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11 (Orsay), 1992.

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In Complex projective geometry (Trieste, 1989/Bergen, 1989), pages 213{240. Cam-

bridge Univ. Press, Cambridge, 1992.

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Sup., 4:181{192, 1971.

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[20] C. Sorger. On moduli of G-bundles of a curve for exceptional G. Ann. Sci. Ecole

Norm. Sup. (4), 32(1):127{133, 1999.

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Villars, Paris, 1962.

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To appear in Comm. Math. Phys.

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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 dene 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 dierent.

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 (unspecied) point, and dene

[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 identied 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

dierent

-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 dene 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 dene 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 dene 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 dene these

maps, we introduce the incidence scheme X [n;n+i] X [n] X [n+i]; where

now i 0. It is dened 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 dier. We dene

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 conned 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 dene the Nakajima creation operators; i.e., the

operators qi [u] with i 0. We dene 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 denition is similar to the classical way of dening 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 dier 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 dier 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 denition 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 dene

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

dier 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 denition 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 denition 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 dierent 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 denition 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 dier 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 dier in one point which we call P, and

at the same time W 0 and W 00 are dierent only in one point that we call

Q. The quotient IW =IW 0 has support fPg and satises 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 dierent. 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

Wâ€™ Wâ€™â€™

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 dierence 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 dened 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 dier in just one

point, and at the same time W 0 and W 00 also dier only in one point. Call

those points P and Q respectively.

The picture now looks like

Wâ€™ Wâ€™â€™

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 dierent only at a point P and W 0 and W 00

dier only at a point Q. The picture looks like

Wâ€™â€™

1

0P

0

1

00 0 0 0

11 1 1 1

0W

1

Wâ€™

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 dene

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 fm gm0 of classes in H (X), with m of weight

m and 0 = 1, which are dened 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 denition of P(z)

=

m>0 Z

uv)zi exp(P (z)) 1

di ci (

= Nakajima relations:

X

By the denition of fm 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 fm g and fm 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 fm g and fm 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 satised.

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 fm g dened by the

identity

m1

n = exp X ( 1)

X

m 1

nz qm []z cm

m>0

n0

satises

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 dierent 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 dierent 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 eort 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 dened for any algebraic variety and fulll 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 stratication 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).

Denition 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 stratied as follows: Let = (n1 ; : : : ; nr ) be a partition of n.

We also write = (1 ; 2 ; : : : nn ), 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 stratication of X (n) and similarly

[n]

the X form a stratication 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 stratication 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 ; : : : ; nn );

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 2jj ;

(n) () ()

P

where jj = i i . Putting this together, we get that

M

R (Q X [2n]) = R( ) Q X 2jj :

(14) [n] ()

P

Here the sum runs through all () = (1 ; : : : ; n ) with i ii = 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+2jj (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 ) )t1 +22 +:::nn

n=0 n=0 1 +22 +:::nn =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.

Denition 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 dene 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 innite, 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

ñòð. 3 |