introduction - pkubicmr.pku.edu.cn/~qizheng/files/motdec.pdf · 2019-12-24 · pects of the...

MOTIVIC DECOMPOSITIONS FOR THE HILBERT SCHEME OF

POINTS OF A K3 SURFACE

ANDREI NEGUT, , GEORG OBERDIECK, AND QIZHENG YIN

Abstract. We construct an explicit, multiplicative Chow–Kunneth decompo-

sition for the Hilbert scheme of points of a K3 surface. We further refine thisdecomposition with respect to the action of the Looijenga–Lunts–Verbitsky

Lie algebra.

1. Introduction

This note is a continuation of [27] by the second author. We study the motivic as-pects of the Looijenga–Lunts–Verbitsky ([20, 31], LLV for short) Lie algebra actionon the Chow ring of the Hilbert scheme of points of a K3 surface. Using a specialelement of the LLV algebra and formulas of [22] by Maulik and the first author, weconstruct an explicit Chow–Kunneth decomposition for the Hilbert scheme, proveits multiplicativity, and show that all divisor classes and Chern classes lie in the cor-rect component of the decomposition. This confirms expectations of Beauville [2]and Voisin [33]. We also obtain a refined motivic decomposition for the Hilbertscheme by taking into account the LLV algebra action.

Both results parallel the case of an abelian variety, which we shall briefly review.

1.1. Abelian varieties. Let X be an abelian variety of dimension g. Recall theclassical result of Deninger–Murre on the decomposition of the Chow motive h(X).

Theorem 1.1 ([5]). There is a unique, multiplicative Chow–Kunneth decomposi-tion

(1) h(X) =

2g⊕i=0

hi(X)

such that for all N ∈ Z, the multiplication [N ] : X → X acts on hi(X) by [N ]∗= N i.

The decomposition (1) specializes to the Kunneth decomposition in cohomology(hence the name Chow–Kunneth), and to the Beauville decomposition [1] in Chow.The latter takes the form

(2) A∗(X) =⊕i,s

Ai(X)s

with

Ai(X)s = Ai(h2i−s(X)) = {α ∈ Ai(X) | [N ]∗α = N2i−sα, for all N ∈ Z}.The multiplicativity of (1) stands for the fact that the cup product

∪ : h(X)⊗ h(X)→ h(X)

Date: December 24, 2019.

1

2 ANDREI NEGUT, , GEORG OBERDIECK, AND QIZHENG YIN

respects the grading, in the sense that

∪ : hi(X)⊗ hj(X)→ hi+j(X)

for all i, j ∈ {0, ..., 2g}. This can be seen by simply comparing the actions of [N ]∗.As a result, the bigrading in (2) is multiplicative, i.e., compatible with the ringstructure of A∗(X).

The Beauville decomposition is expected to provide a multiplicative splittingof the conjectural Bloch–Beilinson filtration on A∗(X). A difficult conjecture ofBeauville (and consequence of the Bloch–Beilinson conjecture) predicts the vanish-ing A∗(X)s = 0 for s < 0 and the injectivity of the cycle class map

cl : A∗(X)0 → H∗(X).

Further, any symmetric ample class α ∈ A1(X)0 induces an sl2-triple (eα, fα, h)acting on A∗(X). A Lefschetz decomposition of h(X) with respect to the sl2-action was obtained by Kunnemann [16], refining (1). More generally, Moonen [23]constructed an action of the Neron–Severi part of the Looijenga–Lunts [20] Liealgebra gNS on A∗(X), which contains all possible sl2-triples above (he actuallyconsidered the slightly larger Lie algebra sp(X ×X∨); see [23, Section 6]). He thenobtained a refined motivic decomposition with respect to the gNS-action.

Theorem 1.2 ([23]). There is a unique decomposition

(3) h(X) =⊕

ψ∈Irrep(gNS)

hψ(X)

where ψ runs through all isomorphism classes of finite-dimensional irreducible rep-resentations of gNS, and hψ(X) is ψ-isotypic under gNS.

Here being ψ-isotypic means that hψ(X) is stable under gNS and that for anyChow motive M , the gNS-representation Hom(M, hψ(X)) is isomorphic to a directsum of copies of ψ.

Again (3) specializes to refined decompositions in cohomology and in Chow.

1.2. Chow–Kunneth. We switch to the Hilbert scheme case. Let S be a projec-tive K3 surface and let X = Hilbn(S) be the Hilbert scheme of n points on S.

In [27], the action of the Neron–Severi part of the LLV algebra gNS was liftedfrom cohomology to Chow. In particular, there is an explicit grading operator

h ∈ A2n(X ×X)

which appears in every sl2-triple (eα, fα, h) in gNS. We normalize h so that it actson H2i(X) by multiplication by i− n.

We regard h as a natural replacement for the operator [N ]∗ in the abelian varietycase. Our first result decomposes the Chow motive h(X) into eigen-submotives of h.

Theorem 1.3. There is a unique Chow–Kunneth decomposition

(4) h(X) =

2n⊕i=0

h2i(X)

such that h acts on h2i(X) by multiplication by i− n.

MOTIVIC DECOMPOSITIONS FOR Hilbn(K3) 3

The mutually orthogonal projectors in the decomposition (4) are written explic-itly in terms of the Heisenberg algebra action [15, 25]. We also show that (4) agreeswith the Chow–Kunneth decomposition obtained by de Cataldo–Migliorini [4] andVial [32].

As before (4) specializes to a decomposition in Chow

(5) A∗(X) =⊕i,s

Ai(X)2s

with

Ai(X)2s = Ai(h2i−2s(X)) = {α ∈ Ai(X) |h(α) = (i− s− n)α}.

1.3. Multiplicativity. In the seminal paper [2], Beauville raised the questionwhether hyper-Kahler varieties behave similarly to abelian varieties in the sensethat the conjectural Bloch–Beilinson filtration also admits a multiplicative split-ting. As a test case, he conjectured that for a hyper-Kahler variety, the cycle classmap is injective on the subring generated by divisor classes.

For the Hilbert scheme of points of a K3 surface, Beauville’s conjecture was re-cently proven in [22]; see also [27] for a shorter proof. But the ultimate goal remainsto find the multiplicative splitting. Meanwhile, Shen and Vial [29, 30] introducedthe notion of a multiplicative Chow–Kunneth decomposition, upgrading Beauville’squestion from Chow groups to the level of correspondences/Chow motives.

