Motivic Multiplicative Mckay Correspondence for Surfaces

Home , Patrick du Val

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Irr′(G) Irr(E) ≃ ρ E . 7→ ρ = Thus E ρ Irr′(G) Eρ. Moreover, this bijection respects the ‘incidence relations’ : precisely, for any S ∈ ρ , ρ Irr′(G), the intersection number (E E ), which is 0 or 1, is equal to the multiplicity of 1 2 ∈ ρ1 · ρ2 ρ2 in ρ1 V (hence is also equal to the multiplicity of ρ1 in ρ2 V), where V is the 2-dimensional natural representation⊗ via G SL(V). All these informations can⊗ be encoded into Dynkin diagrams of A-D-E type, which is on the⊂ one hand the dual graph of the exceptional divisor E and on the other hand the McKay graph of the non-trivial irreducible representations of G, with respect to the preferred representation V. Apart from the original observation of McKay, there are many approaches to construct this correspondence geometrically and to extend it to higher dimensions: K-theory of sheaves [21], G-Hilbert schemes [29], [24], [23], [22], motivic integration [4], [5], [13], [14], [35], [27] and derived categories [8] etc. We refer the reader to Reid’s note of his Bourbaki talk [30] for more details and history. Following Reid [29], one can recast the above McKay correspondence (the bijection) as follows: the isomorphism classes of irreducible representations index a basis of the homology of the resolution Y. This is of course equivalent to say that the conjugacy classes of G index a basis of the cohomology of Y. The starting point of this paper is that the quotient X = V/G is the coarse moduli space of a smooth orbifold/Deligne–Mumford stack := [V/G], and that the (co)homology of the coarse moduli space I of its inertia stack I has aX basis indexed by the conjugacy classes of G. Thus Reid’s McKay| correspondenceX| can beX stated as an isomorphism of vector spaces:

H∗(Y) H∗( I ). ≃ | X| Chen and Ruan defined in [11] the orbifold cohomology and the orbifold product (i.e. Chen–Ruan cohomology) for any smooth orbifold. See Definition 2.1 for a down-to-earth construction in the global quotient case. By definition, the orbifold cohomology ring H ([V/G]) has H ( I ) orb∗ ∗ | X| as the underlying vector space. Therefore it is natural to ask whether there is a multiplicative isomorphism (of algebras)

H∗(Y) H∗ ([V/G]). ≃ orb None of the aforementioned beautiful theories (K-theory, G-Hilbert schemes, motivic integration and derived categories) produces an isomorphism which respects the multiplicative structures. Nevertheless, the existence of such an isomorphism of algebras is known. For example, it is a baby case of the result of Ginzburg–Kaledin [20] on symplectic resolutions of symplectic quotient singularities. An explicit isomorphism between the equivariant orbifold quantum cohomology of [V/G] and the equivariant cohomology of its minimal resolution is proposed by Bryan–Graber– 2 Pandharipande in [10], which is veriﬁed for the C /Z3 case (see also the related work [7], [9]). We will use the same formula to construct our multiplicative isomorphism. This isomorphism ﬁts perfectly into Ruan’s following more general Cohomological Crepant Resolution Conjecture (CCRC) :

Conjecture 1.1 (CCRC [31]). Let M be a smooth projective variety endowed with a faithful action of a ﬁnite group G. Assume that the quotient X := M/G is Gorenstein, then for any crepant resolution Y X, there is an isomorphism of graded C-algebras: →

(1) H∗ (Y, C) H∗ ([M/G], C) . qc ≃ orb MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 3

More generally, given a smooth proper orbifold with underlying singular variety X being Gorenstein, then for any crepant resolution Y X, we have anX isomorphism of graded C-algebras: → H∗ (Y, C) H∗ ( , C) . qc ≃ orb X

Here the left hand side is the quantum corrected cohomology algebra, whose underlying graded vector space is just H∗(Y, C), endowed with the cup product with quantum corrections related to Gromov–Witten invariants with curve classes contracted by the crepant resolution, as defined in [31]. Since we only consider in this paper the two-dimensional situation, the Gromov– Witten invariants always vanish hence there are no quantum corrections involved. See Lemma 2.3 for this vanishing. Conjecture 1.1 suggests that one should consider the existence of such multiplicative McKay correspondence in the global situation (instead of a quotient of a vector space by a finite group), that is, a Gorenstein quotient of a surface by a finite group action, or even more generally a two-dimensional proper Gorenstein orbifold. Our following main result confirms this, which also pushes the (surface) McKay correspondence to the motivic level:

Theorem 1.2 (Motivic multiplicative global McKay correspondence). Let be a smooth proper two- dimensional Deligne–Mumford stack with isolated stacky points. Assume thatX has projective coarse moduli space X with Gorenstein singularities. Let Y X be the minimal resolution.X Then we have an → isomorphism of algebra objects in the category CHMC of Chow motives with complex coeﬃcients: (2) h(Y) h ( ) . C ≃ orb X C In particular, one has an isomorphism of C-algebras:

