Operators of Nakajima’s type

Theorem 2.2.3. The refined Brian¸con variety Br^[n,n+r] is of complex dimension n− ₂^r

. Furt-hermore, the number of top-dimensional components of Br^[n,n+r] is equal to the number of Young diagrams withn+r boxes withr columns.

Proof. Choose a generic one-parameter subgruopC^∗→T so that we have a cell decomposition of Br^[n,n+r]. SinceBr^[n,n+r] is a deformation retract of Hilb^[n,n+r], by equation 2.18 of the Poincar´e polynomial of Hilb^[n,n+r], we have that Br^[n,n+r] is a disjoint union of affine cells of complex dimension

n+^{−r2+3r−2w(4)}₂

, where 4 ∈

Hilb^[n,n+r]T

. We know also each4contains at least rremovable boxes⇒w(4)≥r. We conclude that dim_CBr^[n,n+r] =n+^−r2₂^+r and top-dimensional components are indexed by Young diagrams4with exactlyrcolumns and|4|=n+r.

2.3 Operators on the direct sum of homology groups of Hilb

^[n,n+r]

In the case of Hilb^[n], the generating function ofP

Hilb^[n];t

has a form of the infinite product P

nP

Hilb^[n];t

qⁿ = Q∞ d=1

1

1−t^2d−2q^d (1.17). Grojnowski ([20]) and Nakajima ([29]) show a geometric interpretation as the character of an infinite dimensional Heisenberg algebra action on

⊕_nH_∗

Hilb^[n]

through operators on H.

We observe that in the formula for the generating function ofP

Hilb^[n,n+r];t :

∞

X

n=(^r₂) P

Hilb^[n,n+r];t

qⁿ=q(^r2)

∞

Y

d=1

1 1−t^2d−2q^d

! _r Y

d=1

1 1−t²q^d

! ,

there are two products: an infinite product

∞

Y

d=1

1

1−t^2d−2q^d and a finite productQr d=1

1

1−t²q^d. This observation suggests that there exists an action of Lie algebra of operators on the infinite direct sum of the homology of the refined Hilbert schemes

Her:= M

n≥(^r₂) H_∗

Hilb^[n,n+r]

with the character (2.15). In particular, there should be infinitely many operators of degree 2(d−1) androperators of degree 2dthat correspond toQ∞

d=1 1

1−t^2d−2q^d andQr d=1

1

1−t²q^d respectively.

2.3.1 Operators of Nakajima’s type

We give a construction of the operators of degree (2d−2) through the correspondence (cf. 1.2.2).

We refer to these operators are of Nakajima’s type since our construction is analogous to Nakajima’s construction for Hilb^[n] (cf. [29,30]).

Denoted byπij, πk the projections from Hilb^[n,n+r]×Hilb[n+k,n+k+r]

×C²to (i, j)-th andk-th factors respectively. For a positive integerk, the projectionπ2to Hilb[n+k,n+k+r]

is proper, whereas the projectionπ1to Hilb^[n,n+r] is not.

First, let Z(r, k)n:=n

((I, J),(K, L),p)∈Hilb^[n,n+r]×Hilb[n+k,n+k+r]×C²\{(0,0)} I⊆K,Supp(I\K) ={p}o . (2.21) We defined an incidence varietyZ(r, k)_n in Hilb^[n,n+r]×Hilb[n+k,n+k+r]×C² by taking the closure ofZ(r, k)_n.

Proposition 2.3.1. The varietyZ(r, k)_n has dimension2n−r(r−1) +k+ 1.

Proof. Recall that an element ofZ(r, k) is of the form ((I, J),(K, L),p). We decompose Z(r, k)_n into strata by the multiplicitym_p,I ofI at each pointpin the support ofI:

Z(r, k)_n =

n

[

s=0

Z(r, k)_n(s),

whereZ(r, k)_n(s) :={((I, J),(K, L), p) m_p,I =s}. We claim thatZ(r, k)_n(0) is in fact the generic part ofZ(r, k)_n. Whens= 0,Z(r, k)_n(0) consists of element ((I, J),(K, L),p) such that the point p is not in the support ofI and it gives rise to a birational map

Z(r, k)_n(0)→Hilb^[n,n+r]×Br^[k]

C²

((I, J),(K, L),p)7→((I, J), B_p). Then we get the dimension ofZ(r, k)_n(0)

dim_CZ(r, k)_n(0) = dim_CHilb^[n,n+r]+ dim_CBr^[k]

C²

= 2(n− r

2

) + (k+ 1) = 2n−r(r−1) +k+ 1.