The main result of this paper confirms that (4) provides a multiplicative Chow–Kunneth decomposition for the Hilbert scheme.

Theorem 1.4. Let S be a projective K3 surface and let X = Hilbn(S).

(i) The Chow–Kunneth decomposition (4) is multiplicative, i.e., the cup prod-uct

∪ : h(X)⊗ h(X)→ h(X)

respects the grading, in the sense that

∪ : h2i(X)⊗ h2j(X)→ h2i+2j(X)

for all i, j ∈ {0, ..., 2n}. As a result, the bigrading in (5) is multiplicative.(ii) All divisor classes and Chern classes of X belong to A∗(X)0.

Theorem 1.4 (ii) is related to the Beauville–Voisin conjecture [33], which predictsthat for a hyper-Kahler variety, the cycle class map is injective on the subringgenerated by divisor classes and Chern classes. In the Hilbert scheme case, onemay further ask the vanishing A∗(X)2s = 0 for s < 0 and the injectivity of thecycle class map

cl : A∗(X)0 → H∗(X).

We do not tackle these questions in the present paper.The key to the proof of Theorem 1.4 is the compatibility between the grading

operator h and the cup product. For example, at the level of Chow groups, weshow that the operator

h = h+ n∆X ∈ A2n(X ×X)

acts on A∗(X) by derivation, i.e.,

(6) h(x · x′) = h(x) · x′ + x · h(x′)


for all x, x′ ∈ A∗(X). We achieve this by explicit calculations using the Chowlifts [22] of the well-known machinery for the Heisenberg algebra action [18, 19]. Infact, our argument yields (6) at the level of correspondences; see Section 4. Oncethe compatibility is established, Theorem 1.4 is deduced by simply comparing theeigenvalues.

1.4. Previous work. Theorem 1.4 was previously obtained by Vial [32] basedon Voisin’s announced result [34, Theorem 5.12] on universally defined cycles. Asecond proof, also relying on Voisin’s theorem, was given by Fu and Tian [9]. Theyinterpreted Theorem 1.4 (i) as the motivic incarnation of Ruan’s crepant resolutionconjecture [28]. Our proof has the advantage of being explicit and unconditionalat the moment.

We note that multiplicative Chow–Kunneth decompositions, for both hyper-Kahler and non-hyper-Kahler varieties, have been studied in [6, 7, 8, 10, 11, 12, 17].

1.5. Refined decomposition. We further obtain a refined decomposition of theChow motive h(X) with respect to the action of the Neron–Severi part of the LLValgebra gNS. Both the statement and the proof parallel the abelian variety case.

Theorem 1.5. Let S be a projective K3 surface and let X = Hilbn(S). There is aunique decomposition

(7) h(X) =⊕

ψ∈Irrep(gNS)

hψ(X)

where ψ runs through all isomorphism classes of finite-dimensional irreducible rep-resentations of gNS, and hψ(X) is ψ-isotypic under gNS.

As before (7) specializes to refined decompositions in cohomology and in Chow.We may also consider a Cartan subalgebra t ⊂ gNS and note that the decom-

position (7) in terms of the irreducible representations of gNS implies a similardecomposition in terms of the irreducible representations (i.e., characters) of t.

More precisely, recall the weight decomposition

gNS = gNS,−2 ⊕ gNS,0 ⊕ gNS,2, gNS,0 = gNS ⊕Q · hwhere gNS is the Neron–Severi part of the reduced LLV algebra (terminology takenfrom [14]). Let t ⊂ gNS be a Cartan subalgebra and write

t = t⊕Q · h.(so t differs from t in that the element h was replaced by h = h + n∆X). Hencethere is a motivic decomposition

(8) h(X) =⊕λ∈t∗

hλ(X).

We expect (8) to also be multiplicative, in the sense that the cup product sends

(9) ∪ : hλ(X)⊗ hµ(X)→ hλ+µ(X).

To this end, it would suffice to prove the analogue of (6) when h is replaced byany element in gNS. The generators of gNS are denoted by

hαβ ∈ A2n(X ×X)

and indexed by α ∧ β in ∧2(A1(X)). For n > 1, we have

A1(X) ∼= A1(S)⊕Q · δ


where δ corresponds to the exceptional divisor of the Hilbert scheme. Toward thegoal of proving (9), we obtain the following result.

Theorem 1.6. For all α, β ∈ A1(S) ⊂ A1(X), we have

hαβ(x · x′) = hαβ(x) · x′ + x · hαβ(x′)

for all x, x′ ∈ A∗(X).

We believe that the statement below can be proved with the machinery describedin Section 4, but the combinatorics is quite involved, and we therefore do nottackle it.

Conjecture 1.7. For all α ∈ A1(S) ⊂ A1(X), we have

hαδ(x · x′) = hαδ(x) · x′ + x · hαδ(x′)

for all x, x′ ∈ A∗(X).

An equivalent formulation of Conjecture 1.7 is that gNS acts on A∗(X) by deriva-

tions. Since the modified Cartan subalgebra t is contained in gNS⊕Q·h, Theorem 1.6and Conjecture 1.7 (both at the level of correspondences) would imply the multi-plicativity of the refined decomposition (8). On its own, Theorem 1.6 can onlyimply the weaker statement where the direct sum in (8) goes over the dual of a

modified Cartan subalgebra of dimension 1 smaller than t.

1.6. Conventions. Throughout the present paper, Chow groups and Chow mo-tives will be taken with Q-coefficients. We refer to [24] for the definitions andconventions of Chow motives.

We will often switch between the languages of correspondences and operators onChow groups, in the following sense. Every operator f : A∗(X)→ A∗(Y ) will arisefrom a correspondence F ∈ A∗(X × Y ) by the usual construction

X × Yπ1

{{

π2

##X Y

f = π2∗(F · π∗1)

and any compositions and equalities of operators implicitly entail compositions andequalities of correspondences. For example, the operator

multτ : A∗(X)→ A∗(X)

of cup product with a fixed element τ ∈ A∗(X) is associated to the correspon-dence ∆∗(τ) ∈ A∗(X ×X), where ∆ : X ↪→ X ×X is the diagonal embedding.

