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194, 1982.
Hilbert schemes of points
and Heisenberg algebras
Geir Ellingsrud and Lothar Gttschey
o

y The Abdus Salam International Centre for Theoretical Physics,
Trieste, Italy.




Lecture given at the
School on Algebraic Geometry
Trieste, 26 July { 13 August 1999

LNS001002




 gottsche@ictp.trieste.it
Abstract
Let X [n] be the Hilbert scheme of n points on a smooth projective surface
X over the complex numbers. In these lectures we describe the action of the
Heisenberg algebra on the direct sum of the cohomologies of all the X [n] ,
which has been constructed by Nakajima. In the second half of the lectures
we study the relation of the Heisenberg algebra action and the ring structures
of the cohomologies of the X [n] , following recent work of Lehn. In particular
we study the Chern and Segre classes of tautological vector bundles on the
Hilbert schemes X [n] .
Contents
1. Introduction 63
[n]
2. The Betti numbers of X 65
3. The Fock space and the current algebra 66
4. The Nakajima operators 70
5. The relations 73
6. Indication of how to get the relations 75
7. Vertex operators and Nakajimas computation of the constants 80
8. Computation of the Betti numbers of X [n] 85
9. The Virasoro algebra 88
10. Tautological sheaves 90
11. Geometric interpretation of the Virasoro operators 91
12. Chern classes of tautological sheaves 95
References 100
Hilbert schemes and Heisenberg algebras 63

1. Introduction
In these notes X will be a smooth and projective surface over the complex
numbers. The object of our interest will be the Hilbert scheme of points on
X. For any nonnegative integer n there is such a Hilbert scheme X [n] which
parameterizes nite subschemes of X of length n.
If W  X is a nite subscheme of length n, we shall also denote the
corresponding point in X [n] by W.
There is a universal subscheme Zn  X [n]  X whose underlying set is

given as Zn = (W; P) P 2 W . The rst projection from Zn  X [n]  X
onto X [n] induces a nite and
at map  : Zn ! X [n] . Let O[n] :=  (OZn ).
It is a locally free sheaf on X [n] of rank n.
The Hilbert scheme X [n] enjoys several nice geometric properties, the most
basic one being:
Theorem 1.1. The Hilbert scheme X [n] is smooth, connected and of dimen-
sion 2n.
The rst proof of this result was given in [6]. Once connectedness is
established, that the dimension of X [n] is 2n, is clear: Each of the n points
has two degrees of freedom. S
Any subscheme W 2 X [n] can be written as W = i Wi where the Wi
are mutually disjoint subschemes each having support in just one point. If
Supp Wi = fPi g, we may de ne the 0-cycle
X X
(length OW;P )P:
(W) := (length Wi )Pi =
i W2X

This 0-cycle is an element of the symmetric power X (n) := X n =Sn ; the
quotient of X n by the symmetric group Sn acting on X n by permutation.
In this way we get a map  : X [n] ! X (n) , which turns out to be a morphism
(see [6]). It is called the Hilbert-Chow morphism.
Contrary to X [n] the symmetric power X (n) is singular. Along the diago-
nals, where two or more points come together, the action of the symmetric
group has nontrivial isotropy, and because this happens in codimension two
or more, the quotient will be singular.
It is easy to see that the Hilbert-Chow morphism is birational; indeed it
is an isomorphism between the set of reduced subschemes in X [n] and the
subset of X (n) consisting of 0-cycles all whose points are di erent.
64

Theorem 1.2. The Hilbert-Chow morphism is a resolution of the singular-
ities. In fact it is even a semismall resolution; which means that
codimfzj dim  1 (z)  rg  2r
for any natural number r.
For any point P 2 X we let the closed subscheme Mn (P )  X [n] be the
set of subschemes whose support is the single point P. In other words


Mn(P ) = W 2 X [n] Supp (W ) = fPg :
This is set-theoretically the same as  1 (nP ), and Mn (P ) is indeed closed.
We also give a name to the closed subset of X [n] whose elements are the
subschemes with support in one (unspeci ed) point, and de ne
[n] Supp (W ) contains just one point :

Mn := W 2 X
There is an obvious map Mn ! X which sends a one-point-supported
subscheme to the point where it is supported. The following is a basic result
which now has several proofs. The rst one was given by Brianon in [2],
c
For other proofs see [4] or [5].
Theorem 1.3. Mn(P ) is irreducible of dimension n 1, and Mn is irre-
ducible of dimension n + 1.
When studying the Hilbert schemes X [n] of points, it is often a good
idea to look at all the X [n] at the same time, because they are all related
and therefore there is hope that a new structure emerges. One instance
of this is the fact that there is a nice generating function for all the Betti
numbers of all the X [n] . We shall see that this is a re
ection of the fact
that the direct sum of all the cohomologies of all the X [n] has an additional
structure. It is an irreducible module for a Heisenberg algebra action. This
has been shown by Nakajima [13]. This Heisenberg action is constructed
by means of correspondences between the Hilbert schemes, and the varieties
Mn and Mn (P ) play a big role. In fact the idea is that one can go from
the cohomology of X [k] to that of X [k+n], by adding subschemes of length n
supported in one point of X.
In the second part of these lecture notes we will investigate how this
Heisenberg action is related to the ring structure of the cohomology rings
of the Hilbert schemes. Here we follow the work [11] of Lehn. We are
particularly interested in the Chern classes of so-called tautological vector
bundles on the Hilbert schemes. For every vector bundle V on X one has
an associated tautological vector bundle V [n] on X [n] whose bers over the
Hilbert schemes and Heisenberg algebras 65

points W 2 X [n] are naturally identi ed with H 0 (W; V jW ). In particular, if
V has rank r, then V [n] is a vector bundle of rank nr. The Chern classes and
Chern numbers of these tautological bundles have interesting geometrical
and enumerative interpretations.
We study the operators of multiplication with the Chern classes of the
tautological sheaves, and express them in terms of the operators of the
Heisenberg algebra action. It is easy to see that the Heisenberg algebra
action induces an action of a Virasoro algebra and an important step in the
argument is a geometric interpretation of the Virasoro operators. Finally, we
restrict to the case of tautological vector bundles associated to a line bundle
L on X. We nd a generating function for all the Chern classes in terms of
the Heisenberg operators and, at least conjecturally, a generating function
for the top Segre classes of the L[n].

