X is the critical locus of a regular function f : M C, the motivic invariant νx motivic which -theory people consider, using the motivic Milnor fiber, is the latter case, simply due to the important fact that the Behrend function can be expressed using the Milnor fiber. For example, as done in [9], even for an isolated singularity x 0, the difference T y (X) T y (X) is, up to sign, the χ y -genus of (the Hodge structure of) the Milnor fiber at the singularity x 0, so does depend on the parameter y. So, as long as the Behrend function has some geometric or topological descriptions, e.g., such as Milnor fibers, then one could think of the corresponding motivic elements in a naive or canonical way. We will hope to come back to properties of these two classes td Al (X), L Al (X) and χ a (X), σ (X) and discussion on some relations with other invariants of singularities. Remark In [9] Cappell et al. use the Hirzebruch class transformation MHM T y : K 0 (MHM(X)) H BM (X) Q[y, y 1 ] from the Grothendieck group K 0 (MHM(X)) of the category of mixed Hodge modules (introduced by Morihiko Saito), instead of the Grothendieck group. We could do the same things on MHM T y : K 0 (MHM(X)) H BM (X) Q[y, y 1 ] and get MHM-theoretic analogues of the above. We hope to get back to this calculation. Remark In [14] Göttsche and Shende made an application of the above motivic Hirzebruch class MHM T y. A bit more precisely, for a family π : C B of plane curves they introduce certain invariants N i C/B K 0(MHM(B)) and apply the above functor to these invariant N i C /B : MHM T y : K 0 (MHM(B)) H BM (B) Q[y, y 1 ] N i C/B (y) := MHM T y (N i C/B ), which are used to make some formulations and some conjectures. Remark In a successive paper, we intend to apply the motivic Hirzebruch transformation to the motivic vanishing cycle constructed on the Donaldson Thomas moduli space and announced in [6, 8]. This will hopefully provide the right motivic Donaldson Thomas type Hirzebruch class. 6. A BIVARIANT GROUP OF PULLBACKS OF CONSTRUCTIBLE FUNCTIONS AND A BIVARIANT-THEORETIC PROBLEM In the above section we mainly dealt with the class c δ X (V h X) of h : V X supported on the target space X. In this section we deal with the class c δ X (V h X) of h : V X supported on the source space V. The class c δ X (V h X) is by definition c (h h δ X ) = h c (h δ X ) H BM (X), and can be captured as the image of a homomorphism between two abelian groups assigned to the space X, as done in the previous sections. However, when it comes to the case of c δ X (V h X) H BM (V ), one cannot do so, i.e. one cannot capture it as the image of a homomorphism between two abelian groups assigned to the space V. So we approach this class from a bivariant-theoretic viewpoint as follows.

20 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 45 For a morphism f : X Y and a constructible function δ Y F(Y ), we define F δ Y (X f Y ) as follows: { } F δ Y (X f Y ) := a S i S i Sf δ Y S are closed subvarieties of X, as Z F(X), S where i S : S X is the inclusion map. Thus, using this notation, for a morphism h : V X and for a constructible function δ X F(X), we have that h δ X F δ X (V h X) F(V ). For the sake of simplicity, unless some confusion is possible, we simply denote i S (i S ) f δ Y by (f S ) δ Y (= (i S ) f δ Y ). In particular, let us consider the signed Behrend function ν Y as δ Y, i.e., F ν Y (X f Y ), which shall be denoted by F Beh (X f Y ). It is easy to prove the following lemma. Lemma 6.1. (1) If Y is smooth, then F Beh (X f Y ) = F(X). (2) F Beh (X π pt) = F(X). (3) If X is