Moreover, a family of operators fγ : A∗(X)→ A∗(Y ) labeled by γ ∈ A∗(Z) willarise from a correspondence F ∈ A∗(X × Y × Z), by the assignment

fγ arises from π12∗(F · π∗3(γ)) ∈ A∗(X × Y )

for all γ ∈ A∗(Z). We employ the language of “operators indexed by γ ∈ A∗(Z)”instead of cycles on X × Y ×Z because it makes manifest the fact that γ does notplay any role in taking compositions. For instance, the family of operators

multγ : A∗(X)→ A∗(X)

labeled by γ ∈ A∗(X) is associated to the small diagonal ∆123 ⊂ X ×X ×X.


We will often be concerned with cycles on a variety of the form Sn = S × ...×Sfor a smooth algebraic variety S (most often an algebraic surface). We let

∆a1...ak ∈ A∗(Sn)

denote the diagonal {(x1, ..., xn) |xa1 = ... = xak}, for all collections of distinctindices a1, ..., ak ∈ {1, ..., n}. Moreover, given a class Γ ∈ A∗(Sk), we may chooseto write it as Γ1...k in order to indicate the power of S where this class lives. Thenfor any collection of distinct indices a1, ..., ak ∈ {1, ..., n}, we define

Γa1...ak = p∗a1...ak(Γ) ∈ A∗(Sn)

where we let pa1...ak = (pa1 , ..., pak) : Sn → Sk with pi : Sn → S the projection tothe i-th factor. Finally, if • denotes any index from 1 to k + 1, we write∫

•: A∗(Sk+1)→ A∗(Sk)

for the push-forward map which forgets the factor labeled by •.

1.7. Acknowledgements. We would like to thank Lie Fu, Alina Marian, DaveshMaulik, Junliang Shen, Catharina Stroppel, and Zhiyu Tian for useful discussions.A. N. gratefully acknowledges the NSF grants DMS–1760264 and DMS–1845034,as well as support from the Alfred P. Sloan Foundation. Q. Y. was supported bythe NSFC grants 11701014, 11831013, and 11890661.

2. Hilbert schemes

2.1. Throughout the present paper, S will denote a projective K3 surface over C,and A∗(S) will denote its Chow ring with coefficients in Q, graded by codimension.Beauville and Voisin [3] have studied the class c ∈ A2(S) of any closed point on arational curve in S, and they proved the following formulas in A∗(S):

c2(TanS) = 24c

α · β = 〈α, β〉c

for all α, β ∈ A1(S) (above, we write 〈·, ·〉 : A∗(S)⊗A∗(S)→ Q for the intersectionpairing). Moreover, we have the following identities in A∗(S2):

∆ · c1 = ∆ · c2 = c1 · c2(10)

∆ · α1 = ∆ · α2 = α1 · c2 + α2 · c1(11)

where ∆ ∈ A∗(S2) is the class of the diagonal, and the following identity in A∗(S3):

(12) ∆123 = ∆12 · c3 + ∆13 · c2 + ∆23 · c1 − c1 · c2 − c1 · c3 − c2 · c3 .

As a corollary of (10) and (12), one can prove by induction the following identity:

(13)

k∑i=1

∆1...i−1,i+1...kci = (k − 2)∆1...k +

k∑i=1

c1...ci−1ci+1...ck

in A∗(Sk) for all k.


Proposition 2.1. For any Γ12...k ∈ A∗(Sk), we have the following identity:

(c0 − c1)(Γ02...k − Γ12...k) = ∆01

(∫•

Γ•2...k(c• − c1)− Γ1...k

)(14)

(α0β1 − α1β0)(Γ02...k − Γ12...k) = ∆01

∫•

Γ•2...k(α•β0 − α0β•)(15)

in A∗(Sk+1), for any α, β ∈ A1(S).

Proof. To prove (14), let us consider the identity (12) on S3:

∆01• = ∆0• · c1 + ∆1• · c0 + ∆01 · c• − c0 · c1 − c0 · c• − c1 · c• .

Let us pull this identity back to Sk+2 with indices •, 0, ..., k, multiply it by Γ•2...k,and then push forward by forgetting the factor represented by the index •:

(16) ∆01Γ12...k = c1Γ02...k + c0Γ12...k

+

∫•

(∆01c•Γ•2...k − c0c1Γ•2...k − c0c•Γ•2...k − c1c•Γ•2...k) .

Using the identity (10) we have:∫•c0c•Γ•2...k =

∫•c0∆0•Γ•2...k = c0Γ02...k

and similarly:∫•c0c1Γ•2...k = ∆01

∫•

Γ•2...k ,

∫•c1c•Γ•2...k = c1Γ1...k .

Inserting these into (16) the formula (14) then follows from rearranging the terms.As for (15), we start from the identity:

(α0β1 − α1β0)(∆0• −∆1•) = ∆01(α•β0 − α0β•)

which is a straightforward application of (11). If we pull this identity back to Sk+2

with indices •, 0, ..., k, multiply it by Γ•2...k, then we obtain the following:

(α0β1 − α1β0)(∆0•Γ02...k −∆1•Γ12...k) = ∆01(α•β0 − α0β•)Γ•2...k .

If we push forward by forgetting the factor represented by •, we obtain (15). �

2.2. Consider the Hilbert scheme Hilbn of n points on S and the Chow rings:

Hilb =

∞⊔n=0

Hilbn, A∗(Hilb) =

∞⊕n=0

A∗(Hilbn)

always with rational coefficients. We will consider two types of elements of the Chowrings above. The first of these are defined by considering the universal subscheme:

Zn ⊂ Hilbn × S

(Zn is flat over Hilbn, and its fibers have the property that Zn|{Z}×S ∼= Z for anypoint {Z} ∈ Hilbn corresponding to a closed subscheme Z ⊂ S). For any k ∈ N,

let π : Hilbn × Sk → Hilbn denote the usual projection, and Z(i)n ⊂ Hilbn × Sk

denote the pull-back of Zn ⊂ Hilbn × S via the i-th projection map Sk → S.


Definition 2.2. A universal class is any element of A∗(Hilbn) of the form:

(17) π∗

[P (..., chj(OZ(i)

n), ...)1≤i≤k

j∈N

]for all k ∈ N and for all polynomials P with coefficients pulled back from A∗(Sk).

The following theorem holds for every smooth quasi-projective surface (see [26]),but we only prove it here in the case where S is a K3 surface (the argument hereineasily generalizes to any smooth projective surface using the results of [13]).

Theorem 2.3. Any class in A∗(Hilbn) is universal, i.e., of the form (17).