CH∗(Y) CH∗ ( ) ; C ≃ orb X C H∗(Y, C) H∗ ( , C) ; ≃ orb X K0(Y)C Korb ( )C ; ≃ top X Ktop(Y) K ( ) . C ≃ orb X C This result also confirms the 2-dimensional case of the so-called Motivic HyperKähler Resolu- tion Conjecture studied in [19] and [18]. See §2.1 for the basics of Chow motives. As the definitions of the orbifold theories are particularly explicit and elementary for the global quotient stacks (cf. §2.2), we deliberately treat the global quotient case (§3) and the general case (§4) separately. Convention : All Chow rings and K-theories are with rational coefficients unless otherwise stated. CHM is the category of Chow motives with rational coefficients and h : SmProjop CHM is the (contra-variant) functor that associates a smooth projective variety its Chow motive→ (§2.1). An orbifold means a separated Deligne–Mumford stack of finite type with trivial stabilizer at the generic point. We work over an algebraically closed field of characteristic zero. Acknowledgement : The authors want to thank Samuel Boissière, Cédric Bonnafé, Philippe Caldero, Jérôme Germoni and Dmitry Kaledin for interesting discussions and also the referee for his or her very helpful suggestions. The most part of the paper is prepared when Lie Fu is staying with his family at the Hausdorff Institute of Mathematics for the 2017 trimester program on K-theory. He thanks Bonn University for providing the perfect working condition in HIM and the relaxing living style in such a beautiful city. 4 LIE FU AND ZHIYU TIAN

2. Crepant resolution conjecture

Let us give the construction of the orbifold Chow motive (as an algebra object) and the orbifold Chow ring. we will first give the down-to-earth definition for an orbifold which is a global Gorenstein quotient by a finite group; then we invoke the techniques in [1] to give the construction in the general setting of Deligne–Mumford stacks. We refer to our previous work [19] (joint with Charles Vial), [18] as well as the original sources (for cohomology and Chow rings) [11], [17], [1], [25] for the history and more details. For the convenience of the reader, we start with a reminder on the basic notions of Chow motives.

2.1. The category of Chow motives. Theideaof(pure) motives, proposed initially by Grothendieck, is to construct a universal cohomology theory X h(X) for smooth projective varieties. His construction uses directly the algebraic cycles on the7→ varieties together with some natural categorical operations. On the one hand, motives behave just like the classically considered Weil cohomology theories; on the other hand, they no longer take values in the category of vector spaces but in some additive idempotent-complete rigid symmetric mono¨ıdal category. Although the construction works for any adequate equivalence relation on algebraic cycles, we use throughout this paper the finest one, namely the rational equivalence, so that our results will hold for Chow groups and imply the other analogous ones, on cohomology for instance, by applying appropriate realization functors. Fix a base field k. The category of Chow motives over the field k with rational coefficients, denoted by CHM, is defined as follows (cf. [3] for a more detailed treatment). An object, called a Chow motive, is a triple (X, p, n), where n is an integer, X is a smooth projective variety over k and p CHdim X(X X) is a projector, that is, p p = p as correspondences. Morphisms between two ∈ × ◦ m n+dim X objects (X, p, n)to(Y, q, m) form the following Q-vector subspace of CH − (X Y): × m n+dim X HomCHM (X, p, n), (Y, q, m) := q CH − (X Y) p. ◦ × ◦ The composition of morphisms are defined by the composition of correspondences. We have the following naturally defined contra-variant functor from the category of smooth projective varieties to the category of Chow motives: h : SmProjop CHM → X (X, ∆X, 0) 7→ ( f : X Y) tΓ CHdim X(Y X) → 7→ f ∈ × t where Γ f is the transpose of the graph of f . The image h(X) is called the Chow motive of X. The category CHM is additive with direct sum induced by the disjoint union of varieties. By construction, CHM is idempotent complete (i.e. pseudo-abelian) : for any motive M and any projector of it, that is, φ EndCHM(M) such that φ φ = φ, we have M Im(φ) Im(idM φ). As an example, let us recall the∈ definition of the so-called◦ reduced motive: for a smooth⊕ projective− variety x X together with a chosen k-rational point x, the composition of Spec k X with the structure morphism X Spec k is identity, hence defines a projector φ of h(X).−→ The reduced motive of → the pointed variety (X, x), denoted by h(X), is by definition Im(id φ). One can show that the h(X) − isomorphism class of h(X) is independente of the choice of the point x (cf. [3, Exemple 4.1.2.1]). There is also a naturale symmetric mono¨ıdal structure on CHM, compatible with the additive structure, given by (X, p, n) (Y, q, m):= (X Y, p q, n + m). ⊗ × × MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 5

Hence the Künneth formula: h(X Y) h(X) h(Y) holds for any smooth projective varieties X and Y. Moreover, this tensor category× is rigid, with⊗ the following duality functor t (X, p, n)∨ := (X, p, dim X n). − Given an integer n, the motive (Spec k, ∆Spec k, n) is called the n-th Tate motive and is denoted by 1(n). They are the tensor invertible objects. For any motive M, the tensor product M 1(n) is denoted by M(n) and called the n-th Tate twist of M. In particular, we have the Poicaréduality⊗ : h(X) h(X)(dim X) for any smooth projective variety X. The Lefschetz motive is L := 1( 1). One ∨ − checks that the reduced motive of P1 is isomorphic to L. The functor h is considered as a cohomology theory and it is universal in the sense that any Weil cohomology theory must factorize through h. We can extend the notion of Chow groups from varieties to motives by defining for any integer i and any Chow motive M, CHi(M):= Hom (1( i), M) , CHM − Hence the Chow groups of a smooth projective variety X is recovered as CHi(X) = CHi (h(X)) . In all the above constructions, one can replace for the coefficient field the rational numbers by the complex numbers and obtain the category of complex Chow motives CHMC.