smooth, F Beh (X id X X) = F(X). (4) If Y is singular and f(x) Y sing =, F Beh (X f Y ) = F(X). (5) If Y is singular, f(x) Y sing and there exists a point y f(x) Y sing such that ν Y (y) > 1, F Beh (X f Y ) F(X). Remark 6.2. In an earlier version of the paper, in the above lemma we stated If X is singular, then id X F Beh (X X) F(X) and in particular, the characteristic function 1 X F Beh (X id X X). However the referee pointed out that this is not obvious, and we have realized that F Beh (X id X X) = F(X) is also possible. If X is a plane curve with a node x 0, then ν X (x 0 ) = Eu X (x 0 ) = 2, in which case we get F Beh (X id X X) F(X). Let X be the union of a reduced surface Y with an isolated singular point x 0 such that Eu Y (x 0 ) = m with m > 1 and a reduced curve C with the isolated singular point being the same x 0 such that Eu C (x 0 ) = m 1, where we assume that Y C = {x 0 }. For example, the following is such a (non-pure dimensional) surface: Let Y be a projective cone of a non-singular curve of degree d(> 3) with the cone point x 0. Then Eu Y (x 0 ) = 2d d 2 (see [29, p. 426]). Hence ν Y = ( 1) 2 Eu Y = Eu Y. Now let C be a plane curve with x 0 being a (2d d 2 + 1)-ple point such that Y C = {x 0 }. Then let us set X = Y C. Then we have ν X = ( 1) 2 Eu Y +( 1) 1 Eu C, hence ν X (x 0 ) = 2d d 2 (2d d 2 + 1) = 1, and ν X (y) = 1 for y Y {x 0 } and ν X (y) = 1 for y C {x 0 }. Then we have 1 X = i Y i Y ν X + ( 1)i C i C ν X + i x0 i x0 ν X F Beh (X id X X). If 1 X F Beh (X id X X), then any constructible function belongs to F Beh (X id X X), thus we get F Beh (X id X X) = F(X). In passing, at the moment we do not know an example of a pure dimensional singular variety X which satisfies F Beh (X id X X) = F(X). In order to show that F Beh (X f Y ) is a bivariant theory in the sense of Fulton and MacPherson [13], first we quickly recall some basics about Fulton MacPherson s bivariant theory. Definition 6.3. A bivariant theory B on a category C assigns to each morphism X f Y in the category C a (graded) abelian group B(X f Y ). This bivariant theory is equipped with the following three basic operations:

21 46 VITTORIA BUSSI () AND SHOJI YOKURA () (i) for morphisms X f Y and Y g Z, the product operation : B(X f Y ) B(Y g Z) B(X gf Z) is defined; (ii) for morphisms X f Y and Y g Z with f proper, the pushforward operation is defined; (iii) for a fiber square is defined. X g f Y f : B(X gf Z) B(Y g Z) X f g Y, the pullback operation g : B(X f Y ) B(X f Y ) These three operations are required to satisfy the seven axioms which are natural properties to make them compatible each other: (B1) product is associative; (B2) pushforward is functorial; (B3) pullback is functorial; (B4) product and pushforward commute; (B5) product and pullback commute; (B6) pushforward and pullback commute; (B7) projection formula. Definition 6.4. Let B and B be two bivariant theories on a category C. Then a Grothendieck transformation from B to B γ : B B is a collection of morphisms B(X f Y ) B (X f Y ) for each morphism X f Y in the category C, which preserves the above three basic operations. As to the constructible functions we recall the following fact from [40]: Theorem 6.5. If we define F(X f Y ) := F (X) (ignoring the morphism f), then it become a bivariant theory, called the simple bivariant theory of constructible functions with the following three bivariant operations: (bivariant product) : F(X f Y ) F(Y g Z) F(X gf Z), α β := α f β. (bivariant pushforward) For morphisms f : X Y and g : Y Z with f proper f : F(X gf Z) F(Y g Z) f α := f α.