Proof. Consider the product Hilbn × Sk ×Hilbn, and we will write π1, π2, π3, π12,π23 and π13 for the various projections to its factors. As a consequence of [21] (seealso [13]), the diagonal ∆ ⊂ Hilbn ×Hilbn can be written as follows:

[∆] = π13∗

∑a

π∗2(γa)∏(i,j)

chj

(OZ(i)

n

) ∏(i,j)

chj

(OZ(i)n

)for suitably chosen k ∈ N, where we do not care much about the specific coeffi-

cients γa and indices i, j, i, j which appear in the sum above (we write Zn and Znfor the universal subschemes in Hilbn×S×Hilbn corresponding to the first and sec-ond, respectively, copies of Hilbn). Since the diagonal corresponds to the identityoperator, the equality above implies that:

IdHilbn = π1∗

∑a

π∗2(γa)∏(i,j)

chj

(OZ(i)

n

) ∏(i,j)

chj

(OZ(i)n

)π∗3

=∑a

π∗

∏(i,j)

chj

(OZ(i)

n

)ρ∗

γa · ρ∗∏

(i,j)

chj

(OZ(i)n

)· π∗(18)

where in (18), π, ρ : Hilbn × Sk → Hilbn, Sk denote the standard projections. �

Formula (18) implies the existence of a surjective homomorphism:

(19)

⊕a

A∗(Sk)� A∗(Hilbn)

∑a

Γa ∑a

π∗

∏(i,j)

chj

(OZ(i)

n

)ρ∗(Γa)

where the sums over a are in one-to-one correspondence with the sums in (18).

2.3. Let us present another important source of elements of A∗(Hilbn), based onthe following construction independently due to Grojnowski [15] and Nakajima [25](in the present paper, we will mostly use the presentation by Nakajima). Forany n, k ∈ N, consider the closed subscheme:

Hilbn,n+k ={

(I ⊃ I ′) | I/I ′ is supported at a single x ∈ S}⊂ Hilbn ×Hilbn+k


endowed with projection maps:

Hilbn,n+k

p−

yypS

��

p+

&&Hilbn S Hilbn+k

that remember I, x, I ′, respectively. One may use Hilbn,n+k as a correspondence:

A∗(Hilbn)q±k−−→ A∗(Hilbn±k × S)

given by:

(20) q±k = (±1)k · (p± × pS)∗ ◦ p∗∓.Because the correspondences above are defined for all n, it makes sense to set:

A∗(Hilb)q±k−−→ A∗(Hilb× S).

We also set q0 = 0. The main result of [25] is that the operators qk obey thecommutation relations in the Heisenberg algebra, namely:

(21) [qk, ql] = kδ0k+l (IdHilb ×∆)

as correspondences A∗(Hilb) → A∗(Hilb × S2). In terms of self-correspondencesA∗(Hilb)→ A∗(Hilb), the identity (21) reads, for all α, β ∈ A∗(S):

(22) [qk(α), ql(β)] = k(α, β) IdHilb .

2.4. More generally, we may consider:

qn1 ...qnt : A∗(Hilb)→ A∗(Hilb× St)where the convention is that the operator qni acts in the i-th factor of St = S×...×S.Then associated to any Γ ∈ A∗(St), one obtains an endomorphism of A∗(Hilb):

qn1...qnt(Γ) = π∗(ρ

∗(Γ) · qn1...qnt)

where π, ρ : Hilb× St → Hilb, St denote the standard projections.

Theorem 2.4 ([4]). We have a decomposition:

(23) A∗(Hilb) =

n1≥...≥nt∈N⊕Γ∈A∗(St)sym

qn1...qnt(Γ) · v

where “ sym” refers to the part of A∗(St) which is symmetric with respect to thosetranspositions (ij) ∈ St for which ni = nj, and v is a generator of A∗(Hilb0) ∼= Q.

Proof. Since we will need it later, we recall the precise relationship between Naka-jima operators and the correspondences studied in [4]. Let λ be a partition of nwith k parts, let Sλ = Sk and let Sλ → S(n) be the map that sends (x1, ..., xk) tothe cycle λ1x1 + ... + λkxk in the n-th symmetric product of the surface S. Weconsider the correspondence:

Γλ = (Hilbn ×S(n) Sλ)red

= {(I, x1, ..., xk) |σ(I) = λ1x1 + ...+ λkxk}


where σ : S[n] → S(n) is the Hilbert–Chow morphism sending the subscheme Ito its underlying support. The subscheme Γλ is irreducible of dimension n + kand the locus Γreg

λ ⊂ Γλ, where the points xi are distinct, is open and dense;see [4, Remark 2.0.1]. Similarly, the Nakajima correspondence qλ1

...qλk is a cyclein Hilbn×Sk of dimension n+k supported on a subscheme that contains Γreg

λ as anopen subset and whose complement is of smaller dimension [25, 4(i)]. Moreover themultiplicity of the cycle on Γreg

λ is 1. Hence we have the equality of correspondences:

(24) Γλ = qλ1 ...qλk ∈ A∗(Hilbn × Sλ).

The result follows now from [4, Proposition 6.1.5], which says that:

(25) ∆Hilbn =∑λ`n

(−1)n−l(λ)

|Aut(λ)|∏i λi

(Γλ)t ◦ Γλ

where λ runs over all partitions of size n, and we let l(λ) and λi denote the lengthand the parts of λ. �

Remark 2.5. As shown in [26], there is an explicit way to go between the descrip-tions (17) and (23) of A∗(Hilb). Concretely, for all n1 ≥ ... ≥ nt there exists apolynomial Pn1,...,nt with coefficients in ρ∗(A∗(St)) such that for all Γ ∈ A∗(St):

qn1...qnt(Γ) = π∗

[Pn1,...,nt(..., chj(OZ(i)), ...)

1≤i≤tj∈N · ρ∗(Γ)

]where π, ρ : Hilb × St → Hilb, St are the standard projections. Moreover, loc. cit.gives an algorithm for computing the polynomial Pn1,...,nt .

2.5. We will now present another connection between universal classes and theoperators qn. Let us consider any class of the form:

(26) univ(Γ) = π∗

[t∏i=1

chdi(OZ(i)n

) · ρ∗(Γ)

]∈ A∗(Hilbn)

where π, ρ : Hilbn × St → Hilbn, St are the standard projections, while the natural

numbers d1, ..., dt and the class Γ ∈ A∗(St) are arbitrary. We will write:

univd1,...,dt(Γ) = univ(Γ)

if we wish to emphasize the particular numbers d1, ..., dt which appear in (26),although they will often be inconsequential. Note that the codimension of (26) is:

deg univd1,...,dt(Γ) = deg Γ +

t∑i=1

(di − 2).