2.2. Orbifold theory : global quotient case. Let M be a smooth projective variety and G be a finite group acting faithfully on M. Assume that G preserves locally the canonical bundle : for any x M fixed by 1 G, the differential D1 SL(TxM). This amounts to require that the quotient X ∈:= M/G has only∈ Gorenstein singularities.∈ Denote by M1 = x M 1x = x the fixed locus of 1,h 1 h ∈ | 1 G, Mh i = M M (with the reduced scheme structure) and := [M/G] the quotient smooth Deligne–Mumford∈ ∩ stack. X Definition 2.1 (Orbifold theories). We define an auxiliary algebra object h(M, G) in CHM with G-action, and the orbifold motive h([M/G]) will be its subalgebra of invariants. The definitions for Chow rings, cohomology and K-theory are similar.

(1◦) For any 1 G, the age function, denoted by age(1), is a Z-valued locally constant function on M1, whose∈ value on a connected component Z is r 1 − j age(1) Z := rank(W ), | X r j j=0

where r is the order of 1, Wj is the eigen-subbundle of the restricted tangent bundle TM Z, for 2πi j | the natural automorphism induced by 1, with eigenvalue e r . The age function is invariant under conjugacy. (2◦) We endow the direct sums h(M, G):= h(M1)( age(1)) M − 1 G ∈ CH (M, G):= CH age(1)(M1) ∗ M ∗− 1 G ∈ H (M, G):= H 2 age(1)(M1) ∗ M ∗− 1 G ∈ K(M, G):= K (M1) M 0 Q 1 G ∈ 6 LIE FU AND ZHIYU TIAN

Ktop(M, G):= Ktop(M1) M Q 1 G ∈ with the natural G-action induced by the following action: for any 1, h G, ∈ 1 h : M1 ≃ Mh1h− −→ x hx. 7→ (3◦) Forany 1 G, deﬁne ∈ r 1 − j 1 V1 := [W ] K (M ) , X r j ∈ 0 Q j=0

whose virtual rank is age(1), where r and Wj ’s are as in (1◦).

= 1 1 (4◦) Forany 11, 12 G, let 13 12− 11− , we deﬁne the (virtual class of ) the obstruction bundle on the ∈1 ,1 ﬁxed locus Mh 1 2i by

<11,12> <11,12> (3) F := V + V + V + TM TM <11,12> K M . 11,12 11 M<11,12> 12 M<11,12> 13 M<11,12> M 0 Q − | ∈ <1,h> The orbifold product ⋆orb is deﬁned as follows: given 1, h G, let ι : M ֒ M be the natural (◦5) inclusion. ∈ → For cohomology: • i 2 age(1) 1 j 2 age(h) h i+j 2 age(1h) 1h ⋆ : H − (M ) H − (M ) H − (M ) orb × → (α, β) ι α M<1,h> ⌣β M<1,h> ⌣ ctop(F1,h) 7→ ∗ | | For Chow groups: • i age(1) 1 j age(h) h i+j age(1h) 1h ⋆ : CH − (M ) CH − (M ) CH − (M ) orb × → (α, β) ι α M<1,h> β M<1,h> ctop(F1,h) 7→ ∗ | · | · For K-theory: • ⋆ : K (M1) K (Mh) K (M1h) orb 0 Q × 0 Q → 0 Q (α, β) ι! α M<1,h> β M<1,h> λ 1(F∨ ) 7→ | · | · − 1,h <1,h> <1,h> where λ 1 is obtained from the Lambda operation λt : K0(M ) K0(M )~t by evaluating− t = 1 (cf. [34, Chapter II, §4]). The deﬁnition for topological→ K-theory is similar. − 1 h 1h For motives: ⋆orb : h(M )( age(1)) h(M )( age(h)) h(M )( age(1h)) is determined • by the correspondence − ⊗ − → − dim M1+dim Mh+age(1)+age(h) age(1h) 1 h 1h δ (ctop(F1,h)) CH − (M M M ), ∗ ∈ × × where δ : M<1,h> M1 Mh M1h is the natural morphism sending x to (x, x, x). → × × (6◦) Finally, we take the subalgebra of invariants whose existence is guaranteed by the idempotent completeness of CHM (see §2.1): G horb ([M/G]) := h (M, G) ; = G CHorb∗ ([M/G]) : (CH∗(M, G), ⋆orb) ; and similarly = G Horb∗ ([M/G]) : (H∗(M, G), ⋆orb) ; G Korb ([M/G]) := (K(M, G), ⋆orb) ; MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 7

top G K ([M/G]) := Ktop(M, G), ⋆ . orb orb These are commutative Q-algebras and dependonly on thestack [M/G] (not the presentation).

2.3. Orbifold theory : general case. Let be a smooth proper orbifold with projective coarse moduli space X with Gorenstein singularities.X Recall that under the Gorenstein assumption, the age function takes values in integers. Define the orbifold Chow motive and orbifold Chow group as follows: h ( ):= h(I )( age) := h(I )( age ), orb X X − ⊕i Xi − i age age CH∗ ( ):= CH∗− (I ):= CH∗− i (I ); orb X X ⊕i Xi wherethetheoryofChowring(withrationalcoefficients)aswellastheintersectiontheoryofastack is the one developed by Vistoli in [33]; the theory of Chow motives for smooth proper Deligne– Mumford stacks is the so-called DMC motives4 developed by Behrend–Manin in [6] and reviewed in Toën [32, §2. First construction], which is proven in [32, Theorem 2.1] to be equivalent to the usual category of Chow motives; I = i I i is the decomposition into connected components X ` X while the age function age is the locally constant function whose value on I i is agei which is Chen–Ruan’s degree shifting number defined in [11, §3.2]. Let us also point outX that Toën’s second construction in [32, §3] of Chow motives of Delign–Mumford stacks is very close to the orbifold Chow motive defined above with the only difference being the age-shifting.

