Operators on the direct sum of the homology groups

In this subsection, we explain the construction of a representation of the Heisenberg algebra on the homology group of Hilb^[n] which makes the homology group of Hilb^[n] a Fock space. The idea is to construct correspondencesin Hilb^[n]×Hilb^[n+k] for arbitraryn∈Nandk∈Zand the associated representation on the direct sum of homology groups of all Hilb^[n]

H:=M

n

H_∗

Hilb^[n]

.

These correspondences define operators on the homology group and give generators of the Heisenberg algebra. In this subsection, we explain Nakajima’s construction ([30,31]).

We will first state a general construction of correspondences in Borel-Moore homology groups and give the definition of Heisenberg algebra. After that, we illustrate the construction of the representation of a infinite dimension Heisenberg algebra onH.

Convolution products and correspondences in Borel-Moore homology groups Existence of fundamental class in Borel-Moore homology groups:

We recall some facts about the homology and cohomology of a topological space X. The cohomologyH^∗(X) has a cup product structure

∪:Hⁱ(X)×H^j(X)→H^i+j(X), (α, β)7→α∪β

(sometimes also written with a dot ’ “·”) which makesH^∗(X) a ring. UnlikeH^∗(X), the homology H_∗(X) does not have such a natural product structure, however, it is a module over the cohomology through the cap product

∩:Hⁱ(X)⊗Hj(X)→H_j−i(X) α⊗β 7→α∩β.

The homologyH∗(−) andH^∗(−) have following properties:

• Push-forward and pullback of a continuous map:

Iff:X →Y is a continuous map. Then it determinespush-forwardon homology f_∗:H_i(X)→H_i(Y)

and

emphpull−backon cohomology

f^∗:Hⁱ(X)→Hⁱ(Y).

• Poincar´e duality:

A complexn-dimensional projective non-singular varietyXis a compact oriented 2n-dimensional real manifold. Thus, the top homologyH2n(X) is isomorphic toZ. The image of the generator

1∈ Z, denoted by [X] ∈ H2n(X), is the fundamental class of X. We define the Poincar´e duality mapHⁱ(X)→H_2n−i(X) by capping with the fundamental class ofX:

PD :Hⁱ(X)→H_2n−i(X) α7→α∩[X].

as PD is an isomorphism for compact oriented manifolds.

In general, for X a smooth space, but not necessarily compact, the homology groups are Poincar´e dual with compactly supported cohomology

PD : H_cptⁱ (X)→H_2n−i(X).

In this case, we obtain a push-forward homomorphism off:Xⁿ→Y^mon cohomology f∗:Hⁱ(X)^PD' H2n−i(X)→H2n−i(Y)^PD' H^{2m−(2n−i)}(Y) =H^2(m−n)+i(Y).

Existence of fundamental class of a closed subvariety inside of a non-singular projective variety:

If X is not compact and smooth, it is not clear if the fundamental [X] exists. However, the fundamental class can be defined for a closed subvariety a non-singular projective variety as a refined class in BM homology.

We define, for a topological spaceX embedded as a closed subspace of a oriented manifoldM, a version ofBorel-Moore homology:

H_i^BM(X) :=H^dimX−i(M, M\X).

Intersection pairing:

If X ⊂ M, Y ⊂ N are closed subsets of oriented manifolds M, N of dimension m, n and if f:X →Y is a smooth map withf⁻¹(Y) contained inX. We consider the composite on relative cohomology:

H_k^BM(Y) =H^n−k(N, N\Y) ^f

∗

−→H^n−k(M, M\f⁻¹(Y))

→H^{m−(k−n+m)}(M,(M\X)∪(N\Y)) =H_k−n+m^BM (X∩Y).

This gives rise to the pullback on BM homology groups:

f_∗:H_k^BM(Y)→H_k+(m−n)^BM (X).

Lemma 1.2.8. IfV is ak-dimensional algebraic subset of a non-singular variety. ThenH_i^BM(V) vanishes fori >2k, andH_2k^BM(V)is a free Abelian group with a generator for eachk-dimensional component ofV.

Poincar´e duality:

The cap productH^j(X)⊗H_i^BM(X)→H_i−j^BM(X) gives rise to a non-degenerate pairing H^dimX−i'H_i^BM(X).

Convolution products on Borel-Moore homology groups

LetM, N be oriented smooth varieties of dimensionsm, nandZ⊂M×N a subvariety such that the projection map Z→N is proper. From the previous statement, we know that the fundamental class [Z] of Z exists as an element inH2dim_CZ(M ×N). A convolution product is an operator on the homology groups ofM, N

cZ:Hi(M)→H_{i+2 dim}^BM

CZ−2 dim_CM(N)

y7→c_Z(y) =p_N∗(p^∗_M(PD⁻¹(y))∩[Z]).

Or equivalently on the cohomology groups

cZ:Hⁱ(M)→H^{i+2 dimCN−2 dimCZ}(N)

α7→cZ(α) = PD⁻¹(pN∗(p^∗_M(α)∩[Z])).

We sometimes also refer these constructions of operators ascorrespondences.

Lety∈Hi(M), then PD(y)∈H^{2 dimCM−i}. Since the pullpack of a cohomology class leaves the degree invariant,p^∗_M(P D(y))∩[Z] is an element inH_{2 dim}

CZ−(2 dim_CM−i)(M ×N). ThencZ(y) is a class of degree

2 dim_CZ−(2 dim_CM −i) =i+ 2(dim_CZ−dim_CM). (1.18) That iscZ increases the homological degree by 2(dim_CZ−dim_CM). The cohomological degree is computed in the same way, and we get the degree 2 dim_CN−2 dim_CZ.

LetM1, M2, M3 be orientedC^∞-manifolds of dimensionm1, m2, m3 andpij:M1×M2×M3→ M_i×M_jbe the projection to (i, j) factor. Suppose thatZ₁₂^d12⊂M₁×M₂andZ₂₃^d23 ⊂M₂×M₃are two correspondences induced by [Z₁₂]∈H_d12(M₁×M₂) and [Z₂₃]∈H_d23(M₂×M₃) respectively. We ask in addition that the projectionp₁₃is proper when restricts top⁻¹₁₂(Z₁₂)∩p⁻¹₂₃(Z₂₃)⊂M₁×M₂×M₃.

Then the composition of convolutions equals to [Z₁₂]◦[Z₂₃] = (p₁₃)_∗ p^∗₁₂([Z₁₂])∩p^∗₂₃([Z₂₃])

∈H_d12+d23−m2(M₁×M₃) defines a correspondence (pM₃)_∗(p^∗_M

1(·)∩[Z12]◦[Z23]) :Hi(M1)→Hi+d₁₂+d₂₃−m1−m2(M3) of degree d12+d23−m1−m2.

The infinite dimensional Heisenberg algebra

Definition 1.2.9. The infinite dimensional Heisenberg algebra s is a Lie algebra spanned by {p[k] k∈Z} ∪ {K} such that they satisfy the ”commutation relations”

[p[k], p[l]] =kδ_k+l,0and [p[k], K] = 0. (1.19) For everyc6= 0∈C, the algebrashas a representation as an algebra of operators onC[q₁, q₂,· · ·], on which thes-module structure is given as follows:

P[k]7→







q_−k ifk <0

0 if k=0

c k_∂q^∂

k ifk >0 K7→c·id

This representation has a highest weight vector 1. Moreover C[q1, q2, . . .] are spanned by q^j₁¹q^j₂²· · ·q^j_k^k =P[1]^j1P[2]^j2· · ·P[k]^jk·1 We extendsby defining a derivationd0 by

[d0, p[k]] =−kp[k].

And above representation can be extended by d07→X

k

kqk

∂

∂qk

with the character formula X

i

dim{v∈C[q₁, q₂, . . .] d₀v=iv} tⁱ=

∞

Y

i=1

1

1−tⁱ. (1.20)

Construction of correspondence on ⊕_nH_∗(Hilb^[n])

We introduce H.Nakajima’s contruction of correspondences on ⊕nH∗(Hilb^[n]) (cf. [12,29,30]). He originally constructed these correspondences for smooth quasi-projective surfaces. We introduce an incidence variety fork >0:

P[k] :=a

n

n

(I, J,p)∈Hilb^[n]×Hilb^[n+k]×C² I⊃J , Supp(I\J) ={p}o andk <0 by switching IandJ.

We show thatP[k] has complex dimension 2n−k+ 1. Assume thatk >0, we may decompose P[k] into disjoint union of strataP[k]0∪P[k]1∪ · · · ∪P[k]k where

P[k]l=n

(I, J, x)∈P[k] I∈Hilb^[l]o . We define the Brian¸con variety on the whole spaceC²:

Br^[n]

C² :=

IER ρ(I) =n[p] , for somep∈C² . (1.21) IfJ ∈Hilb^[n+k] is obtained by adding a pointp that is disjoint from the support ofI, thenJ =I∪B for some idealB∈Br^[k]

C². Hence we have a birational map P[k]₀→Hilb^[n]×Br^[k]

C²

and therefore dim_CP[k]0= 2n+k+ 1. Similarly, for eachi∈N, 0< l≤kwe have birational maps P[k]_l→Hilb^[n−l]×Br^[k+l]

C² . But these strata have dimension

dim_CP[k]_l= 2n−2l+k+l= 2n−l+k+ 1,

which are strictly less than 2n+k+ 1 for l > 0. A Similar argument can be applied to the case of k < 0 by exchanging I andJ. Let u∈H_∗(C²) and v ∈H_∗^BM(C²) be homology classes

of the ordinary homology and Borel-Moore homology of C² respectively. We define, fork > 0, Pu[k] :=u∩[Pu[k]]∈H∗

Hilb^[n+k]

andPv[−k] :=v∩[Pv[−k]]∈H∗

Hilb^[n−k]

. Letπn,πn+k

and π_C2 be the projection of Hilb^[n] ×Hilb^[n+k]×C² to Hilb^[n], Hilb^[n+k] and C² respectively, i= 1,2,3. ThenP_u[k],P_v[−k](z) define operators onH=⊕nH_∗(Hilb^[n])

Pu[k](z) := (πn+k)_∗(p^∗_C2(u)∪π_n^∗(z)∩[P[k]])∈H_∗

Hilb^[n+k]

, Pv[−k](z) := (πn−k)∗(p^∗_C2(v)∪π^∗_n(z)∩[P[−k]])∈H∗

Hilb^[n−k]

.

Most importantly, next theorem state that they satisfy the commutation relation and thusHbecomes a representation space of Heisenberg algebra.

Theorem 1.2.10([12,20,29]). For all u, v∈H_∗(C²),k, l∈Z, we have

[Pu[k], Pv[l]] =kδk+l,0hu, viid_H. (1.22) An important part of proof involves a calculation of the intersection number:

Theorem 1.2.11([15,30]).

h[Br^[i]_p ],[Br^[i]

C²]i= (−1)ⁱ⁻¹i. (1.23)

Refined Hilbert scheme of points