Lemma 2.6. The operator of multiplication by univ(Γ) is given by:

(27) multuniv(Γ) =∑

ε1,...,εt∈{0,2}

λbs−1+1≥...≥λbs∈Z,∀s∈{1,...,t}∑λbs−1+1+...+λbs=0,∀s∈{1,...,t}

ct · qλ1 ...qλbt(∆b0+1...b1∆b1+1...b2 ...∆bt−1+1...btΓb1...btφb1 ...φbt

)where in each summand we write, for all s ∈ {0, ..., t}:

(28) bs =

s∑i=1

(ds − εs)


and φbs is either 1 or cbs , depending on whether εs is 0 or 2. The constants “ ct”in (27) depend on the particular numbers di, εs and λk but not on Γ.

Proof. In the course of this proof, let π, ρ : Hilb×S → Hilb, S denote the standardprojections. Let us recall the operators of multiplication by universal classes:

Gd : A∗(Hilb)π∗−→ A∗(Hilb× S)

multchd(OZ )−−−−−−−−→ A∗(Hilb× S).

The following formulas were proved in cohomology by [19] and in Chow by [22]:

(29) Gd =

λ1≥...≥λd∑λ1+...+λd=0

ct ·qλ1 ...qλd

∣∣∣∆1...d

+

λ1≥...≥λd−2∑λ1+...+λd−2=0

ct ·qλ1...qλd−2

∣∣∣∆1...d

·ρ∗(c) .

The constants “ct” that appear in the formulas above are certain rational numbersthat will not be important to us. The meaning of the notation |∆1...d

is that werestrict the target of the operator qλ1 ...qλd from Hilb×Sd to Hilb×S via the smalldiagonal. The meaning of the notation “·ρ∗(c)” is that after this restriction, wealso multiply by the pull-back of the class c from the second factor of Hilb × S.Formula (27) simply entails composing t of the operators (29), multiplying with thepull-back of Γ ∈ A∗(St), and pushing forward to Hilb. �

2.6. In this section, ∆ will refer to the smallest diagonal of any St. Two interestingcollections of elements of A∗(Hilbn) can be written as universal classes: divisors andChern classes of the tangent bundle. It is well-known that:

A1(S)⊕Q · δ ∼= A1(Hilbn)

(with the convention that δ = 0 if n = 1) where:

l ∈ A1(S) univ2(∆∗(l))(30)

δ univ3(∆∗(1)).(31)

Similarly, the Chern character of the tangent bundle to Hilbn is given by the fol-lowing well-known formula (let π, ρ : Hilbn × S → Hilbn, S denote the standardprojections):

ch(TanHilbn) = π∗[(

ch (OZn) + ch (OZn)′ − ch (OZn) ch (OZn)

′)ρ∗(1 + 2c)

]where ( )′ is the operator which multiplies a codimension d class by (−1)d. There-fore, the Chern character of the tangent bundle is a linear combination of thefollowing particular universal classes:

(32) univd(∆∗(φ)) and univd,d′(∆∗(φ))

where φ ∈ {1, c}, and the natural numbers d and d′ are arbitrary.

3. Motivic decompositions

3.1. Let us recall the Lie algebra action gNS y A∗(Hilbn) from [27], which liftsthe classical construction of [20, 31] in cohomology. To this end, note that theBeauville–Bogomolov form is the pairing on

V = A1(Hilbn) ∼= A1(S)⊕Q · δ

which extends the intersection form on A1(S) and satisfies

(δ, δ) = 2− 2n, (δ, A1(S)) = 0.


Let U = ( 0 11 0 ) be the hyperbolic lattice with fixed symplectic basis e, f . We have

gNS = ∧2(V⊥⊕ UQ)

where the Lie bracket is defined for all a, b, c, d ∈ V ⊕ UQ by

[a ∧ b, c ∧ d] = (a, d)b ∧ c− (a, c)b ∧ d− (b, d)a ∧ c+ (b, c)a ∧ d.

Consider for all α ∈ A1(S) the following operators:

eα = −∑n>0

qnq−n(∆∗α)

eδ = −1

6

∑i+j+k=0

: qiqjqk(∆123) :

fα = −∑n>0

1

n2qnq−n(α1 + α2)

(33)

fδ = −1

6

∑i+j+k=0

: qiqjqk

(1

k2∆12 +

1

j2∆13 +

1

i2∆23 +

2

jkc1 +

2

ikc2 +

2

ijc3

): .

Here : − : is the normal ordered product defined by

: qi1 ...qik : = qiσ(1) ...qiσ(k)

where σ is any permutation such that iσ(1) ≥ ... ≥ iσ(k). We define operators eα

and fα for general α ∈ A1(Hilbn) by linearity in α. By [22] we have that eα is

the operator of cup product with α. If (α, α) 6= 0, the multiple fα/(α, α) acts oncohomology as the Lefschetz dual of eα. In [27], it was show that the assignment

(34)act : gNS → A∗(Hilbn ×Hilbn)

act(e ∧ α) = eα, act(α ∧ f) = fα

for all α ∈ V , induces a Lie algebra homomorphism. In particular, the ele-ment e ∧ f ∈ gNS acts by

(35) h =∑k>0

1

kqkq−k(c2 − c1).

The operator h specializes in cohomology to the Lefschetz grading operator, whichby our normalization acts on H2i(Hilbn) by multiplication by i− n.

From a straightforward calculation (see [27, Lemma 3.4]), one obtains the com-mutation relations

(36) [h, qλ1...qλk(Φ)] = qλ1

...qλk(Φ)

for all Φ ∈ A∗(Sk), where we write

Φ =

k∑i=1

(IdSi−1 ×h× IdSk−i)(Φ)(37)

=

k∑i=1

∫•

Φ1...i−1,•,i+1...k(ci − c•)︸︷︷︸this class lies in A∗(Sk×S)

with the last factor in Sk × S represented by the index •.


3.2. Proof of Theorem 1.3. We start with the decomposition of the diagonalinto Nakajima operators:

(38) ∆Hilbn =∑λ`n

(−1)l(λ)

z(λ)qλq−λ(∆)

where λ runs over all partitions of n,

z(λ) = |Aut(λ)|∏i

λi

is a combinatorial factor, and for any π ∈ A∗(S × S) we write

qλq−λ(π) = qλ1 ...qλl(λ)q−λ1 ...q−λl(λ)(π1,l(λ)+1π2,l(λ)+2...πl(λ),2l(λ)

)= :qλ1q−λ1(π)...qλl(λ)q−λl(λ)(π) : .