Nowthe keypointis toputaproductstructureon horb( ) and CHorb∗ ( ). Considerthe moduli space K ( , 0), constructed by Abramovich–Vistoli [2], of 3-pointedX twistedX stable maps of genus 0,3 X zero with trivial curve class. It comes equipped with a virtual fundamental class [K ( , 0)]vir 0,3 X ∈ CHdim X K0,3( , 0) together with three (proper) evaluation maps: ei : K0,3( , 0) I with target being the inertiaX stack ([1]). Note that in general, the evaluation morphism hasX target→ Xin a diﬀerent stack, the rigidiﬁed cyclotomic inertial stack ([1, Section 3.4]). However, in the smooth orbifold case, one can prove that the evaluation morphisms of the degree 0 twisted stable maps land in the inertial stack [16, Section 1.3.1]. Pushing forward the virtual fundamental class gives the class

vir 3 γ := (e1, e2, eˇ3) [K0,3( , 0)] CHdim X(I ), ∗ X ∈ X where eˇ3 is the composition of the evaluation map e3 and the involution I I inverting the group element (cf. [1]); and we are using again Vistoli’s Chow groups ([33]).X → TheXorbifold product for the orbifold Chow ring is deﬁned as the action of the correspondence γ:

CH∗ ( ) CH∗ ( ) CH∗ ( ) orb X × orb X → orb X

age kage k age CH∗− (I ) CH∗− (I ) CH∗− (I ) X × X → X (α, β) pr3, pr1∗ (α) pr2∗ (β) γ 7→ ∗ · · It can be checked (cf. [1, Theorem 7.4.1]) that the age shifting makes the above orbifold product additivewithrespecttothedegrees(otherwise,itisnot!). Similarly, we can deﬁne the multiplicative structure on h ( )to be orb X 3 γ CHdim X(I ) = HomCHM h(I )( age) h(I )( age), h(I )( age) ∈ X X − ⊗ X − X − = Hom (h ( ) h ( ), h ( )) . CHM orb X ⊗ orb X orb X

4DMC stands for Deligne–Mumford Chow. 8 LIE FU AND ZHIYU TIAN

Thanks to [1, Theorem 7.4.1], this product structure is associative. On the other hand, when is a ﬁnite group global quotient stack, the main result of [25, §8] implies that the elementary constructionX in §2.2 actually recovers the above abstract construction.

2.4. Crepant resolution conjectures. With orbifold theories being defined, we can speculate that a motivic or K-theoretic version of the Crepant Resolution Conjecture 1.1 should hold. But the problem is that in the definition of the quantum corrections, there is the subtle convergence property which is difficult to make sense in general for Chow groups / motives or for K-theory. Therefore, we will look at some cases that these quantum corrections actually vanish a priori : Case 1: Hyper-Kähler resolution. Thefirstoneis whentheresolution Y is holomorphic symplectic, which implies that all (Chow-theoretic, K-theoretic or cohomological) Gromov–Witten invariants vanish (see the proof of [18, Lemma 8.1]). In this case, we indeed have the following Motivic Hyper-Kähler Resolution Conjecture (MHRC), proposed in [19]: Conjecture 2.2 (MHRC [19], [18]). Let M be a smooth projective holomorphic symplectic variety endowed with a faithful symplectic action of a finite group G. If the quotient X := M/G has a crepant resolution Y X, then there is an isomorphism of algebra object in the category CHM of complex Chow motives: → C h(Y) h ([M/G]). ≃ orb In particular, we have an isomorphism of graded C-algebras:

CH∗(Y) CH∗ ([M/G]) . C ≃ orb C Thanks to the orbifold Chern character isomorphism constructed by Jarvis–Kaufmann– Kimura in [25], MHRC also implies the K-theoretic Hyper-K¨ahler Resolution Conjecture of loc.cit. . Conjecture 2.2 is proven in our joint work with Charles Vial [19] for Hilbert schemes of abelian varieties and generalized Kummer varieties and in [18] for Hilbert schemes of K3 surfaces. Case 2: Surface minimal resolution. The second one is the main purpose of the article, namely the surface case, i.e. dim(Y) = 2. In this case, the vanishing of quantum corrections is explained in the following lemma. Lemma 2.3. Let X be a surface with Du Val singularities and π : Y X be the minimal resolution. Then → the virtual fundamental class of M0,3 Y,β is rationally equivalent to zero for any curve class β which is contracted by π.

Proof. Consider the forgetful-stabilization morphism

f : M0,3 Y,β M0,0 Y,β . → By the general theory, the virtual fundamental class of M0,3 Y,β is the pull-back of the virtual fundamental class of M0,0 Y,β . However,thevirtualdimensionof M0,0 Y,β is (β KY)+(dim Y 3) = 1 since π is crepant. Therefore, both moduli spaces have zero virtual fundamental · class in Chow− −group, cohomology or K-theory.