Now, fors >1, we have dimZ(r, k)(s)<2n−r(r−1) +k+ 1. It is because that if we add/subtract one point, dimension of Hilb^[n,n+r] increases/drops by 2, Br^[k]

C² increases/drops only by 1. We

conclude that dim_CZ(r, k) = 2n−r(r−1) +k+ 1.

It follows thatZ(r, k)_n defines a class [Z(r, k)_n]∈H2n−r(r−1)+k+1

Z(r, k)_n . Ifα∈H_∗(C²),β ∈H_∗(C²)^BM andu∈H_i

Hilb^[n,n+r]

, then the operatorsβ_k(·) of adding k points and the operatorsαk(·) of subtractingkpoints are defined by

αk(u) =π2∗ π^∗₁(PD⁻¹u)∪π₃^∗(PD⁻¹α)

∩[Z(r, k)_n]

∈H_i+(k−1)

Hilb[n+k,n+k+r]

(2.22) and

βk(y) =π1∗ π^∗₂(PD⁻¹u)∪π₃^∗(β)

∩[Z(r, k)_n]

∈H∗

Hilb[n−k,n−k+r]

(2.23) respectively, whereαk has degree degα+k−1.

Here, the map ((I, J),(K, L),p)7→ρ(I\K) restricted onZ(r, k)_n is the relative Hilbert-Chow morphism, which is proper. These operators are well-defined onHer:

To ensure that we do get a homology class, we need at least one of pull-back classesπ_j^∗(·) to be compactly supported. π₁^∗(PD⁻¹u) is not the pull-back of a proper map whereas π^∗₃(PD⁻¹α) is compactly support pull-back by a proper map. π₂^∗(PD⁻¹u) is a pull-back through a proper map, so there is no further requirements onπ^∗₃(β).

Proposition 2.3.2. Let β ∈ H_∗(C²), α ∈ H_∗(C²). The operators αk, βk satisfy the Heisenberg relation

[αi, βj] =iδi+j,0hα, βiid

Her

where the pairing hα, βistands for the push-forward a pointf_∗(α∩β),f:C²→ {pt}.

We note that the only nontrivial choice of suchαandβ isα= [pt], β= [C²].

Proof. We proceed a proof that is analogous to the Nakajima’s proof for Hilb^[n] by considering separated cases fori+j = 0 andi+j6= 0 (cf. [30, Proposition 2.3.2]). The main idea of the proof is that when i6=j, the support of the corresponding correspondence has a dimension strictly less than the expected dimension and the only contribution comes from the case ofi=j.

Byαiβj, we mean the composition which applies first the operatorαi and then the operatorsβj

and [αi, βj] =αiβj−βjαi. We define the following notations:

Y_i,j:= Hilb^[n,n+r]×C²×Hilb[n+i,n+i+r]

×C²×Hilb[n+i+j,n+i+j+r], cij := [Z(r, i)_n][Z(r, j)_n+i] =π_1245∗

π₁₂₃^∗ [Z(r, i)_n]π^∗₃₄₅[Z(r, j)_n+i] , Cij :=π1245

π₁₂₃⁻¹(Z(r, i)_n)π₃₄₅⁻¹(Z(r, j)_n+i) , {p6=q}:=a

i,j

{(I,p,q, J)∈π₁₂₄₅(Y_ij)|p6=q}, {p=q}:=a

i,j

{(I,p,q, J)∈π1245(Yij) p=q},

whereπ₋ are the projections to (−)-th factors.

Note that the mapC²×C²→C²×C²,(p,q)7→(q,p) on{p6=q} is bijective. This induces an isomorphism

C_ij∩ {p6=q} →C_ji∩ {p6=q}

by ((I, J),p,q,(I⁰J⁰))7→((I, J),q,p,(I⁰J⁰)). Thus,π⁻¹₁₂₃

Z(r, i)_n

andπ⁻¹₃₄₅

Z(r, j)_n+i

intersect transversally over{(p,q),p6=q}, where the dimension is given by

(2n−r(r−1) +i+ 1) + (2(n+i)−r(r−1) +j+ 1)−2(n+i− r

2

)

= 2n+i+j+ 2−r(r−1).