The formula (38) follows directly from (24), (25), and the fact that qtm = (−1)mq−m(which is incorporated in the definition (20)).

Consider the decomposition of the diagonal of S as

(39) ∆ = π−1 + π0 + π1

where

π−1 = c1, π0 = ∆− c1 − c2, π1 = c2.

It is easy to note that π−1, π0, π1 are the projectors onto the −1, 0,+1 eigenspacesof the action of h on A∗(Hilb1) = A∗(S).

To define projectors corresponding to the action of h on A∗(Hilbn), we insert thedecomposition (39) into (38), and then expand and collect the terms of degree i.Concretely, for every integer i, we let

Pi =

λ,µ,ν∑|λ|+|µ|+|ν|=n−l(λ)+l(ν)=i

(−1)l(λ)+l(µ)+l(ν)

z(λ)z(µ)z(ν):qλq−λ(πt−1)qµq−µ(πt0)qνq−ν(πt1) : .

Let us check that Pi are indeed projectors onto the eigenspaces of h.

Claim 3.1. For all i, j ∈ Z we have the following equalities in A∗(Hilbn ×Hilbn):

(a) Pi ◦ Pj = Piδij

(b) h ◦ Pi = iPi.

Proof. (a) We determine Pi◦Pj by commuting all Nakajima operators with negativeindices to the right, and then using that we act on Hilbn so all products of Nakajimaoperators with purely negative indices of degree > n vanish. Since every summandin Pj contains such a product of degree n, we find that for a term to contributeall operators with negative indices coming from Pi have to interact with operators(with positive indices) from the second term. The interactions are described asfollows. For a single term (let a, b > 0 and r, s ∈ {−1, 0, 1}) we have

qaq−a(πtr)qbq−b(πts) = qa[q−a, qb]q−b

((πtr)12(πts)34

)+ qaqbq−aq−b((π

tr)13(πts)24)


where by the commutation relations (21) the first term on the right is

qa[q−a, qb]q−b((πtr)12(πts)34

)= (−a)δabqaq−b

(π14∗((π

tr)12(πts)34∆23)

)= (−a)δabqaq−a

(πts ◦ πtr

)= (−a)δabqaq−a

((πr ◦ πs)t

)= (−a)δabδrsqaq−a(πtr).

Hence for a composition

:qλq−λ(πt−1)qµq−µ(πt0)qνq−ν(πt1) : ◦ :qλ′q−λ′(πt−1)qµ′q−µ′(π

t0)qν′q−ν′(π

t1) :

(with λ, µ, ν as in the definition of Pi, and the same for the primed partitions) toact non-trivially on Hilbn we have to have λ = λ′, µ = µ′ and ν = ν′. Moreover,if we write λ multiplicatively as (1l12l2 ...) where li is the number of parts of size i,then there are precisely |Aut(λ)| =

∏i li! different ways to pair the negative factors

in qλq−λ(πt−1) with the positive factors qλ′q−λ′(πt−1), and similarly for µ, ν. Hence

:qλq−λ(πt−1)qµq−µ(πt0)qνq−ν(πt1) : ◦ :qλ′q−λ′(πt−1)qµ′q−µ′(π

t0)qν′q−ν′(π

t1) :

= δλλ′δµµ′δνν′(−1)l(λ)+l(µ)+l(ν)z(λ)z(µ)z(ν) :qλq−λ(πt−1)qµq−µ(πt0)qνq−ν(πt1) :

which implies the claim.(b) To determine h ◦ Pi we commute h into the middle, i.e., to the right of all

Nakajima operators with positive indices, and to the left of all with negative ones.In the middle position h acts on the Chow ring of Hilb0 where it vanishes. Henceagain we only need to compute the commutators. For this we use (36) and that πiare the projectors onto the eigenspaces of h so that

(h× Id)(πtr) = ((Id×h)(πr))t = (h ◦ πr)t = rπtr.

As desired we find

h ◦ Pi = (−1 · l(λ) + 0 · l(µ) + 1 · l(ν))Pi = iPi. �

Using the claim it follows that the motivic decomposition

h(Hilbn) =

2n⊕i=0

h2i(Hilbn)

with h2i(Hilbn) = (Hilbn, Pi−n) has the stated properties. The uniqueness of thedecomposition follows from the uniqueness of the decomposition of ∆Hilbn underthe action of h on A∗(Hilbn ×Hilbn); see the proof of the refined decomposition inSection 3.3 below. �

By (25), an alternative way to write the projector Pi is

Pi =∑λ`n

(−1)n−l(λ)

z(λ)(Γλ)t ◦ Pi ◦ Γλ

where Pi ∈ A∗(Sλ × Sλ) is the projector

Pi =∑

i1+...+il(λ)=i

πi1 × ...× πil(λ) .

Hence the decomposition of Theorem 1.3 is precisely the Chow–Kunneth decompo-sition constructed by Vial in [32, Section 2].


3.3. Refined decomposition. Let U(gNS) be the universal enveloping algebraof gNS. The Lie algebra homomorphism (34) extends to an algebra homomorphism

act : U(gNS)→ A∗(Hilbn ×Hilbn).

Lemma 3.2. The image W ⊂ A∗(Hilbn ×Hilbn) of act is finite-dimensional.

Proof. For every fixed k ≥ 1 the subring of R∗(Sk) ⊂ A∗(Sk) generated by

• αi for all i and α ∈ A1(S)• ci for all i• ∆ij for all i, j

is finite-dimensional, and preserved by the projections to the factors. Hence the

space of operators W ⊂ A∗(Hilbn ×Hilbn) spanned by

qλ1...qλl(λ)q−µ1

...q−µl(µ)(Γ)

for all partitions λ, µ of n and all Γ ∈ R∗(Sl(λ)+l(µ)) is finite-dimensional. The

commutation relations (21) show that W is closed under compositions of correspon-dences. Moreover, by inspecting the expressions for the generators of gNS in (33)(and using (38) to bring them into the desired form), we see that all generators

of gNS lie in W . Hence g ∈ W for all g ∈ U(gNS), i.e., W ⊂ W . �

We find that W is a finite-dimensional vector space which is preserved by theaction of U(gNS), and hence defines a finite-dimensional representation of gNS.Since gNS is semisimple, this representation decomposes into isotypic summands

W =⊕

ψ∈Irrep(gNS)

Wψ.