Thanks to the vanishing of quantum corrections, the motivic version of the Crepant Reso- lution Conjecture 1.1 for surfaces is exactly the content of our main Theorem 1.2. See the precise statement in Introduction. We will ﬁrst give the proof for stacks which are ﬁnite group global quotients in §3, then the proof in the general case in §4. MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 9

3. Proof of Theorem 1.2 : global quotient case

In this section, we show Theorem 1.2 in the following setting: S is a smooth projective surface, G is a ﬁnite group acting faithfully on S such that the canonical bundle is locally preserved (Gorenstein condition), X := S/G is the quotient surface (with Du Val singularities) and Y X is the minimal (crepant) resolution. Recall that L := 1( 1) is the Lefschetz motive in CHM (§2.1→). − For any x S, let ∈ Gx := 1 G 1x = x { ∈ | } be the stabilizer. Let Irr(Gx) be the set of isomorphism classes of irreducible representations of Gx and Irr′(Gx) be that of non-trivial ones. We remark that by assumption, there are only ﬁnitely many points of S with non-trivial stabilizer.

3.1. Resolution side. We first compute the Chow motive algebra (or Chow ring) of the minimal resolution Y. For any x S, we denote by x its image in S/G. The Chow motive of Y has the following decomposition in∈ CHM: G G (4) h (Y) h(S) Lx,ρ h(S) Lx,ρ , ≃ ⊕ M M ≃  ⊕ M M  x S/G ρ Irr′(Gx)  x S ρ Irr′(Gx)  ∈ ∈  ∈ ∈    where Lx,ρ and Lx,ρ are both the Lefschetz motive L corresponding to the irreducible component of the exceptional divisor over x, indexed by the non-trivial irreducible representation ρ of Gx via the classical McKay correspondence. The second isomorphism in (4) being just a trick of reindexing, let usexplain a bit more onthefirstone. Let f : Y S/G be the minimal resolution of singularities. By the classical McKay correspondence, over each→ singular point x S/G, the exceptional divisor = 1 ∈ Ex : f − (x) is a union (with A-D-E configuration) of smooth rational curves ρ Irr′(Gx)Ex,ρ. As f ∪ ∈ is obviously a semi-small morphism, we can invoke the motivic decomposition of De Cataldo– Migliorini [12, Theorem 1.0.1], with the stratification being S/G = (S/G)re1 Sing(S/G), to obtain directly the first isomorphism in (4). It is then not hard to follow the proof∪ in loc.cit. to see that the first isomorphism in (4) is induced by the pull-back f = tΓ : h(S)G = h(S/G) h(Y) together ∗ f → ix,ρ, with the push-forward along the inclusions L = h(E ) ∗ h(Y), where h is the reduced motive x,ρ −−−→ (see §2.1). We remark that the inverse of the isomorphisme (4) is more complicatede to describe and involves the inverse of the intersection matrix (cf. the definition of Γ′ in the end of [12, §2]). The product structure of h(Y) is determined as follows via the above decomposition (4), .which also uses the classical McKay correspondence. Let ix : x ֒ S be the natural inclusion { } → δ h(S) h(S) S h(S) is the usual product induced by the small diagonal of S3. • ⊗ −→ For any x with non-trivial stabilizer Gx and any ρ Irr′(Gx), • ∈ ix∗ h(S) Lx Lx ⊗ ,ρ −→ ,ρ is determined by the class x CH2(S) = Hom(h(S) L, L). ∈ ⊗ For any ρ Irr′(Gx) as above, • ∈ 2ix, Lx Lx − ∗ h(S), ,ρ ⊗ ,ρ −−−−→ is determined by 2x CH2(S). The reason is that each component of the exceptional divisor is a smooth− rational∈ curve of self-intersection number equal to 2. − 10 LIE FU AND ZHIYU TIAN

For any ρ , ρ Irr′(Gx), • 1 2 ∈ – If they are adjacent, that is, ρ1 appears (with multiplicity 1) in the Gx-module ρ2 TxS, then by the classical McKay correspondence, the componentsin the exceptional divisor⊗ over x indexed by ρ1 and by ρ2 intersect transversally at one point. Therefore

ix, Lx Lx ∗ h(S), ,ρ1 ⊗ ,ρ2 −−→ is determined by x CH2(S). – If they are not adjacent,∈ then again the classical McKay correspondence tells us that the two components indexed by ρ1 and ρ2 of the exceptional divisor do not intersect; L L 0 hence x,ρ1 x,ρ2 h(S) is the zero map. The other multiplication⊗ −→ maps are zero. • The G-action on (4) is as follows:

The G-action of h(S) is induced by the original action on S. • For any h G, it maps for any x S and ρ Irr′(Gx), the Lefschetz motive Lx isomor- • ∈ ∈ ∈ ,ρ phically to Lhx,hρ, where hρ Irr′(Ghx) is the representation which makes the following diagram commutes: ∈

1 1 h1h− (5) G 7→ / G x ❆ hx ❆❆ ≃ ⑤⑤ ❆❆ ⑤⑤ ρ ❆❆ ⑤⑤ ❆ ⑤~ ⑤ hρ Vρ

3.2. Orbifold side. Now we compute the orbifold Chow motive algebra of the quotient stack [S/G]. The computation is quite straight-forward. Here L := 1( 1) is the Lefschetz motive. − First of all, it is easy to see that age(1) = 1 for any element 1 , id of G, and age(id) = 0. By Deﬁnition 2.1,

(6) h(S, G) = h(S) Lx,1 = h(S) Lx,1, ⊕ M M1 ⊕ M M 1 G x S x S 1 Gx 1,∈id ∈ ∈ 1∈,id where Lx 1 is the Lefschetz motive 1( 1) indexed by the fixed point x of 1. , − Lemma 3.1 (Obstruction class). For any 1, h G different from id, the obstruction class is ∈ 1 if 1 = h 1 = − c1,h  1 0 if 1 , h−  1 Proof. For any 1 , id and any x S , the action of 1 on TxS is diagonalizable with a pair of ∈ 1 conjugate eigenvalues, therefore V1 in Definition 2.1 is a trivial vector bundle of rank one on S . Hence for any 1, h G different from id and x S fixed by 1 and h, the dimension of the fiber of ∈ ∈ the obstruction bundle F1,h at x is

dim F1 h(x) = dim V1(x) + dim Vh(x) + dim V 1 (x) dim TxS, , (1h)− − 1 1 which is 1 if 1 , h− and is 0 if 1 = h− . The computation of c1,h follows.