On the other hand, the subset{(p,q)|p=q}ofC²×C² is isomorphic toC²under the diagonal morphism ∆ :C²→C²×C²,p7→(p,p). This induces a map

C_ij∩ {p=q} →Z(n, i+j)

((I, J),p,q,(K, L))7→((I, J),p=q,(K, L)),

which is compatible with the extended Hilbert-Chow morphism ((I, J),(K, L))7→ρ(I\K) by the fact thatα∩β= ∆^∗(α×β). Thus, the componentCij∩{p =q}has dimension 2n−r(r−1)+i+j+1, which is less than the expected dimension.

Case of i+j6= 0 and ij≥0: Suppose that i >0 andj >0, we first consider the case of [αi, βj].

We observe that the set-theoretical support of the compositionαiβj is given by the Cij ⊂ Hilb^[n,n+r]×C²×C²×Hilb[n+i+j,n+i+j+r], where

C_ij =n

((I, J),p,q,(I⁰J⁰)) ∃(K, L)∈Hilb[n+i,n+i+r], ρ(K) =ρ(I) +i[p], ρ(I⁰) =ρ(K) +j[q]o . Considering C²×C² = {p 6= q} ∪ {p = q}, we will discuss the cases C_ij ∩ {p 6= q} and C_ij∩ {p =q} separately. As we pointed out previously, a generic element ofZ(r, i)_n is an element (I, J) of Hilb^[n,n+r] where the support ofIis disjoint from the point adding. One can identifyCij∩ {p6=q}with the closure of the image of the map

Hilb^[n,n+r]×Br^[i]

C²×Br^[j]

C² →Hilb^[n,n+r]×C²×C²×Hilb[n+i+j,n+i+j+r]

((I, J), Bp, Bq)7→((I, J),p,q,(I∩Bp∩Bq, J∩Bp∩Bq))) with dimension 2(n− ^r₂

) + (i+ 1) + (j+ 1).

For the case ofCij∩ {p =q}, under the identification ofC²×C² withC² by the diagonal

∆(p,p) =p∈C², there is a birational map

C_ij∩ {p=q} →Z(r, i+j)_n ((I, J),p,p,(I⁰, J))7→((I, J),(I⁰, J⁰),p).

Since dimZ(r, i+j)_n= 2n−r(r−1) +i+j+ 1 is less than the expected dimension ofc_ij, C_ij∩ {p=q}does contribute to c_i,j. And similarly,C_ji∩ {p=q}does not contribution to the intersection for the same dimensional reason.

Now, for the case of [α−i, β−j], bothC−i−j∩ {p6=q}andC−j−i∩ {p6=q}can be identified with the closure of the image of the map

Hilb[n−i−j,n−i−j+r]×Br^[i]

C²×Br^[j]

C² →Hilb^[n,n+r]×C²×C²×Hilb[n−i−j,n+i+j+r]

((I, J), Bp, Bq)7→((I∩Bp∩Bq, J∩Bp∩Bq),p,q,(I, J))), where Hilb[n−i−j,n−i−j+r]

×Br^[i]

C²×Br^[j]

C² has dimension 2(n−i−j− ₂^r

) + (i+ 1) + (j+ 1) = 2n−i−j+ 2−r(r−1), which is the same as the dimension of c−i−j andc−j−i. Whereas C_−i−j∩ {p =q}andC_−j−i∩ {p =q}has strictly smaller dimensions after a similar argument as above.

Thusc_−i−j−c_−j−i = 0.

Case of i+j6= 0 and ij≤0: Without losing generality, we compute for the case [α_i, β_−j].

C_i−j∩ {p6=q}andC_−ji∩ {p6=q}can be both identified by the closure of the image of the map

Hilb[n−j,n−j+r]

×Br^[i]

C²×Br^[j]

C²→Hilb^[n,n+r]×C²×C²×Hilb[n+i−j,n+i−j+r]

((I, J), Bp, Bq)7→((I∩Bq, J∩Bq),p,q, I∩Bp, J∩Bp))) where (I, J), Bp, Bq have disjoint supports. Moreover, Hilb[n−j,n−j+r]×Br^[i]

C² ×Br^[j]

C² has dimension 2(n−j− ^r₂

) + (i+ 1) + (j+ 1) = 2n−j+i+ 2−r(r−1).

C_i−j∩ {p=q}can be parametrized by Hilb[n−j,n−j+r]×Br^[i]

C²×Br^[j]

C²→Hilb^[n,n+r]×∆⁻¹(C²×C²)×Hilb[n+i−j,n+i−j+r]

(I, J), B_p, B⁰_p

7→((I, J),p,(I∩B_p, J∩B_p)) Case of i+j = 0: Let

Y⁺:= Hilb^[n,n+r]×C²×Hilb[n+i,n+i+r]×C²×Hilb^[n,n+r]

Y⁻ := Hilb^[n,n+r]×C²×Hilb[n−i,n−i+r]×C²×Hilb^[n,n+r],

and projectionsπ₍₋₎to (−)-factors. The compositions αiα−i andα−iαi are induced by the classes

c_i,−i:=π_1245∗

π^∗₁₂₃([Z(r, i)_n])∩π^∗₃₄₅([τ Z(r, i)_n]) c_−i,i:=π_1245∗

π^∗₁₂₃([τ Z(r, i)_n−i])∩π₃₄₅^∗ ([Z(r, i)_n−i]) and are supported on the sets

C_i,−i :=π1245

π₁₂₃⁻¹(Z(r, i)_n)∩π₃₄₅⁻¹(τ Z(r, i)_n)

=n

((I, J),p,q,(I⁰, J⁰)) ∃(K, L)∈Hilb[n+i,n+i+r]

:I⊃K⊂I⁰,Supp(I/K) =p,Supp(I⁰/K) ={q}o C−i,i:=π1245

π₁₂₃⁻¹(τ Z(r, i)_n−i)∩π⁻¹₃₄₅(Z(r, i)_n−i)

=n

((I, J),p,q,(I⁰, J⁰)) ∃( ¯K,L)¯ ∈Hilb[n−i,n−i+r]

:I⊂K¯ ⊃I⁰,Supp( ¯K/I) ={p},Supp( ¯K/I⁰) ={q}o

with expected dimension 2 (2n−r(r−1) +i)−(2(n+i)−r(r−1)) = 2n−r(r−1).

Hilb^[n,n+r]

C_i,−i⊂π145(Y⁺)

π1



π5

C−i,i⊂π145(Y⁻)

π5

??

π1

__ Hilb^[n,n+r]

C_i,−i has 2 irreducible components of dimension 2n−r(r−1):

• The component Ci,−i∩ {p6=q}=φ

Hilb[n−i,n−i+r]

×Br^[i]

C²×Br^[i]

C²

where φ: Hilb[n−i,n−i+r]×Br^[i]

C²×Br^[i]

C²→C_i,−i∩ {p6=q}

((I, J), Bp, Bq)7→((I∪Bp, J∪Bp), ρ(Bp), ρ(Bq),(I∪Bq, J∪Bq)) for the ideals ((I, J), Bp, Bq)∈Hilb[n−i,n−i+r]×Br^[i]

C²×Br^[i]

C² having disjoint supports.

• The componentC_i,−i∩ {p=q}

=n

((I, J),p,p,(I⁰, J⁰)) ∃( ¯K,L)¯ ∈Hilb[n−i,n−i+r]:I⊂K¯ ⊃I⁰,Supp( ¯K/I) ={p}= Supp( ¯K/I⁰)o , which is ∆

Hilb^[n,n+r]

the diagonal of Hilb^[n,n+r]. We proceed the same decomposition forC_−i,i

C_−i,i= (C_−i,i∩ {p6=q})∪(C_−i,i∩ {p=q}). C−i,i∩ {p6=q} can be parametrized by the same map

φ: Hilb[n−i,n−i+r]

×Br^[i]

C²×Br^[i]

C²→Ci,−i∩ {p6=q}

((I, J), Bp, Bq)7→((I∪Bp, J∪Bp), ρ(Bp), ρ(Bq),(I∪Bq, J∪Bq)).

As for the pieceC−i,i∩ {p =q}, note that ((I, J),q,p,(I⁰, J⁰))∈C−i,i∩ {p =q} implies (I, J) = (I⁰, J⁰)∈Hilb^[n,n+r] s≥i+ ^r₂

which is non-generic. Thus, it does not contribute to this class.

SinceC_i,−i andC_−i,i intersects transversally over{p6=q}, we have C_i,−i−C_−i,i=C_i,−i∩ {p=q}=ch

∆

Hilb^[n,n+r]i

for some constantN ∈C. In order to determine the constantN, we consider locally near a generic point of

π₁₂₃⁻¹(Z(r, i)_n)∩π⁻¹₃₄₅(τ Z(r, i)_n)

{p=q}⊂Hilb^[n,n+r]×C²×Hilb[n+i,n+i+r]

, which are elements ((I, J),p,(K, L)) such that Supp(I)∩ {p} = ∅. Near a generic point ((I, J),p,(K, L)), the product Hilb^[n,n+r]×C²×Hilb[n+i,n+i+r]×C²×Hilb^[n,n+r] can be

replaced byy

Hilb^[n,n+r]×C²×(Hilb^[n,n+r]×Br^[i]

C²)×C²×Hilb^[n,n+r]

⊂Hilb^[n,n+r]×C²×(Hilb^[n,n+r]×Hilb^[i])×C²×Hilb^[n,n+r]

Then the degreeN of the diagonal [Ci,−i∩ {p=q}] is independent ofn, r chosen as there is a rational map

Hilb^[n,n+r]×C²×(Hilb^[i])×C²→

Hilb^[n,n+r]×C²

Hilb^[n,n+r]×C²×(Hilb^[n,n+r]×Hilb^[i])×C²×Hilb^[n,n+r]

(π₁,π₄)



(π4,π5)

C²×Hilb^[n,n+r]

such that the original projectionsπ1 andπ5 are identity mapsid: Hilb^[n,n+r]→Hilb^[n,n+r]. We have degπ1245= degπ45.

The inclusion

(ρ, id) :Br_p^[i]→C²×Br^[i]

C², Bp 7→(ρ(Bp), Bp) gives a local expression ofZ(r, i)_nof the form Hilb^[n,n+r]×C²×Br^[i]

C²in Hilb^[n,n+r]×C²×Hilb^[i]

by ((I, J), ρ(Bp), Bp)7→((I, J),p, Bp). Furthermore,

π₁₂₃^∗ ([Z(r, i)_n])∩π^∗₃₄₅([τ Z(r, i)_n]) {p=q}

is supported on the intersection π₁₂₃⁻¹

Hilb^[n,n+r]×C²×Br^[i]

C²

\π₃₄₅⁻¹ Br^[i]

C²×C²×Hilb^[n,n+r]

⊂

Hilb^[n,n+r]×C²×Br^[i]

C²×C²×Hilb^[n,n+r]

We obtain a simpler picture after lettingn= ^r₂

, where Hilb^[n,n+r] is a just a point.

Hilb^[n,n+r]×C²

Hilb^[n,n+r]×C²×Hilb^[i]×C²×Hilb^[n,n+r]

˜ π₁₂



˜ π₄₅

C²×Hilb^[n,n+r] → {pt} ×C²

{pt} ×C²×Hilb^[i]×C²× {pt}

˜ π1



˜ p₃

X× {pt}

The diagonal in this case is given by N

∆(C²)

=π13∗

h

C²×(ρ, id)^∗(Br^[i]

C²)i

∩h

(ρ, id)^∗(Br^[i]

C²)×C² i

,

for some constantN ∈C. After intersecting withπ₃^∗([q]), the class of a pointq∈C^∗, we have N[{q}] =π_1∗h

C²×(q, Br_p^[i])i \ h

(ρ, id)^∗(Br^[i]

C²)× {q}i

=π_1∗h

C²×Br^[i]_p i \ h

(ρ, id)^∗(Br^[i]

C²)i

From this, we see that N equals to intersection number of the classesh Brp^[i]

i and h

Br^[i]

C²

i

inside Hilb^[n]. These numbersN were calculated by various persons ([15,30]) which is equal toi. We conclude thatc_i,−i−c_−i,i=i∆

Hilb^[n,n+r]

and we obtain the final result α_iα_−i−α_−i, α_i= [α_i, α_−i] =iidH˜.

Dans le document Geometry and topology of refined structures on the Hilbert scheme of points on the plane (Page 53-59)