2. The Betti numbers of X [n]
If one is interested in the cohomology of X [n] , the rst question to ask is
what are the Betti numbers of X [n]; i.e., what are the dimensions bi (X [n] ) :=
dim H i (X [n] )? (In these notes we will only be interested in homology and
cohomology with coecients in Q , so for any space Y we write H i (Y ) for
H i(Y; Q ) and Hi (Y ) for Hi(Y; Q ).)
The Betti numbers of the Hilbert schemes X [n] were determined in [8].
There the following generating series for the Betti numbers was obtained:
Theorem 2.1.
X Y
2+i qm )( 1)i+1 bi (X [n] ) :
bi(X [n] ) ti qn = (1 ( 1)i t2m
n0;i0 m>0;i0
There are several proofs of this formula. The original proof is by using
the Weil-conjectures and counting subschemes over nite elds. A second
proof, based on intersection cohomology, was given by Gttsche and Soergel
o
in [9], and nally in [3] Cheah gave a third proof using the so-called virtual
Hodge polynomials. In addition to the Betti numbers, the last two proofs
also give the Hodge numbers of the Hilbert schemes.
If one puts t = 1 in Theorem 2.1, one gets an expression for the topo-
logical Euler characteristic e(X [n] ) of the spaces X [n]:
X Y
e(X [n] ) qn = (1 qm ) e(X)
m>0
n0
66

and by putting t = 1 one gets the generating series for the total dimensions
of the cohomology of X [n] :
(1 + qm )d :
X Y
dimQ H  (X [n] ) qn =
(1)
m>0 (1 q )
md +
n0
Here d and d+ are respectively the dimensions of the even and odd part of
H (X), i.e.,
X X
dim H 2i (X); dim H 2i+1 (X):
d+ = d=
i i
Later in these notes we shall come back to these formulas and give indi-
cations on how one can prove them.
One should note that we got nice generating functions for the Betti num-
bers and Euler numbers by looking at all the Hilbert schemes X [n] at once.
This is a rst indication that one should also look at all the cohomologies of
the Hilbert schemes at the same time.

3. The Fock space and the current algebra
Let
M
H  (X [n] )
H (X) =
n0
be the direct sum of all the cohomologies of all the Hilbert schemes X [n] .
This is a bigraded vector space over Q whose homogeneous parts are the
cohomology groups H i (X [n] ) for n  0 and i  0. For any class 2 H i (X [n] )
we will call n the weight of and i the cohomological degree or for short the
degree of . Sometimes we will write deg = (n; i).
The Hilbert scheme X [0] is just one point | the empty set is the only
subscheme of length zero. Hence H  (X [0] )  Q in a canonical way. We let
=
1 denote the fundamental class [X [0] ]. It corresponds to 1 2 Q , and we call
it the vacuum vector.
The space H (X) has a parity structure, or a super structure as many call
it: There is a decomposition
= H + (X)  H (X)
H (X)
Hilbert schemes and Heisenberg algebras 67

where H + (X) and H (X) are respectively the sums of the even and odd
part of the cohomology H  (X [n] ); that is
+ (X) = M H 2i (X [n] );
H
n0;i0
M
H 2i+1 (X [n] ):
H (X) =
n0;i0
The intersection form
Z
=: h ; i
X [n]


induces an intersection form on H (X) which respects the parity structure,
which means that it is symmetric on H + (X) and antisymmetric on H (X),
and that the two spaces H + (X) and H (X) are orthogonal.
The Poincar series of H (X) with respect to the weight-grading is given
e
by Gttsche's formula with t = 1 as in (1):
o
md
 (X [n] ) qn = Y (1 + q ) :
X
dimQ H
m>0 (1 q )
md +
n0
This series also appears naturally in a construction in the theory of Lie
algebras: Let V be a Q -vector space with a parity structure, or a super
space if you want; that is a decomposition V = V +  V of V into an odd
and an even part. Assume that V comes equipped with a bilinear form h ; i
respecting the parity structure. The cohomology H  (X) with the pairing
R
X  is our prototype of such a V .
Associated to V one constructs the Fock space F(V ) in the following way:
First we take a look at V
Q t Q [t]. A typical element of this space looks like
Pm
i=1 vi
t . Let T be the full tensor algebra on V
Q t Q[t]. To construct
i
F(V ) we impose in T the (super-)commutation relations:

[u
ti ; v
tj ] := (u
ti )(v
tj ) ( 1)p(u)p(v) (v
tj )(u
ti ) = 0
(2)
where u and v are any homogeneous elements in V , i.e., elements either in
V + or V , and where i  1 and j  1 are any integers. By p(w) we mean
the parity of a homogeneous element w, i.e., p(w) = 0 when w 2 V + and
p(w) = 1 when w 2 V . In order not to get confused with having two
di erent
-signs around, one from V
Q t Q [t] and one from T, we have
suppressed the
-signs from the tensor algebra T in equation (2).
68

The formal way to impose the relations above, is to divide T by the two-
sided ideal generated by the relations in (2). Clearly F(V ) is an algebra.
The unit element 1 2 F 0 (V ) is called the vacuum vector.
There is a natural grading on V
Q t Q [t] for which the degree of v
ti is i.
This grading induces, in an obvious way, a grading on the tensor algebra T.
As the relations (2) are homogeneous of degree i + j, the Fock space F(V )
is graded.
The elements of F(V ) are linear combinations of monomials of the form
(v1
tj ) (v2
tj ) : : : (vp
tjp )
1 2



where each vP is either an even or an odd element. The degree of such a
m
monomial is jm . The Fock space also has a parity structure. A monomial
as the one above is even (resp. odd) if the number of odd vm 's is even (resp.
odd).
One may then easily check that there is an isomorphism of graded vector
spaces
1
O
) S(V +
tm )
(V
tm):
=
F(V
m=0
Here M M
S i (V ); i (V );
S(V ) := (V ) :=
i0 i0
are the symmetric and alternating algebra on V .
From this the Poincar series of F(V ) is readily found to be
e
m dim V
m (V ) = Y (1 + q )
X
:
dimQ F
(1 qm )dim V +
m>0
m0
There is another algebra one may associate to V called the current algebra.
To construct this we start by setting V [t; t 1 ] = V
Q [t; t 1 ]. The elements
of V [t; t 1 ] are linear combinations of the elements qi [v] := v
ti for v 2 V
and i 2 Z.
Let now T be the full tensor algebra on V [t; t 1 ]. Elements of T are linear
combinations of monomials qi [v1 ] qi [v2 ] : : : qip [vp ] where we again suppress
1 2
the
-signs.
By declaring the degree (or weight) of qi [v] to be i, we get a grading on
T. There is also a parity structure on T: We declare qi [v] to be even if v
is even and odd if v is odd; and a monomial qi [v1 ] qi [v2 ] : : : qip [vp ] is even
1 2
(resp. odd) if it contains an even (resp. odd) number of odd qi [v]'s.
Hilbert schemes and Heisenberg algebras 69