Once the obstructionclasses are computed, wecan write down explicitly the orbifold product from Deﬁnition 2.1, which is summarized in the following proposition. MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 11

Proposition 3.2. The orbifold product on h(S, G) is given as follows via the decomposition (6): δ h(S) h(S) S h(S); ⊗ −→ ix∗ h(S) Lx 1 Lx 1 1x = x; ⊗ , −→ , ∀ ix, Lx,1 L 1 ∗ h(S). ⊗ x,1− −−→ where the ﬁrst morphism is the usual product given by small diagonal; the second and the third morphisms 2 are given by the class x CH (S) and ix : x ֒ S is the natural inclusion; all the other possible maps are zero. ∈ { } →

The G-action on (6) is as follows by Deﬁnition 2.1: The G-action on h(S) is the original action. • For any h G, it maps for any x S and 1 , id Gx, the Lefschetz motive Lx 1 isomorphi- • ∈ ∈ ∈ , cally to L 1 . hx,h1h−

3.3. The multiplicative correspondence. With both sides of the correspondence computed, we can give the multiplicative McKay correspondence morphism, which is in the category CHMC of complex Chow motives. Consider the morphism

(7) Φ : h(S) Lx h(S) Lx 1, ⊕ M M ,ρ → ⊕ M M , x S ρ Irr′(Gx) x S 1 Gx ∈ ∈ ∈ 1∈,id which is given by the following ‘block diagonal matrix’: id : h(S) h(S); • → For each x S (with non-trivial stabilizer Gx), the morphism • ∈ Lx Lx 1 M ,ρ → M , ρ Irr′(Gx) 1 Gx ∈ 1∈,id 1 is the ‘matrix’ with coeﬃcient χρ0 (1) 2 χρ(1) at place (ρ, 1) Irr′(Gx) (Gx id ), √ Gx | | p − · ∈ × \{ } where χ denotes the character, ρ0 is the natural 2-dimensional representation TxS of Gx.

Note that ρ0(1) has determinant 1, hence its trace χρ0 (1) is a real number. The other morphisms are zero. • To conclude the main theorem, one has to show three things: (i) Φ is compatible with the G-action; (ii) Φ is multiplicative and (iii) Φ induces an isomorphism ΦG of complex Chow motives on G-invariants. Lemma 3.3. Φ is G-equivariant.

Proof. The G-action on the first direct summand h(S) is by definition the same, hence is preserved by Φ = id. For the other direct summands, since it is a matrix computation, we can treat the |h(S) Lefschetz motives as 1-dimensional vector spaces: let Ex,ρ be the ‘generator’ of Lx,ρ and ex,1 be the ‘generator’ of Lx,1. Then the G-actions computed in the previous subsections say that for any x and any h Gx, ∈ h.E = E and h.e = e 1 , x,ρ hx,hρ x,1 hx,h1h− 12 LIE FU AND ZHIYU TIAN where hρ is defined in (5). Therefore

Φ(h.Ex,ρ)

= Φ(Ehx,hρ) 1 = χρ (1) 2 χhρ(1) ehx,1 √ X q 0 − Ghx 1 G | | ∈ hx 1 = 1 χρ0 (1) 2 χhρ(h1h− ) ehx,h1h 1 √ G X q − − x 1 Gx | | ∈ 1 = χρ0 (1) 2 χρ(1) ehx,h1h 1 √ G X q − − x 1 Gx | | ∈ 1 = χρ (1) 2 χρ(1) h.ex,1 √ G X q 0 − x 1 Gx | | ∈ = h.Φ(Ex,ρ),

1 where the third equality is a change of variable: replace 1 by h1h− , the fourth equality follows from the deﬁnition of hρ in (5)

Proposition 3.4 (Multiplicativity). Φ preserves the multiplication, i.e. Φ is a morphism of algebra objects in CHMC.

Proof. The cases of multiplying h(S) with itself or with a Lefschetz motive Lx,ρ are all obviously preserved by Φ. We only need to show that for any x S with non-trivial stabilizer Gx, the morphism ∈

Lx Lx 1 M ,ρ → M , ρ Irr′(Gx) 1 Gx ∈ 1∈,id

1 given by the matrix with coeﬃcient χρ0 (1) 2 χρ(1) at place (ρ, 1) is multiplicative (note √ Gx p − · that the result of the multiplication could| go| outside of these direct sums to h(S)). Since this is just a matrix computation, let us treat Lefschetz motives as 1-dimensional vector spaces (or equivalently, we are looking at the corresponding multiplicativity of the realization of Φ for Chow rings): let Ex,ρ be the ‘generator’ of Lx,ρ and ex,1 be the ‘generator’ of Lx,1. Then the computations of the products in the previous two subsections say that:

2x if ρ1 = ρ2; − = (8) Ex,ρ1 Ex,ρ2 x if ρ1,ρ2 are adjacent; ·  0 if ρ ,ρ are not adjacent;  1 2  x if 1 = h 1; = − (9) ex,1 ex,h  1 · 0 if 1 , h− ;  MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 13

Therefore for any ρ ,ρ Irr′(Gx), we have 1 2 ∈ Φ(Ex ) Φ(Ex ) ,ρ1 · ,ρ2 1 = χρ0 (1) 2 χρ0 (h) 2 χρ1 (1)χρ2 (h) ex,1 ex,h Gx X X q − q − · 1 Gx h Gx | | ∈ ∈ 1 1 1 = χρ0 (1) 2 χρ0 (1− ) 2 χρ1 (1)χρ2 (1− ) x Gx X q − q − · 1 Gx | | ∈ 1 = (χρ0 (1) 2) χρ1 (1)χρ2 (1) x Gx X − · 1 Gx | | ∈

1   = χρ0 ρ1 (1)χρ2 (1) 2 χρ1 (1)χρ2 (1) x Gx X ⊗ − X  · 1 Gx 1 Gx  | |  ∈ ∈    = ρ0 ρ1,ρ2 2 ρ1,ρ2 x h ⊗ i− h i · = Φ Ex,ρ Ex,ρ 1 · 2 where the first equality is the definition of Φ (and we add the non-existent ex,1 with coefficient 0), the second equality uses (9) the orthogonality among ex,1’s (i.e. Lx,1’s), the third equality uses the fact that χρ0 takes real value; the last equality uses all three cases of (8). Proposition 3.5 (Additive isomorphism). Taking G-invariants on both sides of (7), ΦG is an isomorphism of complex Chow motives between h(Y) and horb([S/G]).

Proof. We should prove the following morphism is an isomorphism: G G   G G  L  G  L  Φ : h(S) x,ρ h(S)  x,1 . ⊕ M M  → ⊕ M M   x S ρ Irr′(Gx)   x S 1 Gx   ∈ ∈   ∈ ∈,     1 id    Since Φ is given by ‘block diagonal matrix’, it amounts to show that for each x S (with Gx non trivial), the following is an isomorphism : ∈

Gx   L  L  (10) x,ρ  x,1 . M → M  ρ Irr′(Gx) 1 Gx  ∈  ∈,   1 id    which is equivalent to say that the following square matrix is non-degenerate:

(11) χρ0 (1) 2 χρ(1) , q − · (ρ,[1]) where ρ runs over the set Irr′(Gx) of isomorphism classes of non-trivial irreducible representations and [1] runs over the set of conjugacy classes of Gx diﬀerent from id. As this is about a matrix, it is enough to look at the realization of(10):

Gx     Ex,ρ  ex,1 , M → M  ρ Irr′(Gx) 1 Gx  ∈  ∈,   1 id    14 LIE FU AND ZHIYU TIAN where both sides come equipped with non-degenerate quadratic forms given by intersection numbers and degrees of the orbifold product respectively. More precisely, by (8) and (9):

2 if ρ1 = ρ2; − = (Ex,ρ1 Ex,ρ2 ) 1 if ρ1,ρ2 are adjacent; ·  0 if ρ ,ρ are not adjacent;  1 2  1 if 1 = h 1; = − (ex,1 ex,h)  1 · 0 if 1 , h− ;  which are both clearly non-degenerate. Now Proposition 3.4 shows that our matrix (11) respects the non-degenerate quadratic forms on both sides, therefore it is non-degenerate. Let us note here also an elementary proof which does not use the orbifold product. We first remark that for any 1 , id, ρ (1) SL (C) which is of finite order and different from the identity, 0 ∈ 2 hence its trace χ0(1) , 2. Therefore the nondegeneracy of the matrix (11) is equivalent to the nondegeneracy of the matrix

χρ(1) , (ρ,[1]) which is obtained from the character table of the finite group Gx by removing the first row (corresponding to the trivial representation) and the first column (corresponding to id Gx). The nondegeneracy of this matrix is a completely general fact, which holds for all finite groups.∈ We will give a proof in Lemma 3.6 at the end of this section.

The combination of Lemma 3.3, Proposition 3.4 and Proposition 3.5 proves the isomorphism of algebra objects (2) in the main Theorem 1.2 in the global quotient case. For the isomorphisms for the Chow rings and cohomology rings, it is enough to apply realization functors. For the isomorphisms for the K-theory and topological K-theory, it suffices to invoke the construction of orbifold Chern characters in [25] which induce isomorphisms of algebras from (orbifold) K-theory to (orbifold) Chow ring as well as from (orbifold) topological K-theory to (orbifold) cohomology ring. The proof of Theorem 1.2 in the global quotient case is now complete. The following lemma is used in the second proof of Proposition 3.5. The elegant proof below is due to Cédric Bonnafé. We thank him for allowing us to use it. Recall that for a finite group G, its character table is a square matrix whose rows are indexed by isomorphism classes of irreducible complex representations of G and columns are indexed by conjugacy classes of G. Lemma 3.6. Let G be any finite group. Then the matrix obtained from the character table by removing the first row corresponding to the trivial representation and the first column corresponding to the identity element, is non-degenerate.