22 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 47 (bivariant pullback) For a fiber square X g f X f Y g Y, g : F(X f Y ) F(X f Y ) g α := (g ) α. Theorem 6.6. Here we consider the category of complex algebraic varieties. Then the above group F Beh (X f Y ) becomes a bivariant theory as a subtheory of the above simple bivariant theory F(X f Y ), provided that we consider smooth morphisms g for the bivariant pullback. Proof. All we have to do is to show that those three bivariant operations are well-defined on the subgroup F Beh (X f Y ). Below, as to bivariant product and bivariant pushforward, we do not need the requirement that δ Y is the signed Behrend function ν Y, but we need it for bivariant pullback. (1) (bivariant product) It suffices to show that (f S ) δ Y (g W ) δ Z = (f S ) δ Y f (g W ) δ Z F δ Z (X gf Z). = V a V 1 V where V s are subva- Since (f S ) δ Y is a constructible function on S, (f S ) δ Y rieties of S, hence subvarieties of X. Thus we get (f S ) δ Y f (g W ) δ Z = V a V 1 V (gf f 1 (W )) δ Z = V a V (gf f 1 (W ) V ) δ Z Since f 1 (W ) V is a finite union of subvarieties, it follows that (f S ) δ Y f (g W ) δ Z F δ Z (X gf Z). (2) (bivariant pushforward) It suffices to show that f ((gf S ) δ Z ) F δ Z (Y g Z). More precisely, f ((gf S ) δ Z ) = f (i S ) (f S ) g δ Z ) = (f S ) (f S ) g δ Z. Now it follows from Verdier s result [37, (5.1) Corollaire] that the morphism f S : S Y is a stratified submersion, more precisely there is a filtration of closed subvarieties V 1 V 2 V m Y such that the restriction of f S to each strata V i+1 \ V i, i.e., (f S ) 1 (V i+1 \ V i ) V i+1 \ V i is a fiber bundle. Hence the operation (f S ) (f S ) is the same as the multiplication ( m i=1 a i 1 Vi ) with some integers a i s, i.e., (f S ) (f S ) g δ Z = ( i a i 1 Vi ) g δ Z = i a i (g Vi ) δ Z F δ Z (Y g Z). Here we remark that the above integer a i is expressed as follows. Let χ i denote the Euler- Poincaré characteristic of the fiber of the above fiber bundle (f S ) Vi\V i 1. Then a m = χ m and a i = χ i m χ j for 1 i < m. j=i+1

23 48 VITTORIA BUSSI () AND SHOJI YOKURA () (3) (bivariant pullback) Here we show that the following is well-defined Consider the following fiber squares: g : F δ Y (X f Y ) F g δ Y (X f Y ). i S S g S i S X g f X f Indeed, Y g ((f S ) δ Y ) = (g ) ((f S ) δ Y g Y. (by definition) = (g ) ((i S ) (f S ) δ Y (more precisely) = (i S ) (g ) (i S ) f δ Y = (i S ) (i S ) (f ) g δ Y F g δ Y (X f Y ). Hence, if δ Y is the signed Behrend function ν Y, then for a smooth morphism g : Y Y we have ν Y = g ν Y, thus the pullback g : F Beh (X f Y ) F Beh (X f Y ) is well-defined. Here we note that for any constructible functions δ Y which are preserved by smooth morphisms g : Y Y, i.e. δ Y = g δ Y, the subgroups F δ Y (X f Y ) give rise to a bivariant theory. Problem 6.7. Define a bivariant homology theory H(X Y ) such that (1) H(X f Y ) H BM (X) for any morphism f : X Y, (2) H(X Y ) = H BM (X) for a smooth Y, (3) the homomorphism c : F Beh (X f Y ) H(X f Y ) defined by c (i S i S f ν Y ) := i S c (i S f ν Y ) H BM (X) and extended linearly, becomes a Grothendieck transformation. (4) if Y is a point pt, then c : F (X) = F Beh (X f pt) H(X f pt) = H BM (X) is equal to the original MacPherson s Chern class homomorphism. Remark 6.8. One simple-minded construction of such a bivariant homology theory H(X Y ) could be simply the image of F Beh (X f Y ) under MacPherson s Chern class c : F(X) H BM (X). It remains to see whether the image H(X Y ) := c (F Beh (X f Y )) gives rise to a bivariant theory. Before closing this section, we mention a bivariant-theoretic analogue of the covariant functor L of conical Lagrangian cycles. For the covariant functor of conical Lagrangian cycles, see [33, 21, 22]. In [21] Kennedy proved that Ch : F (X) = L(X) is an isomorphism. In general, suppose we have a correspondence H such that H assigns an abelian group H(X) to a variety X there is an isomorphism Θ X : F (X) = H(X).