We get the current algebra S(V ) by imposing the following relations in T:
 
qi[u]; qj [v] = ii+j hu; vie
(3)
where e is the unit element in T 0 V [t; t 1 ] = Q , and where u and v are any
elements either in V + or in V . The bracket is the supercommutator
[A; B] = AB ( 1)p(A)p(B) BA:
We also use the convention that m = 0 if m 6= 0 and 0 = 1.
The current algebra S(V ) acts on the Fock space F(V ) in the following
way. Recall that the Fock space is an algebra.
If i > 0, we let the element qi [u] act as multiplication by u
ti in the
algebra F(V ), i.e., qi[u] w = (u
ti ) w for any w 2 F(V ). In particular
qi[u]1 = u
ti .
For i = 0, we simply put q0 [u] w = 0 for any u and w.
To de ne the action of the operators q i [u], with i > 0, it is sucient to
state that q i [u]1 = 0 for any i > 0 and any u. Indeed by the relations (3)
we get
q i[u] (v
tj ) = q i[u] qj [v]1
= qj [v] q i [u]1 ij i hu; vi1
= ij i hu; vi1:
Thus the action is given by the formula
q i [u] (v
tj ) = ij ihu; vi1:
(4)
We call the operators qi [u] creation operators if i > 0 and annihilation
operators if i < 0. One has the following lemma:
Lemma 3.1. If the pairing h ; i is non-degenerate, the S(V )-module F(V ) is
irreducible, i.e., there is no proper, nonzero subspace invariant under S(V ).
Proof. It is clear that the vacuum vector 1 is a generator for F(V ) as a
module over S(V ). On the other hand, by applying an appropriate sequence
of annihilation operators q i [u] to any element w of F(V ), we may bring it
back to the vacuum 1. Indeed if fv g and fv g are dual bases for V , then
0
by equation (4) above we get
q ip [vi0p ] q [vi0p ] : : : q i [vi0 ](vi
ti ) (vi
ti ) : : : (vip
tip ) =
ip 1 2
1 1 1 2
1 1

= ( 1)p i1  i2  : : :  ip 1
70

where the vi 's and the vi0 's are elements from the bases fv g and fv g. The
0
operator
q ip [uip ]q ip [uip ] : : : q i [ui ] kills any other monomial made from ele-
1 1 1 1
ments in fv g, again by the relation (4). Hence any nonzero and invariant
subspace contains the vacuum, and consequently equals F(V ) because the
vacuum generates F(V ) as an S(V )-module.

4. The Nakajima operators
We now come back to our space H (X). It has the same Poincar seriese
 (X) of X. The aim of
as the Fock space modelled on the cohomology H 
this section is to de ne an action of the current algebra S H  (X) on the

space H (X) in a geometric way, making H (X) and F H  (X) isomorphic as

S H  (X) -modules.
We need to de ne operators qi [u] for i 2 Z and u 2 H  (X) satisfying
the relations (3). The operator qi [u] changes the weight by i, hence is given
by a map H  (X [n] ) ! H  (X [n+i]) for any n  0. In order to de ne these
maps, we introduce the incidence scheme X [n;n+i]  X [n]  X [n+i]; where
now i  0. It is de ned as


X [n;n+i] := (W; W 0 ) W  W 0; W 2 X [n] and W 0 2 X [n+i]
Here, as also in future W  W 0 means that W is a subscheme of W 0. This is
easily seen to be a closed subset of the product, and we give it the reduced
scheme structure.
The two projections induce two maps pn : X [n;n+i] ! X [n] and qn+i :
X [n;n+i] ! X [n+i] . There also is a morphism  : X [n;n+i] ! X (i) which is a
variant of the Hilbert-Chow-map. If W  W 0, then for the ideals IW and
IW 0 of IW and IW 0, we do have the inclusion IW 0  IW , and the quotient
IW =IW 0 is an OX -module of nite length which is supported at the points
where the two subschemes W and W 0 di er. We de ne
X
(W; W 0) := length(IW =IW 0 ) P 2 X (i) :
P2X
One may show that  is a morphism.


P 2 X which is
Inside X (i) there is the small diagonal  = iP
isomorphic to X.
Hilbert schemes and Heisenberg algebras 71

We have the following diagram:
X =   Xx (i)
x
f? ?
? ?
X [n;n+i] qn+i
Zn;i  !
(5) X [n+i]
?
pn ?
y
X [n]
where Zn;i is the component1 of
1 () = (W; W 0 ) W  W 0 ; I =I 0 is supported in one point
 WW
which is the closure of the subset where Supp (IW =IW 0) is disjoint from W.
We give it the reduced scheme structure.2 One easily checks that
dim Zn;i = 2n + i + 1;
(6)
indeed W is arbitrary in X [n] , but W 0 W is con ned to Mi .
We may pull back any class u 2 H  (X) along f to get a cohomology
class f  u on Zn;i . Applying this to the fundamental class [Zn;i ], we get the
homology class f  u \ [Zn;i ]. This in turn we may push forward to X [n;n+i]
via the inclusion j : Zn;i ! X [n;n+i], and in this way we get the homology
class
Qn;i (u) := j (f  u \ [Zn;i])
on X [n;n+i].
Now we are ready to de ne the Nakajima creation operators; i.e., the
operators qi [u] with i  0. We de ne their action on an element 2 H  (X [n] )
by
qi[u] := qn+i  (p \ Qn;i(u));
n
which we regard as an element in H  (X [n+i] ) by Poincar duality.
e
This de nition is similar to the classical way of de ning the correspondence
between X [n] and X [n+i] associated to a class on their product | if one insists
on qi [u] being a correspondence, one has

qi [u] = pr2  (pr1 \  Qn;i(u))
where  : Zn;i ! X [n]  X [n+i] is the inclusion map, and where pr1 and pr2
are the two projections.
To our knowledge it is unknown whether  1 () is irreducible or not.
1

The scheme-theoretical inverse image  1 () is not reduced.
2
72

In order to get some geometric feeling for what these operators do, we
assume that u and are represented by submanifolds U  X and A  X [n] .
Then qi [u] is represented by the subspace


(7) W 0 2 X [n+i] there is a W 2 A with W  W 0 ;