Proof. Denote by 1 the trivial representation and by ρ1, ,ρn the set of isomorphism classes of ··· n non-trivial representations of G. Suppose we have a linear combination i=1 ciχρi , with ci C, which vanishes for all non-identity conjugacy class, hence for all non-identityP elements of G: ∈ n (12) c χ (1) = 0, 1 , id G. X i ρi ∀ ∈ i=1 Set n 1 c := c dim(ρ ), 0 − G X i i | | i=1 MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 15 and denote by χre1 be the character of the regular representation,then (12) implies that the following linear combination vanishes for all 1 G: ∈ n c χ + c χ = 0. 0 re1 X i ρi i=1

If c0 , 0, it contradicts to the fact that the trivial representation should appear (with multiplicity 1) in the regular representation. Hence we have c0 = 0. Then by the linear independency among the characters of irreducible representations, we must have c = = cn = 0. 1 ···

4. Proof of Theorem 1.2 : general orbifold case

In this section, we give the proof of Theorem 1.2 in the full generality. As the proof goes essentially in the same way as the global quotient case in §3, we will focus on the diﬀerent aspects of the proof and refer to the arguments in §3 whenever possible. Recall the setting: is a two-dimensional Deligne–Mumford stack with only ﬁnitely many points with non-trivial stabilizersX ; X is the underlying (projective) singular surface with only Du Val singularities and Y X is the minimal resolution. For each x X, denote by Gx its stabilizer, → ∈ which is contained in SL2. Throughout this section, Chow groups of stacks are as in [33] and Chow motives of stacks or singular Q-varieties are as in [32, §2].

4.1. Resolution side. Similar to (4), we have the following decomposition given by the classical McKay correspondence (see Introduction):

(13) h(Y) h(X) Lx ≃ ⊕ M M ,ρ x X ρ Irr′(Gx) ∈ ∈ and the multiplication is the following:

δ h(X) h(X) X h(X) is the usual intersection product. • ⊗ −−→ ix∗ 2 For any ρ Irr′(Gx), h(X) Lx Lx is given by the class x CH (X) = Hom(h(X) L, L). • ∈ ⊗ ,ρ −→ ,ρ ∈ ⊗ 2ix, 2 For any ρ Irr′(Gx), Lx Lx − ∗ h(X), is determined by 2x CH (X). • ∈ ,ρ ⊗ ,ρ −−−−→ − ∈ For any ρ1 , ρ2 Irr′(Gx), • ∈ 2 – If they are adjacent, that is, ρ1 appears (with multiplicity 1) in the Gx-module ρ2 C , 2 2 ⊗ where C is such that C /Gx is the singularity type of x, then

ix, Lx Lx ∗ h(X), ,ρ1 ⊗ ,ρ2 −−→ is determined by x CH2(X). ∈ L L 0 – If they are not adjacent, then x,ρ1 x,ρ2 h(X) is the zero map. The other multiplication maps are zero.⊗ −→ • 16 LIE FU AND ZHIYU TIAN

4.2. Orbifold side. Similar to (6), we have

Gx    L  (14) h( ) = h(X)  x,1 , X ⊕ M M  x X 1 Gx  ∈  ∈,   1 id    where the action of Gx is by conjugacy. Note that degree 0 twisted stable maps with 3 marked points to are either untwisted stable maps to or a twisted map to one of the stacky points of . In theX latter case, the irreducible componentsX of the moduli space around these twisted stableX maps and the obstruction bundle are the same as those of the twisted stable maps to the orbifold [C2/G]. It is then clear that the orbifold product can be described as if is a global quotient. Therefore the orbifold product on h( ) is given by the following, via (14):X X

δ h(X) h(X) S h(X) is the usual intersection product. • ⊗ −→ ix∗ For all 1 Gx, h(X) Lx 1 Lx 1 determined by the class of x X. • ∈ ⊗ , −→ , ∈ ix, For all 1 Gx, Lx,1 L 1 ∗ h(X) determined by the class of x X. • ∈ ⊗ x,1− −−→ ∈ The other multiplication maps are zero. •

4.3. The multiplicative isomorphism. Similar to (7), we deﬁne

(15) φ : h(X) Lx h(X) Lx 1, ⊕ M M ,ρ → ⊕ M M , x X ρ Irr′(Gx) x X 1 Gx ∈ ∈ ∈ 1∈,id which is given by the following ‘block matrix’:

id : h(X) h(X); • → For each x X (with non-trivial stabilizer Gx), the morphism • ∈

Lx Lx 1 M ,ρ → M , ρ Irr′(Gx) 1 Gx ∈ 1∈,id

1 is the ‘matrix’ with coeﬃcient χρ0 (1) 2 χρ(1) at place (ρ, 1) Irr′(Gx) (Gx id ), √ Gx − · ∈ × \{ } | | p 2 where χ denotes the character, ρ0 is the natural 2-dimensional representation C of Gx such 2 that C /Gx is the singularity type of x. Note that ρ0(1) has determinant 1, hence its trace

χρ0 (1) is a real number. The other morphisms are zero. • To conclude Theorem 1.2, on the one hand, the same proof as in Proposition 3.4 shows that φ is multiplicative. On the other hand, one sees immediately that φ factorizes through

Gx    L  h(X)  x,1 . ⊕ M M  x X 1 Gx  ∈  ∈,   1 id    MULTIPLICATIVE MCKAY CORRESPONDENCE FOR SURFACES 17

It is thus enough to show that the following induced map is an (additive) isomorphism:

Gx   L  L  ψ : h(X) x,ρ h(X)  x,1 , ⊕ M M → ⊕ M M  x X ρ Irr′(Gx) x X 1 Gx  ∈ ∈ ∈  ∈,   1 id    However this follows from the proof of Proposition 3.5, where one shows that (10) is an isomorphism. The proof of Theorem 1.2 is complete.

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Institut Camille Jordan, Universite´ Claude Bernard Lyon 1, France E-mail address: [email protected]

CNRS, Institut Fourier, Universite´ Grenoble Alpes,France E-mail address: [email protected]