Let us look at the image of ∆Hilbn ∈W under this decomposition

(40) ∆Hilbn =∑

ψ∈Irrep(gNS)

Pψ

where Pψ ∈Wψ.

Claim 3.3. The elements Pψ ∈ A∗(Hilbn ×Hilbn) are orthogonal projectors.

Proof. Let us first show that left-multiplication by Pψ maps W to Wψ, i.e.,

(41) Pψ ◦W ⊂Wψ.

Indeed, for all a ∈ W , right multiplication by a is a gNS-intertwiner and thussends Wψ to Wψ. In other words, we have Wψ ◦ a ⊂ Wψ, hence Wψ ◦W ⊂ Wψ,which implies (41). If we multiply any a ∈W by relation (40), we obtain

a =∑

ψ∈Irrep(gNS)

Pψ ◦ a.

By (41), the summands in the right-hand side each lie in Wψ. If a = Pψ′ , then bycomparing summands the equality above implies

Pψ′ ◦ a = a and Pψ ◦ a = 0

for all ψ 6= ψ′. In particular, taking a = Pψ′ implies the relations Pψ ◦Pψ′ = δψψ′Pψ.

Moreover, this implies that the inclusion (41) is actually an identity, hence leftmultiplication by Pψ projects W onto Wψ. �


From Claim 3.3 we obtain the decomposition

(42) h(Hilbn) =⊕

ψ∈Irrep(gNS)

hψ(Hilbn)

where hψ(Hilbn) = (Hilbn, Pψ). We can now prove the main result of this section.

Proof of Theorem 1.5. It remains to show that the summands hψ(Hilbn) are ψ-isotypic and that the decomposition (42) is unique. Let M be a Chow motive.The action of gNS on Hom(M, h(Hilbn)) is defined by g act(g) ◦ ... . Henceif f ∈ Hom(M,M ′) is a morphism of Chow motives, the pullback

f∗ : Hom(M ′, h(Hilbn))→ Hom(M, h(Hilbn))

is equivariant with respect to the gNS-action. Now, for any v ∈ Hom(M, hψ(Hilbn))we have v = Pψ ◦ w for some w ∈ Hom(M, h(Hilbn)) and thus

U(gNS)v = U(gNS)w∗(Pψ) = w∗(U(gNS) ◦ Pψ).

Since U(gNS) ◦ Pψ ⊂ Wψ this implies that U(gNS)v is finite-dimensional and ψ-isotypic. Since v was arbitrary we conclude that Hom(M, hψ(Hilbn)) is ψ-isotypic.

The decomposition (42) is unique because (40) is unique. Indeed, suppose wehad any other decomposition

(43) ∆Hilbn =∑

ψ∈Irrep(gNS)

P ′ψ

where P ′ψ ∈Wψ, for all ψ. Then we would need P ′ψ = Pψ◦aψ for some aψ ∈W . But

multiplying (43) on the left with Pψ and using the orthogonality of the projectorswould imply Pψ = Pψ ◦ Pψ ◦ aψ = Pψ ◦ aψ, hence P ′ψ = Pψ. �

As in [23, Proof of Theorem 7.2], we could also have used Yoneda’s Lemma toconclude the existence of the decomposition (42). Our presentation above has theadvantage of being constructive. It also shows that the projectors Pψ can be writtenin terms of the Nakajima operators applied to elements in R∗(Sk).

4. Multiplicativity

4.1. Let us recall the operator (35) (in the present section, we will find it usefulto use the language of operators when referring to correspondences, as explainedin Section 1.6). This operator was observed in [27] to lift the (shifted) gradingoperator from cohomology to Chow. Let us undo this shift by considering:

h = h+ n · IdHilbn .

The main purpose of the present section is to prove Theorem 1.4. In the languageof operators, the multiplicativity of the Chow–Kunneth decomposition boils downto the identity (6). Alternatively, we could restate this identity as:

(44) [h,multx] = multx′

of correspondences A∗(Hilbn)→ A∗(Hilbn) indexed by x ∈ A∗(Hilbn). By applyingthe equality (44) to the fundamental class, we must have x′ = h(x)−x ·h([Hilbn]).

Lemma 4.1. We have h([Hilbn]) = −n.

With the lemma above in mind, we conclude that x′ = h(x) in (44). There-fore, (44) is actually equivalent to (6), thus implying part (i) of Theorem 1.4.


Proof of Lemma 4.1. It is well-known that:

[Hilbn] =1

n!q1(1)n[Hilb0].

Because the only operator qk which fails to commute with q1 is q−1, formula (35)implies that:

h([Hilbn]) = h

(1

n!q1(1)n[Hilb0]

)=

=

[h,

1

n!q1(1)n

][Hilb0] =

[q1(1)q−1(c)− q1(c)q−1(1),

1

n!q1(1)n

][Hilb0]

=

n∑i=1

1

n!q1(1)i−1 · q1(1)[q−1(c), q1(1)] · q−1(1)n−i · [Hilb0] = −n[Hilbn]

where the fact that [q−1(c), q1(1)] = −1 is a consequence of (22). �

4.2. Due to the surjectivity property (19), formula (44) remains equivalent if wereplace x ∈ A∗(Hilbn) by a class of the form (26):

(45) univ(Γ) = univd1,...,dt(Γ)

(for any d1, ..., dt, which will be fixed in the present section), which is indexedby Γ ∈ A∗(St). Then instead of proving (44), it suffices to prove the following:

Proposition 4.2. We have the identity of correspondences A∗(Hilbn)→ A∗(Hilbn)

(46) [h,multuniv(Γ)] = (d1 + ...+ dt − t)multuniv(Γ) + multuniv(Γ)

parametrized by Γ ∈ A∗(St) (the bar notation is defined in (37)).

Proof. By combining (36) with (27), we have:

(47) [h,multuniv(Γ)] =∑

ε1,...,εt∈{0,2}

λbs−1+1≥...≥λbs∈Z,∀s∈{1,...,t}∑λbs−1+1+...+λbs=0,∀s∈{1,...,t}

ct · qλ1...qλbt

(∆b0+1...b1∆b1+1...b2 ...∆bt−1+1...btΓb1...btφb1 ...φbt

).

To compute the overlined class on the second row, we will use the following:

Claim 4.3. For any natural numbers k, l and any Φ ∈ A∗(Sl), we have:

(48) ∆1...kΦk∗ = ∆1...k

[(k − 1)Φk∗ +

∫•

Φ•∗(ck − c•)].

The notation ∗ stands for the indices k+ 1, ..., k+ l− 1, and it reflects the fact thatthe bar notation is only defined as in (37) with respect to the indices 1, ..., k only.

Proof. By definition, the LHS of (48) equals:

LHS of (48) =

k−1∑i=1

∫•

∆1...i−1,•,i+1...k(ci − c•)Φk∗ +

∫•

∆1...k−1,•(ck − c•)Φ•∗

= Φk∗

k∑i=1

∫•

∆1...i−1,•,i+1...k(ci − c•) + ∆1...k−1,•(ck − c•)(Φ•∗ − Φk∗)

= RHS of (48)

where the last equality is due to (13) and (14). �


By applying (48) a number of t times, we obtain (let ∆ = ∆b0+1...b1 ...∆bt−1+1...bt):

(49) ∆Γb1...btφb1 ...φbt = ∆

[(bt − t)Γb1...btφb1 ...φbt

+

t∑i=1

∫•

Γb1...bi−1•bi+1...btφb1 ...φbi−1φ•φbi+1

...φbt(cbi − c•)

].

Using (10) we have the following simple identities:∫•

Γb1...•...btφb1 ...φ•...φbt(cbi − c•) =

∫•

Γb1...•...bt(cbi − c•)φb1 ...φbt if φbi = 1

Γb1...btφb1 ...φbt if φbi = cbi∫•

Γb1...•...btφb1 ...φbi ...φbt(cbi − c•) = −Γb1...btφb1 ...φbt if φbi = cbi

which we plug into (49):

∆Γb1...btφb1 ...φbt = ∆[(bt − t+ 2#{i |φbi = cbi})Γb1...bt + Γb1...bt

]φb1 ...φbt .

Since the coefficient of Γb1...bt in the right-hand side is equal to d1 + ... + dt − t,according to (28), we obtain precisely formula (46). �

4.3. Let us now prove part (ii) of Theorem 1.4. In order to show that a cer-tain class x ∈ A∗(Hilbn) lies in the appropriate direct summand, we must showthat h(x) = (deg(x)− n) · x. We are interested in the situation when x is a divisorclass or a Chern class of the tangent bundle, in which case we have:

x = univd1,...,dt(Γ)

for some d1, ..., dt ∈ N and Γ ∈ A∗(St). We have shown in Proposition 4.2 that:

h(x) = (d1 + ...+ dt − n− t)univd1,...,dt(Γ) + univd1,...,dt(Γ)

so the class x lies in the appropriate direct summand if:

(50) Γ = Γ · (deg univd1,...,dt(Γ)− d1 − ...− dt + t) = Γ · (deg Γ− t).Since divisors and Chern classes of the tangent bundle are linear combinations ofthe classes (30), (31), (32), it suffices to prove (50) in the particular case Γ = ∆∗(γ),where ∆ is the small diagonal and γ ∈ {1, l, c}l∈A1(S). In this case, we have:

Γ(48)= ∆∗

((t− 1)γ +

∫•γ•(c− c•)

)= (t− 2 + deg γ) · Γ

where the last equality is a simple case-by-case study for all γ ∈ {1, l, c}l∈A1(S).Since deg Γ = 2t− 2 + deg γ, this implies formula (50).

4.4. Let α, β ∈ A1(S) ↪→ A1(Hilbn). As a straightforward application of formu-las (33), the element α ∧ β ∈ gNS acts on A∗(Hilbn) by the correspondence:

hαβ =

∞∑k=1

1

kqkq−k(α2β1 − α1β2).

In the present section, we will prove Theorem 1.6, that is show that hαβ acts byderivations. As in the setup of Proposition 4.2, it boils down to statement (51)below, for all α, β ∈ A1(S) and all universal classes of the form (45).


Proposition 4.4. We have the identity of correspondences A∗(Hilbn)→ A∗(Hilbn)

(51) [hαβ ,multuniv(Γ)] = multuniv(Γ)

parametrized by Γ ∈ A∗(St), where for any k and any Φ ∈ A∗(Sk), we write:

(52) Φ =

k∑i=1

∫•

Φ1...i−1,•,i+1...k(αiβ• − α•βi)︸︷︷︸this class lies in A∗(Sk×S)

where the last factor in Sk × S is the one represented by the index •.

Proof. The proof follows that of Proposition 4.2 very closely, so we will only pointout the differences. We start from the following identity:

[hαβ , qλ1 ...qλk(Φ)] = qλ1 ...qλk(Φ)

which is a straightforward analogue of (36). Therefore, formulas (27) and (47)for multuniv(Γ) and [hαβ ,multuniv(Γ)] continue to hold, but with the definition (52)for the double bar operation instead of the single bar operation in (37). We leavethe following analogue of (48) as an exercise to the interested reader:

∆1...kΦk∗ = ∆1...k

∫•

Φ•∗(αkβ• − α•βk).

By iterating the formula above t times, we obtain (let ∆ = ∆b0+1...b1 ...∆bt−1+1...bt):

∆Γb1...btφb1 ...φbt = ∆

t∑i=1

∫•

Γb1...bi−1•bi+1...btφb1 ...φbi−1φ•φbi+1

...φbt(αbiβ•−α•βbi).

As a consequence of the following simple formulas:∫•

Γb1...•...btφb1 ...φ•...φbt(αbiβ• − α•βi)

=

∫•

Γb1...•...btφb1 ...φbt(αbiβ• − α•βbi) if φbi = 1∫•

Γb1...•...btφb1 ...(φbi or φ•)...φbt(αbiβ• − α•βbi) = 0 if φbi = cbi

we have ∆Γb1...btφb1 ...φbt = ∆Γb1...btφb1 ...φbt . This concludes the proof of (51). �

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MIT, Department of Mathematics, Cambridge, MA, USA

http://arxiv.org/abs/1911.06580

http://math.univ-lyon1.fr/~fu/articles/MotivicCrepantHilbK3.pdf







Simion Stoilow Institute of Mathematics, Bucharest, Romania

Email address: [email protected]

University of Bonn, Institut fur Mathematik, Bonn, GermanyEmail address: [email protected]

Peking University, BICMR, Beijing, ChinaEmail address: [email protected]

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