W and W 0 such that they di er in one point in U :
To put it loosely, the creation operator qi [u] sends A to the set of subschemes
which we obtain by adding a subscheme of length i supported in just one
point from U to a subscheme in A. As an illustration we prove the following
lemma
Lemma 4.1.
qi[pt]1 = [Mi (P )]:
qi [X]1 = [Mi ]:
Proof. To explain the rst equality, we observe that 1 is represented by the
empty set. Hence by (7) the class qi[pt]1 is represented by
[i] ;  W 0; ; and W 0 di er only in P ;
0
W 2X
where P is any point in X, and this is clearly Mi (P ); we are just adding
subschemes supported at P to the empty set.
The second equality is similar. We add subschemes of length i supported
in one point to the empty set, but this time without any constraint on the
point.
We now come to the de nition of the Nakajima annihilation operators
q i[u], where i > 0. We shall, except for a sign factor, literally go the other
way around in the diagram (5). For any class 2 H  (X [n+i]) we de ne


q i[u] := ( 1)i pn  qn+i \ Qn;i(u) :
The geometrical interpretation of these annihilation operators is analogous to
that of the creation operators. If the class is represented by a submanifold
B  X [n+i], then q i[u] will be represented by the subspace


(8) W 2 X [n] there is a W 0 2 B with W  W 0 such that they

di er in just one point in U :
In other words, the annihilation operator q i[u] sends B to the set of the
subschemes we get by throwing away subschemes supported in one point in
U from subschemes in B. Of course this is possible only for some of the
subschemes in B.
Hilbert schemes and Heisenberg algebras 73

We will give one example. Let C  X be a smooth curve, and let  = [C]
be its fundamental class in H 2 (X). For every n  0 the symmetric product
C (n) is naturally embedded in the Hilbert scheme X [n]. Put n = [C (n) ]. Let
C 0 be another smooth curve, and assume that hC; C 0 i = a. Let 0 = [C 0].
Lemma 4.2.
q i[0 ]n = ( 1)i an i
Proof. We assume for simplicity that C and C 0 intersect transversally in just
one point. Because C is smooth, a subscheme W  C is uniquely determined
P
by the associated 0-cycle P2C length (WP )P . Hence there is just one sub-
scheme W 0 of length i in C (i) , whose support is C \C 0. Splitting o W 0 from
the subschemes in C (n) containing it, obviously gives an isomorphism from


W [ W 0 2 C (n) W 2 C (n i) to C n i. This concludes the proof.
The operators qi [u] and q i [u] behave very well with respect to the inter-
section pairings on X [n] and X [n+i]:
Lemma 4.3. For classes 2 H  (X [n]) and 2 H  (X [n+i]) we have the
equality
Z Z
i  q i[u] = (qi [u] )  :
( 1)
X [n] X [n+i]
Proof. By the de nition of the operators and the projection formula, both
are equal to Z
p  qn+i \ Qn;i (u):

n
X [n;n+i]

The following lemma is easily deduced from the de nition of the Nakajima
operators
Lemma 4.4. The operator qi[u] is of bidegree (i; deg u + 2(i 1)).
5. The relations
The basic result of Nakajima in [13] is that his creation and annihilation
operators satisfy the relations of the current algebra. Below we shall sketch
a proof of that, closely following the proof that Nakajima gave in [14].
Theorem 5.1. (Nakajima, Grojnowski) For all integers i and j and all
classes u and v in H  (X) the following relation holds
 
qi[u]; qj [v] = ii+j hu; viid:
74

The proof is in two steps. The rst is to establish
Proposition 5.2. There are universal non-zero constants ci such that
 
qi[u]; qj [v] = ci i+j hu; viid:
Here by universal we mean that the ci 's neither depend on u or v nor on
the surface X. A sketch of the proof of this proposition, will occupy section
6. The next step is | naturally enough | to establish
Proposition 5.3. ci = i.
The last proposition can be proved in two di erent ways. The constants ci
have a natural interpretation as intersection numbers on the Hilbert scheme.
Recall that dim Mi = i+1 and dim Mi (P ) = i 1. Therefore Mi and Mi (P )
are of complementary dimension, and their intersection gives a number.
R
However Mi (P )  Mi so they do not intersect properly and X [Mi (P )][Mi ] [i]

is not entirely obvious to compute. By induction one may prove (see [5]):
Proposition 5.4. (Ellingsrud{Strmme)
Z
[Mi (P )][Mi ] = ( 1)i 1 i:
X [i]
The following lemma then proves Proposition 5.3.
Lemma 5.5. If i > 0 then ci = ( 1)i 1 RX [Mi(P )][Mi ]. [i]


Proof. Recall that by Lemma 4.1 we have [Mi (P )] = qi [pt]1 and [Mi ] =
qi[X]1. The Nakajima relation for the operators q i[X] and qi[X] reads
qi[X] q i [pt] q i[pt] qi [X] = ci  id:
When we apply this to the vacuum vector, we obtain
q i[pt] qi [X]1 = ci
because any annihilation-operator kills the vacuum. Now, by Lemma 4.3,
we get
Z Z
qi [pt]1  qi [X]1 =
[Mi (P )][Mi ] =
X [i] X [i] Z
1  q i[pt]qi[X]1 =
= ( 1)i
ZX
[0]


= ( 1)i ( ci )1 = ( 1)i 1 ci :
X [0]
Hilbert schemes and Heisenberg algebras 75

There is also another and very elegant approach to Proposition 5.3 due to
Nakajima where he uses vertex operators. We shall give this later on.
The main consequence of the Nakajima-Grojnowski theorem is the follow-
ing:
Theorem 5.6. The space H (X) and the Fock-space F (H  (X)) are isomor-
phic as S(H (X))-modules.
Proof. There is a map as S(H (X))-modules from F (H  (X)) to H (X) de-
ned by sending u
ti to qi[u]1. The two spaces have the same Poincar e
series, and F(H  (X)) is an irreducible S(H (X))-module.
6. Indication of how to get the relations
In this section we explain in a sketchy way why the commutation relations
in Theorem 5.1 hold.
We will rst treat the case when i and j have the same sign, for exam-
ple both are positive. This is the case of the composition of two creation-
operators.
Then i+j = 0, and we have to prove that qi [u] and qj [v] commute up to
the correct sign. For simplicity we also assume that u = [U] and v = [V ]
where U and V are submanifolds of X intersecting transversally.
In the de nition of the Nakajima operators we made use of the subvariety
Zn;i  X [n]  X [n+i]: Recall that it was given as
0 ) W  W 0 di er in one point :

Zn;i = (W; W
We are going to compare the two operators qj [v]qi [u] and qi [u]qj [v], which
both map the cohomology of X [n] to the cohomology of X [n+i+j]. The natu-
ral place to describe the operator qj [v]qi [u], which is the composition of two
correspondences, is on the product X [n] X [n+i] X [n+i+j]: In the description
the following subvariety of this product will play a role:
Z1 = p121 (Zn;i ) \ p231 (Zn+i;j ):
(9)
It consists of triples (W; W 0 ; W 00 ) of nested subschemes | i.e., W  W 0 
W 00 | such that W and W 0 just di er in one point which we call P, and
at the same time W 0 and W 00 are di erent only in one point that we call
Q. The quotient IW =IW 0 has support fPg and satis es length IW =IW 0 = i:
Similarly, the quotient IW 0 =IW 00 has support fQg and is of length j.
There is a map f1 : Z1 ! X  X sending the triple (W; W 0 ; W 00 ) to the
pair (P; Q).
76

In a similar manner we let Z2  X [n] X [n+j] X [n+i+j] be the subvariety
given by
Z2 = p121 Zn;j \ p231 Zn+j;i:
(10)
Its elements are the triples (W; W 0 ; W 00 ) of nested subschemes with IW =IW 0
and IW 0 =IW 00 both having one-point-support in, say, Q and P respectively;
the rst one of length j and the other one of length i. As above there is
a morphism f2 : Z2 ! X  X, sending the triple (W; W 0 ; W 00 ) to the pair
(Q; P).
Lemma 6.1. Let be a class on X [n].

qi[u] qj [v] = p3 p  f2 (v  u) \ [Z2 ] ;
(11) 1

qj [v] qi [u] = p3 p  f1 (u  v) \ [Z1 ] ;
(12) 1
where pi denotes the restriction of the i-th projection to Z1 in the rst line,
and of Z2 in the second.
Proof. This is just the formula for composing correspondences; the only point
to check is that the intersections in (9) and (10) are both proper.
0 0
Let Z1  Z1 and Z2  Z2 be the two open subsets where the two points
0
P and Q are di erent. A typical element of Z1 , for example, may be drawn
as

0 W 0 00 P
11
1
00 1
11 01
0
1
0Q
1
W™ W™™

It has a 'central' part W and two 'fuzzy' ends, one in P and one in Q.
The 'fuzzy' end at P is a subscheme of length i supported there, and the
other 'fuzzy' end is a subscheme supported at Q of length j. The subscheme
W 0 is the union of the 'central' part and the 'fuzzy' end at P. Of course P
or Q may belong to the central part, but still the above statement makes
sense if interpreted in the right way.
0
The drawing above might as well represent a typical element in Z2 . The
only di erence being that in that case the 'fuzzy' part of length j at Q would
belong to W 0 instead of the one of length i at P. Hence to any nested triple
(W; V; W 00 ) in Z1 we may associate the triple (W; V 0 ; W 00 ) where we get V 0
0
Hilbert schemes and Heisenberg algebras 77

from V by swapping the 'fuzzy' parts at P and Q. With a little thought one
may convince oneself that this swapping is well de ned even if the 'central'
part touches P or Q. In this way we get an isomorphism g : Z1  Z2 : 0= 0
Clearly this isomorphism respects both p1 and p3 | it doesn't change the
extreme subschemes W and W 00 | and up to permutation of the two factors
of X  X, it respects f1 and f2 . By the projection formula we therefore get
the following equality
0
g p  f1 (u  v) \ [Z1 ] = ( 1)deg u deg v p  f2(v  u) \ [Z2 ]:
0
1 2
The sign comes from the following: u  v = pr1 u  pr2 v and via g this is
 
 
mapped to pr2 u  pr1 v = ( 1)deg u deg v v  u.
It only remains to see that there is no contribution from the boundaries,
i.e., when P = Q. The easy case is when U \ V = ;, then the boundary is
empty | indeed P 2 U and Q 2 V .
In general, a dimension estimate will show that all components of the
boundary are | with good margin | of too small dimension to contribute.
We shall need
0 0
dim Z1 = dim Z2 = 2n + i + j + 2:
Indeed, the n points in the 'central' part each have 2 degrees of freedom,
and we are free to choose the 'fuzzy' ends from Mi and Mj , and these two
varieties are of dimension i + 1 and j + 1 respectively.
By the transversality of U and V we know that
dimR U \ V = dimR U + dimR V 4
We now give the dimension count for f 1 (U  V ) \ (Z Z 0 ), where we
have suppressed the indices and only write f, Z, Z 0; the suppressed index
can be either 1 or 2. The 'central' part is of length n and gives a contribution
of 4n to the (real) dimension. Now P = Q, so the two 'fuzzy' parts live at
the same point. If they could be chosen independently, their contribution to
the dimension would be
dimR (Mi (P )  Mj (P ) = 2(i 1) + 2(j 1)
as long as P is xed, and P can only move in U \ V . As this gives an upper
bound of their contribution, we get
dimR (f 1 (U  V ) \ (Z Z 0 ))  dimR Mi (P )  Mj (P ) + dimR U \ V
 4n + 2i + 2j + dimR U + dimR V 8
< 4n + 2i + 2j + dimR U + dimR V 4:
78

The class f  (u  v) \ [Z] lives in Hr (Z) where
r = dimR Z (4 dimR U) (4 dimR V )
= 4n + 2i + 2j + dimR U + dimR V 4:
After the dimension count, we know that the map Hr (Z Z 0 ) ! Hr (Z)
induced by the inclusion is an isomorphism. Hence
g f1 (u  v) \ [Z1 ] = ( 1)deg u deg v f2(u  v) \ [Z2 ];
and we are done.
Now we shall treat the perhaps more interesting | at least more subtle
| case of the composition of one creation and one annihilation operator.
That is, the composition of one operator of the form q i [u] and one of the
form qj [v] where i  0 and j  0.
We have to explain why
q i [u] qj [v] + ( 1)deg u deg v qj [v] q i [u] = ihu; vij i id;
and we start by examining the composition q i [u] qj [v]: For any n  0 it
induces a map from H  (X [n] ) to H  (X [n+j i]): As in the preceding case, it
is natural to look at the subvariety
Z1 = p121 Zn;j \ p231 Zn+j i;i  X [n]  X [n+j]  X [n+j i]:
It may be described as the variety of triples (W; W 0 ; W 00 ) 2 X [n]  X [n+j] 
X [n+j i] with W  W 0 and W 00  W 0 | this time the one in the middle is
bigger than the two on the sides | such that W 0 and W 00 di er in just one
point, and at the same time W 0 and W 00 also di er only in one point. Call
those points P and Q respectively.
The picture now looks like
W™ W™™
0P
1
11
00 0
1
00 0 0
11 1 1
1
0
00
11Q W


This time the big one in the middle | W 0 | is the whole subscheme.
The one to the left | W | is the whole except the 'fuzzy' part at P, and
the one to the right | W 00 | is the whole except the 'fuzzy' part supported
at Q. As before there is a map f1 : Z1 ! X  X sending a triple to the two
points (P; Q) and there is the lemma
Hilbert schemes and Heisenberg algebras 79

Lemma 6.2.

q i[u] qj [v] = p3 p  f2 (v  u) \ [Z1 ] :
1
To understand the composition q i[u] qj [v]; we introduce the subvariety
Z2 = p121 Zn i;i \ p231 Zn i;j  X [n]  X [n i]  X [n+j i]:
This time the points in Z2 are triples (W; W 0 ; W 00 ) of subschemes with W 0 
W and W 0  W 00 | the one in the middle is smaller than the other two
| and as usual W 0 and W are di erent only at a point P and W 0 and W 00
di er only at a point Q. The picture looks like

W™™
1
0P
0
1
00 0 0 0
11 1 1 1
0W
1
W™
00
11Q


The little one in the middle | W 0 | is the 'central' part, and the two
extremes | W and W 00 | are subschemes we get by adding the 'fuzzy' part
located at P respectively Q.
Just as before one checks that
dim Z1 = dim Z2 = 2n + i + j + 2;
for the complex dimensions, and there is the usual map f2 : Z2 ! X  X:
We follow the same track as in the creation-creation process, and de ne
Z 0  Z | where the missing index is either 1 or 2 | as the open subsets
where P 6= Q. Then there is an isomorphism g : Z1  Z2 : Indeed we keep
0= 0
the two extremes and exchange the smallest 'central' part with the whole.
Writing WP for the part of W supported at P and similarly for Q and W 0 ,
W 00 , this amounts to sending the biggest one, W 0, to (W 0 nWP nWQ )[WP [
0 0
00
WQ which has a meaning as long as P 6= Q. In the same way, it is easy to
write down the inverse of g.
Lemma 6.3.
0
g p  f1(u  v) \ [Z1 ] = ( 1)deg u deg v p  f2(v  u) \ [Z2 ]:
0
1 2
Now we come to the more subtle point of analyzing the boundaries where
P = Q. Because when we compute the composition, we apply p13, what
really matters is the dimension of p13 (ZnZ 0 ) | for missing index equal 1 and
80

0
2. In the case of p13 (Z2 n Z2 ) everything works as in the creation-creation
case, and there will be no contribution from the boundary, so let us turn
0
our attention to the subtle case p13 (Z1 n Z1 ). The case U \ V = ; gives
no boundary at all, but if U \ V = fPg something happens. If in addition
i = j we may take W = W 00 . There always exists a subscheme of length
n + j containing any subscheme of length n which is supported at p. Hence
0
in this case p13 (Z1 n Z1 ) will be supported along the diagonal in X [n]  X [n].
One may check by dimension count as before that this is the only possible
contribution from the boundary. It follows that
 
q i [u]; qi [v] =  id
for some number .
7. Vertex operators and Nakajimas computation of the
constants
For any class u 2 H  (X) and any sequence d = fdm gm0 of numbers we
introduce the following operator, often called a vertex operator,
X 
dm qm [u]zm = exp(P (z)):
Ed;u(z) = exp
m>0
P
dm qm [u]z m .
where P(z) = m>0 When we apply Ed;u (z) to the vacuum
vector, we obtain a sequence f m gm0 of classes in H (X), with m of weight
m and 0 = 1, which are de ned by the expression
X
m := exp X dm z m qm [u]1 = exp(P (z))  1:
m z
m>0
m0
We have
Proposition 7.1. For any two classes u; v in H (X), and any natural num-
ber i, R element exp(P (z))  1 is an eigenvector for qi [v] with eigenvalue
the
ci di ( X u  v)zi . That is, for m  0, we have the equality
Z 
u  v m i:
qi[v] m = ci di
X
In the proof of the proposition we shall need the following easy lemma:
Lemma 7.2. If A and B are two operators commuting with their commu-
tator, then for any p  1
[A; B p ] = p[A; B]B p 1 :
Hilbert schemes and Heisenberg algebras 81

Furthermore
[A; exp B] = [A; B] exp B:
Proof. Exercise.
To prove Proposition 7.1 we do the following computation:
 
q i[v] exp(P (z))  1 = q i[v]; exp(P (z)) 1 ann. oper. kill vacuum
=[q i [v]; P(z)] exp(P (z))  1 Lemma 7.2
X 
 
dm q i[v]; qm [u] zm exp(P (z))  1 de nition of P(z)
=
m>0 Z
uv)zi exp(P (z))  1
di ci (
= Nakajima relations:
X
By the de nition of f m g, this completes the proof.
The property in Proposition 7.1 is very strong. In fact, it determines the
sequence m completely.
Lemma 7.3. Let the two sequences f m g and f m g from H (X) be given,
with m and m both of weight m and 0 = 0 = 1. Assume that for any
i > 0 and any class v in H  (X), there is a number ei;v such that both m
and m satisfy the equation
qi[v]xm = ei;v xm i
for all n  0. Then m = m for all m  1.
Proof. The proof goes by induction on m. We assume that j = j for
j < m. Then for any i  0 and any class v on X we have
q i [v]( m m ) = ei;v ( m i m i ) = 0
by induction. Hence S(H (X))( m m ) will be a sub S(H (X))-module
all of whose elements are of weight greater than or equal to m. Now if
m  1, the vacuum, being of weight 0, cannot be in this module which
consequently must be trivial, since H (X) is an irreducible S(H (X))-module.
Hence m = m , and we are done.
We shall need the following variant of the above lemma:
Lemma 7.4. Let f m g and f m g be two sequences in H (X) with m and
m both of weight m and 0 = 0 . Assume that for all i  0 and all classes
v in H (X) there are numbers ei;v with ei;v = 0 if deg v < 2, such that the
following two conditions are satis ed.
82

1. q i[v] m = ei;v m i for all i  0 and all classes v in h (X),
2. deg m = 2m and
q i [v] m = ei;v m i
whenever deg v  2 and i > 0.
Then m = m for all m  0.
Proof. Again we use induction on m and assume that m i = m i for all
i > 0. Just as in the proof above, it is sucient to see that the vacuum
vector is not contained in the S(H (X))-module spanned by m m . In
other words we must check that any sequence of 'backwards' moves kills
m m ; to that end let
z = q i [v1 ]q i [v2 ] : : : q ip [vp]( m m )
1 2


be the result of p 'backwards' moves applied to m m . If one of the vi 's
is of degree greater than or equal to 2, we know that z = 0. Indeed, this
follows by induction from two conditions in the lemma since the annihilation
operators involved all commute | we can move the annihilation qij [vj ] with
deg vj  2 to the right in the 'backwards' sequence. Hence we may assume
that all the vi 's are of degree less than 2. Then by condition 1. in the lemma,
we have q i [v1 ]q i [v2 ] : : : q ip [vp ] m = 0 and hence
1 2


z = q i [v1 ]q i [v2 ] : : : q ip [vp] m :
1 2


We want to see that the case z = 1 cannot happen. Indeed, if z = 1, then
Pp
j=1 ij = m. By computing the degree of z from the expression above, we
obtain
X 
deg z = deg m + deg vj 2(ij + 1) =
X X
(deg vj 2) 2 ij =
=2m +
X
(deg vi 2);
=
from which it follows that deg z < 0, and thus z = 0.
Let now C  X be a smooth curve whose class in H  (X) is . Let n
denote the class of the n-th symmetric power C (n) of C in X [n] . The classes
n may be computed in terms of the Nakajima creation operators as in the
following theorem which appeared in [13] and [10].
Hilbert schemes and Heisenberg algebras 83

Theorem 7.5. (Nakajima, Grojnowski)
1)m 1 q []z m   1:
n = exp X (

X
nz m
cm
m>0
n0
Proof. By Proposition 7.1 we know that the sequence f m g de ned by the
identity
m1
n = exp X ( 1)
 
X
m 1
nz qm []z cm
m>0
n0
satis es
Z 
1)i
qi[v] m = ( v m i
X
for all i > 0 and all v 2 H  (X). From Lemma 4.2 we know that q i [v]n =
R
( 1)i an i for any curve class v satisfying X v = a. It is also clear that if
v = [V ] for V a submanifold of X with C \ V = ;, then q i[v]n = 0; hence
we know that
Z
qi[v]n = ( 1)i ( v)n i
X
holds for all i > 0 and all classes v on X of degree 2 or more. The theorem
then follows from Lemma 7.4.
Finally we will give the second computation | due to Nakajima | of the
constants ci as we promised. We start by computing derivatives in Theorem
7.5 to obtain
d exp P(z)1 = d P(z) exp(P (z))  1 =

X
nnzn 1 = dz dz
n1
1
 X ( 1)m 1 m X 
n z n  1:
m1 
cm qm []z 
=
m>0 n=0
From this we obtain
n
( 1)m 1 m q [] :
X
nn =
(13) m nm
cm
m=1
As the constants cm are universal, we may very well assume that X = P2
and that C is a line.
84

Lemma 7.6. Let C and C 0 be two curves in X intersecting transversally in
one point; e.g., two di erent lines in P2 . Then
(
Z
n  n = 1 if n  1
0
0 else
X [n]



Proof. If t = 0 and t0 = 0 are local equations for C and C 0 at the common
point, a subscheme in C (n) supported at this point is necessarily of the form
n (n) must be of the form C [t; t0 ]=(tn ; t0 ). If a
C [t; t0 ]=(t; t0 ) and one in C 0
subscheme W simultaneously is of these two forms, necessarily n  1.
Finally we prove
Theorem 7.7.
ci = i:
Proof. The idea is to intersect (13) with n . For n = 1 we get
Z
1 Z   q []
1 =  = 1 1 0
c1 ZX
X
= c1 ( q 1 []1 )  0
1 ZX
= c1 0  0 = c1 :
1X 1
This gives c1 = 1. Assume now that n  2. Then we obtain
n
Z X ( 1)m 1 m Z
n  n = n  qm[]n m
0= c m
X [n] X [n]
m=1
n
( 1)m 1 m ( 1)m Z
X
q m[]n  n m
= cm [n m]
X
m=1
n
X ( 1)m 1 m Z
n m  n m
= cm X [n m]
m=1
( 1)n 1 n ( 1)n 2 (n 1) :
= +
cn cn 1
Hence
cn = cn 1
n n1
from which we get cn = n.
Hilbert schemes and Heisenberg algebras 85

8. Computation of the Betti numbers of X [n]
As before, let X be a smooth projective surface over C . We will now show
formula (1) for the Betti numbers of the Hilbert scheme X [n] of points. We
needed it in the rst part to show that
M
H  (X [n] )
H (X) :=
n0
is an irreducible representation of the Heisenberg algebra. There are at
least three possible di erent approaches which have been used to prove this
result; using the Weil conjectures [8], using perverse sheaves and intersection
cohomology [9], or nally one can use the so-called virtual Hodge polynomials
[3]. The last two approaches will in addition give the Hodge numbers of the
Hilbert schemes. In these notes we will use the second approach. It has the
advantage of leading to the shortest and most elegant proof, and to almost
completely avoid any computations. The disadvantage is that it requires
very deep results about intersection cohomology and perverse sheaves. We
will rst brie
y describe these results and then show how one can use them
as a black box, which with rather little e ort gives the desired result.
Let Y be an algebraic variety over C . In this section we only use the
complex (strong) topology on Y . We want to stress again that all the coho-
mology that we consider is with Q -coecients. In particular H i (Y ) stands
for H i (Y; Q ). There exists a complex ICY of sheaves on Y (for the strong
topology), such that
IH (Y ) := H  (Y; ICY )
is the intersection homology of Y (strictly speaking ICY is an element in
the derived category of Y ). Recall that the intersection cohomology groups
IHi (Y ) are de ned for any algebraic variety and ful ll Poincar duality (be-
e
i (Y ) and IHi (Y )). ICY is called the intersection cohomology com-
tween IH
plex of Y . If Y is smooth and projective of dimension n, then
ICY = Q Y [n];
is just the constant sheaf Q on Y put in degree n. Therefore IHi n (Y ) =
H i (X; Q ). More generally, if Y = X=G is a quotient of a smooth vari-
ety of dimension n by a nite group, then ICY = Q Y [n], and thus again
IHi n (Y ) = H i (X; Q ).
Let now f : X ! Y be a projective morphism of varieties over C . Suppose
that Y has a strati cation a
Y = Y

86

into locally closed strata. Let X := f 1 (Y ). Assume that f : X ! Y is
a locally trivial bundle with ber F (in the strong topology).
De nition 8.1. f is called strictly semismall (with respect to the strati -
cation), if, for all ,
2dim(F ) = codim(Y ):
We will use the following facts:
: Fact 1. Assume that f : X ! Y is strictly semismall, and that the F
are irreducible, then
X
Rf (ICX ) = ICY :

(see [9]). Here Rf is the push-forward in the derived category, and
Y is the closure of Y . This is a consequence of the Decomposition
Theorem of Beilinson-Bernstein-Deligne [1].
: Fact 2. Let  : X ! Y be a nite birational map of irreducible algebraic
varieties, then
R (ICX ) = ICY
(see [9]).
Now we want to see how these facts about the intersection cohomology
complex can be applied to compute the Betti numbers of the Hilbert schemes
of points.
Let  : X [n] ! X (n) be the Hilbert-Chow morphism. The symmetric
power X (n) is strati ed as follows: Let  = (n1 ; : : : ; nr ) be a partition of n.
We also write  = (1 ; 2 ; : : : n n ), where i is the number of l such that
1 2

nl = i. We put

n o
(n) := X n x 2 X (n) the x are distinct ;
X ii i

[n] (n) (n)
and X :=  1 (X ). The X form a strati cation of X (n) and similarly
[n]
the X form a strati cation of X [n]. The smallest stratum

n o
[n] := W 2 X [n] Supp(W ) is a point
X(n)

is just the variety Mn . It is a locally trivial ber bundle (in the strong
topology) over X (n) ' X, with ber
F(n) := Mn(P ):
In particular the ber is independent of X. This is because nite length
subschemes concentrated in a point depend only on an analytic neighborhood
Hilbert schemes and Heisenberg algebras 87

[n]
of the point. It follows that each stratum X(n ;:::;nr ) is a locally trivial ber
1
(n)
bundle over the corresponding stratum X(n ;:::;nr ) , with ber F(n )  : : :  1
F(nr ) .
1



By Theorem 1.3 Mn (P ) is irreducible of dimension (n 1), which is half
(n)
the codimension of X(n) in X (n) . It follows that  : X [n] ! X (n) is strictly
semismall with respect to the strati cation by partitions. Therefore we ob-
tain by Fact 1. above
M
R (Q X [2n]) = R (ICX ) = ICX  :
[n] [n] (n)


We write
 = (n1 ; : : : ; nr ) = (1 ; 2 ; : : : ; n n );
1 2


and denote ( ) := ( 1 ; 2 ; : : : ; n ). Then there is a morphism
 : X ( ) := X ( )  : : :  X ( n ) ! X (n) 
1

n
X
(1 ; : : : ; n ) 7! i  i :
i=1
X (n) .
It is easy to see that  is the normalization of Therefore Fact 2.

above implies
 
ICX  = R( ) (ICX ) = R( ) Q X 2j j ;
(n) ( ) ( )

P
where j j = i i . Putting this together, we get that
M  
R (Q X [2n]) = R( ) Q X 2j j :
(14) [n] ( )


P
Here the sum runs through all ( ) = ( 1 ; : : : ; n ) with i i i = n. Finally
we take the cohomology of relation (14). We recall that taking the coho-
mology of a complex of sheaves commutes with push-forward. Therefore we
obtain
i+2n (X [n] ) = M H i+2j j (X ( ) ):
H

So with this we have completely determined the additive structure of the
cohomology of the Hilbert schemes X [n] in terms of that of the symmetric
powers X (k) . The cohomology of the symmetric powers is well known. As
X (n) is the quotient of X n by the action of the symmetric group Sn by
permuting the factors, we see that H i (X (n) ) = H i (X n )Sn is the invariant
part of the cohomology of X n under the action of Sn .
88

Now we want to turn this into a generating function for the Betti numbers
of the Hilbert schemes X [n] .
P
Let p(Y ) := i dim(H i+dim(Y ) (Y ))z i be the (shifted) Poincar polyno-
e
mial of a variety Y . The description above of the cohomology of the sym-
metric powers leads, by Macdonald's formula [12], to a generating function
for their Poincar polynomials.
e
1 (1 + z 1 t)b (X) (1 + zt)b (X)
X
(n) )tn = 1 3
p(X :
(1 z 2 t)b (X) (1 t)b (X) (1 z 2 t)b (X)
0 2 4
n=0
Here the bi (X) = dim(H i (X)) are the Betti numbers of X. We are now able
to put all the ingredients together to get our desired generating function for
the Betti numbers of the Hilbert schemes.
1 1
X X X
p(X [n])tn = p(X ( 1 ) )p(X ( 2 ) ) : : : p(X ( n ) )t 1 +2 2 +:::n n
n=0 n=0 1 +2 2 +:::n n =n
!
1
Y X
p(X (l) )tkl
=
k=1 l
1 (1 + z 1 tk )b (X) (1 + ztk )b (X)
Y 1 3
:
=
(1 z 2 tk )b (X) (1 tk )b (X) (1 z 2 tk )b (X)
0 2 4
k=1
This (keeping track of the shift in the Poincar polynomial) is the formula
e
of Theorem 2.1.
9. The Virasoro algebra
The rest of these lectures is mostly based on the paper [11] of Lehn.
Before we got a nice description of the additive structure (+ the intersection
pairing) of the Hilbert schemes, which put all the Hilbert schemes together
into one structure. Our aim now is to get some insight into the ring structure
of the cohomology rings of the Hilbert schemes of points X [n]. We want to
see how the ring structure is related to the action of the Heisenberg algebra.
That is; for any cohomology class 2 H  (X [n] ) we can look at the operator
of multiplying by . We want to try to express these operators in terms of
the Heisenberg operators. In particular we will be interested in the Chern
classes of tautological sheaves on the Hilbert schemes, which are useful in
many applications of Hilbert schemes.
As a rst step we will construct an action of a Virasoro algebra on the
cohomologies of the Hilbert schemes. This is not such a surprising result:
There is a standard construction, which associates to a Heisenberg algebra a
Hilbert schemes and Heisenberg algebras 89

Virasoro algebra. This construction is essentially translated into geometric
terms. One of the main technical results will be a geometric interpretation
of the Virasoro generators.
We will, in the future, ignore all signs coming from odd-degree cohomology
classes.
De nition 9.1. Let  : H  (X) ! H (X  X) = H  (X)
H (X) be the
push-forward via the diagonal embedding  : X ! X  X. If ( ) =
P
i i

i , we write
X
qnqm( ) := qn[ i ]qm[
i ]:
i
We de ne operators Ln : H  (X) ! End(H (X)) by
1 X q q  , if n 6= 0
Ln := 2 n
2Z
X
L0 := q q  :
>0
The sums appear to be in nite, but, for xed y 2 H (X) and 2 H  (X),
only nitely many terms contribute to Ln [ ]y.
 
Theorem 9.2. 1. Ln[u]; qm [w] = mqm+n[uw].
2. ! Z
n3 n
 
c2(X)uw 1:
Ln [u]; Lm [w] = (n m)Ln+m [uw] 12 X
Part 2. can be viewed as saying that the Virasoro algebra given by the
Ln [X] acts on H (X) with central charge c2 (X).
The proof of the theorem is mostly formal. We will show part 1. in case
n 6= 0. Writing X
(u) = si
ti ;
i
we get
    
q [si ]qn  [ti ]; qm [w] = q [si] qn  [ti ]; qm [w] + q [si]; qm [w] qn  [ti ]
! !
Z Z
= ( m)n+m  qn+m [si ] ti w + ( m)m+ wsi qn+m[ti]:
X X

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