24 NAIVE MOTIVIC DONALDSON THOMAS TYPE HIRZEBRUCH CLASSES 49 Then, by transfer of structure using the above isomorphism Θ, we can get the corresponding bivariant theory. Here we go into a bit more details. If we define the pushforward f : H(X) H(Y ) for a map f : X Y by f H := Θ Y f F Θ 1 X : H(X) H(Y ), then the correspondence H becomes a covariant functor via the covariant functor F. Here f F : F (X) F (Y ), emphasizing the covariant functor F. Similary, if we define the pullback f : H(Y ) H(X) by f H := Θ X f F Θ 1 Y : H(Y ) H(X), then the correspondence H becomes a contravariant functor via the contravariant functor F. Here ff : F (Y ) F (X). Furthermore, if we define BH(X f Y ) := H(X) then we get the simple bivariant-theoretic version of the correspondence H as follows: (Bivariant product) BH : BH(X f Y ) BH(Y g Z) BH(X gf Z) is defined by ( α BH β := Θ X Θ 1 Y (α) F Θ 1 X ). (β) (Bivariant pushforward) f BH : BH(X gf Z) BH(Y g Z) is defined by f BH := Θ Y f F Θ X H 1. (Bivariant pullback) g BH : BH(X f Y ) BH(X f Y ) is defined by g BH := Θ X f F Θ 1 X. Clearly we get the canonical Grothendieck transformation γ Θ = Θ : F(X f Y ) BH(X f Y ). If we apply this argument to the conical Lagrangian cycle L(X) we get the simple bivariant theory of conical Lagrangian cycles L(X f Y ) and also we get the canonical Grothendieck transformation γ Ch = Ch : F(X f Y ) L(X f Y ). This simple bivariant theory L(X f Y ) can be defined or constructed directly, which would be however harder. Indeed, it is done in [7] and one has to go through many geometric and/or topological ingredients. Fulton MacPherson s bivariant theory F F M (X f Y ) is a subgroup (or a subtheory) of the simple bivariant theory F(X f Y ) = F (X). Then if we define L F M (X f Y ) := γ Ch (F F M (X f Y )) then we can get a finer bivariant theory of conical Lagrangian cycles, putting aside the problem of how we define or describe such a finer bivariant-theoretic conical Lagrangian cycle; it would be much harder than the case of the simple one L(X f Y ) done in [7].

25 50 VITTORIA BUSSI () AND SHOJI YOKURA () 7. SOME MORE QUESTIONS AND PROBLEMS 7.1. A categorification of Donaldson Thomas type invariant of a morphism. The cardinality c(f ) of a finite set F, i.e., the number of elements of F, satisfies that (1) X = X (set-isomorphism) = c(x) = c(x ), (2) c(x) = c(y ) + c(x \ Y ) for a subset Y X (a scissor relation), (3) c(x Y ) = c(x) c(y ), (4) c(pt) = 1. Now, let us suppose that there is a similar cardinality on a category T OP of certain reasonable topological spaces, satisfying the above four properties, except for the condition (1) and (2), (1) X = X (T OP-isomorphism) = c(x) = c(x ), (2) c(x) = c(y ) + c(x \ Y ) for a closed subset Y X. (3) c(x Y ) = c(x) c(y ), (4) c(pt) = 1. If such a topological cardinality exists, then we can show that c(r 1 ) = 1, hence c(r n ) = ( 1) n (e.g. see [41]). Thus, for a finite CW -complex X, c(x) is exactly the Euler Poincaré characteristic χ(x). The existence of such a topological cardinality is guaranteed by the ordinary homology theory, more precisely c(x) = χ c (X) := ( 1) i dim R H i c(x; R) = i ( 1) i dim R Hi BM (X; R). Here H BM (X) is the Borel Moore homology group of X. Similarly let us suppose that there is a similar cardinality on the category V C of complex algebraic varieties: (1) X = X (V C -isomorphism) = c(x) = c(x ), (2) c(x) = c(y ) + c(x \ Y ) for a closed subvariety Y X (i.e., a closed subset in Zariski topology), (3) c(x Y ) = c(x) c(y ), (4) c(pt) = 1. The complex affine line C 1 is corresponding to the real line R 1. But we cannot do the same trick for C 1 as we do for R 1. The existence of such an algebraic cardinality is guaranteed by Deligne s theory of mixed Hodge structures. Let u, v be two variables, then the Deligne Hodge polynomial χ u,v is defined by χ u,v (X) = ( 1) i dim C Gr p F GrW p+q(h i c(x; C))u p v q. In particular, χ u,v (C 1 ) = uv. The particular case when u = y, v = 1 is the important one for the motivic Hirzebruch class:χ y (X) := χ y,1 (X) = ( 1) i dim C Gr p F (Hi c(x; C))( y) p. This is called χ y -genus of X. Similarly let us consider the Donaldson Thomas type invariant of morphisms: (1) X f Y = X f Y (isomorphism) = χ (X f Y ) = χ (X f Y ), (2) χ (X f Y ) = χ (Z f Z Y ) + χ (X \ Z f X\Z Y ) for a closed subvariety Z X. (3) χ f 1 f 2 (X 1 X 2 Y1 Y 2 ) = χ f 1 (X 1 Y1 ) χ f 2 (X 2 Y2 ), (4) χ (pt) = 1. So, just like the above two cardinalities or counting χ c (X) and χ u,v (X), we pose the following problem, which is related to the above Problem 6.7: