Chern Classes of Tautological Sheaves on Hilbert Schemes of Points on Surfaces

Manfred Lehn

Abstract We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface

within the framework of Nakajima’s oscillator algebra. This leads to an iden-

A tiﬁcation of the cohomology ring of Hilb with a ring of explicitly given

differential operators on a Fock space. We end with the computation of the top n

Segre classes of tautological bundles associated to line bundles on Hilb up to n , extending computations of Severi, LeBarz, Tikhomirov and Troshina and give a conjecture for the generating series.

Introduction

n X Hilbert schemes X of -tuples of points on a complex projective manifold are

natural compactifications of the configuration spaces of unordered distinct n-tuples X of points on X . Their geometry is determined by the geometry of itself and the geometry of the ‘punctual’ Hilbert schemes of all zero-dimensional subschemes in affine space that are supported at the origin. Thus one is naturally led to the following problem:

Determine explicitly the geometric or topological invariants of the Hilbert schemes

X such as the Betti numbers, the Hodge numbers, the Chern numbers, the cohomol-

ogy ring, from the corresponding data of the manifold X itself.

This problem is most attractive when X is a surface, since then the Hilbert schemes are themselves irreducible projective manifolds, by a result of Fogarty [12], whereas for higher dimensional varieties the Hilbert schemes are in general neither irreducible

nor smooth nor pure of expected dimension.

The answer to the problem above for the Betti numbers was given for P and rational ruled surfaces by Ellingsrud and Strømme [6] and for general surfaces by Göttsche in [14]. The answer turns out to be particularly beautiful (cf. Theorem 2.1 below). The problem for the Hodge numbers was solved by Sörgel and Göttsche [15]. For a different approach to both results see [3]. A partial answer for the Chern classes will be given in a forthcoming paper by Ellingsrud, Göttsche and the author [5].

The question for the ring structure of the cohomology is more difﬁcult. In general,

X X

X is the quotient of the blow-up of along the diagonal by the canonical

X n involution that exchanges the factors. Thus the case of interest is H , .

X X The ring structure for H , smooth projective of arbitrary dimension, was

found by Fantechi and G¨ottsche [11]. In another direction, Ellingsrud and Strømme

P Z n [7] gave generators for H , arbitrary, and an implicit description of the relations. Vafa and Witten [31] remarked that G¨ottsche’s Formula for the Betti numbers is identical with the Poincar´e series of a Fock space modelled on the cohomology of

X . Nakajima [24] succeeded in giving a geometric construction of such a Fock space structure on the cohomology of the Hilbert schemes, leading to a natural ‘explanation’ of G¨ottsche’s result. Similar results have been announced by Grojnowski [16].

Following the presentation of Grojnowski, this can be made more precise as fol-

n n

lows: sending a pair of subschemes of length and , respectively, and of

disjoint support to their union deﬁnes a rational map

n n n n

m X X X

This map induces linear maps on the rational cohomology

n n n n

m H X Q H X Q H X Q

n n

and

n n n n n n

m H X Q H X Q H X Q

H H X Q

If we let n , then these maps deﬁne a multiplication and a comul-

tiplication

m H H H m H H H

which make H a commutative and cocommutative bigraded Hopf algebra. The result

of Nakajima and Grojnowski says that this Hopf algebra is isomorphic to the graded

X Q tQ t symmetric algebra of the vector space H .

More explicitly, Nakajima constructed linear maps1

q H X Q End H n Z

n Q

and proved that they satisfy the ‘oscillator’ or ‘Heisenberg’ relations

q q n id

n m nm H X

Here the commutator is to be taken in a graded sense.

The multiplication and the comultiplication of H are not obviously related to the

quite different ring structure of H , which is given by the usual cup product on each

X Q H

direct summand H . (Strictly speaking, contains a countable number of

H X Q

idempotents but not a unit unless we pass to some completion). X This paper attempts to relate the Hopf algebra structure and the cup product structure. More precisely:

1Our presentation differs in notations and conventions slightly from Nakajima’s.

r X X

Let F be locally free sheaf of rank on . Attaching to a point , i.e.

X C F O

a zero-dimensional subscheme , the -vector space deﬁnes a locally

n n n

rn X X

free sheaf F of rank on . The Chern classes of all sheaves on of this H

type generate a subalgebra A . We will describe a purely algebraic algorithm to

H Q H

determine the action of A on in terms of the -basis of provided by Nakajima’s

n F results. We collect the Chern classes of all sheaves F for a given sheaf into

operators

ch F H H cF H H and geometrically compute the commutators of these operators with the oscillator op-

erators deﬁned by Nakajima.

d c O X

A central rˆole is played by the operator , which — up to a factor

— can also be interpreted as the intersection with the ‘boundaries’ of the

n n

Hilbert schemes, i.e. the divisors X of all tuples which have a multiple

EndH f d f

point somewhere. The derivative of any operator f is deﬁned by .

Our main technical result then says that for n

n n

q q q q K

n n

(1)

H X Q H X Q H X Q

where is the map induced by the diagonal X embedding and K is the canonical class of . An immediate algebraic consequence

of this relation is

q nm q q

m (2)

n m q d

for . By induction one concludes that the operators and sufﬁce to

q n

generate all n , .

The commutator of the Chern character operator ch with the standard operator

q q

can be expressed in terms of higher derivatives of :

ch F q q chF

(3)

Equations (1), (2) and (3) together give a complete description of the action of A on

H . Here are some applications:

1. We prove the following formula conjectured by G¨ottsche: If L is a line bundle

on then

X X

m n n

q cLz cL z exp

n m

2. We give a general algebraic solution to Donaldson’s question for the integral

N L L n

n of the top Segre class of the bundles associated to a line bundle for any

N n

and explicitly compute n for . From an analysis of this computational material N

we derive a conjecture for the generating function for all n .

3. We identify the Chow ring of the Hilbert scheme of the afﬁne plane with an

Q q q algebra of explicitly given differential operators on the polynomial ring of countably many variables. This paper is organised as follows: In Section 1 we recall the basic geometric

notions used in the later parts. Section 2 provides an introduction to Nakajima’s results. L

Section 3 contains the core of this paper: we first define Virasoro operators n in analogy to the standard construction in conformal field theory and show how these

arise geometrically. We then introduce the operator d and compute the derivative of q

n . Finally, in Section 4 we apply these results to compute the action of the Chern classes of tautological bundles. Discussions with A. King were important to me in clarifying and understanding the picture that Nakajima draws in his very inspiring article. I am very grateful to G. Ellingsrud for all the things I learned from his talks and conversations with him about Hilbert schemes. To some extent the results in this article are a reflection on an induction method entirely due to him. I thank W. Nahm for pointing out a missing factor in Theorem 3.3 and D. Zagier for a very instructive correspondence on power series. Most of the research for this paper was carried out during my stay at the SFB 343 of the University of Bielefeld. I owe special thanks to S. Bauer for his continuous encouragement, interest and support. This article is a slightly modified version of my Habilitationsschrift [22] presented to the Fakultät für Mathematik der Georg-August-Universität Göttingen in October 1997.

1 Preliminaries

In this section we introduce the basic notations that will be used throughout the paper and collect some results from the literature without proof. All varieties and schemes

are of ﬁnite type over the complex numbers. X will always denote a smooth irreducible

f S S f

projective surface. If is a morphism of schemes, I will write X

f id S X S X

X .

1.1 Hilbert schemes of points

n X For any smooth projective surface X let the Hilbert scheme of zero-dimensional

closed subschemes of length n. It has a natural structure as a projective scheme by a n

result of Grothendieck [17], and is in fact a smooth irreducible variety of dimension

n n

X S X

by a theorem due to Fogarty [12]. There is a natural morphism

X S

n , the Hilbert-Chow morphism, to the symmetric product, which maps a point

X C O x

to the cycle x (cf. Iversen [20]). For a smooth curve , the

n n

C S C

map is an isomorphism.

p X X np X

Fix a point and let p red denote the closed subset

X Supp fpg of all subschemes with (with the reduced induced subscheme

n X

structure). By a theorem of Brianc¸on [1], p is an irreducible variety of dimension

n n for . A new proof with a more geometric and conceptual argument was

recently given by Ellingsrud and Strømme [9].

X x X

Recall that a zero-dimensional subscheme is called curvilinear at ,

C X O x

if is contained in some smooth curve . Equivalently, is curvilinear if x

C C z z O x is isomorphic to the -algebra , where is the length of x.In any ﬂat family of zero-dimensional subschemes the points in the base space which

correspond to curvilinear subschemes form an open subset, and Brianc¸on’s theorem is

n X

equivalent to saying that the curvilinear subschemes are dense in p .

X S X

Generalising the deﬁnition of p slightly, let denote the small diago-

nal, and let X , endowed with the reduced induced subscheme structure.

X n

Thus X consists of all subschemes of length which are supported at some

X X

point in X . The ﬁbres of the surjective morphism are the schemes

n n

X X

p considered above. It follows immediately from Brianc¸on’s theorem that is

irreducible and of dimension n .

1.2 Incidence schemes

Hilb X Since X in fact represents the functor of ﬂat families of subschemes of

relative dimension 0 and length n, there is a universal family of subschemes

X X

For small values of there are explicit descriptions: is empty, is the diagonal

X X Bl X X X X

in , and is the blow-up of the diagonal in . The

Bl X X X Bl X X S

identiﬁcation is given by the quotient map

Bl X X X

and any of the two projections .

Assume that n . Then there is a uniquely determined closed subscheme

n n n n

X X

X with the property that any morphism

n n

f f f T X X

n n n n

f X f X n

factors through if and only if n . Closed points in

X X

correspond to pairs of subschemes with . Let

p p

n n n n

X X X

n n

denote the two projections. Then X parametrises two ﬂat families

p p

X X

Consider the corresponding exact sequence

I p p O O

n n

(4)

X X n n

n n n n

I X X X n

The ideal sheaf n is a coherent sheaf on which is ﬂat over

n and ﬁbrewise zero-dimensional of length n . It therefore induces a classifying morphism to the symmetric product, analogously to the Hilbert-Chow morphism, which

we will also denote by

n n n n

X S X

n n

X S X

As before let , where is the small diagonal. A point

n n

X x SuppI fxg

in is a triple with and .

n n

X Z

We may decompose into locally closed subsets , , with

Z f xj g

Z Z n n n n

Lemma 1.1 — and are irreducible of dimension and , respec-

dimZ n n Z

tively, and for all . Moreover, is contained in the closure Z

of .

Proof. This follows from Brianc¸on’s theorem by an easy dimension count.

Deﬁnition 1.2 — For any pair of nonnegative integers deﬁne subvarieties

n n n n n n

E Q X X X

n n n n

n n Q E Z Z

as follows: if let and be the closure of and , respec-

n n nn nn

X E Q E

tively. Moreover, Q , and , whereas

n n nn

f x jx g n n Q T Q

n . On the other hand, if , let and

n n nn

T E

E under the twist

n n n n

T X X X X X X

nn nn E

By construction Q and are empty or irreducible varieties of dimension

n n n

n and , respectively.

Let us return to the particular case n : consider the projectivisation

nn n

X PI X X PI

. There is a natural isomorphism such that

n n

the diagram

PI X

X X

commutes. It has independently been proved by Cheah [4], Ellingsrud (unpublished),

nn and Tikhomirov [30], that X is a smooth irreducible variety.

An immediate corollary is the following: there is a natural closed immersion

Bl X X PI

; since both are irreducible varieties, this must be an iso-

n n

nn E morphism. The exceptional divisor E is precisely the variety deﬁned above.

Hence in this situation we may write the sequence (4) as

id O E p O p O

(5)

X n X X

2 The structure of the cohomology

The motivating problem in this study is to understand the cohomology rings H

in terms of the cohomology ring H . For the symmetric product Grothendieck

n n S

H S X Q H X Q

[18] showed that , whence Macdonald deduced a

n X formula for the Betti numbers of S [23]. For the Hilbert schemes the corresponding

question is much more difﬁcult. This problem was solved by G¨ottsche [14]:

b X

Theorem 2.1 (Gottsche)¬ — The Betti numbers i are determined by the Betti

b X

numbers j . More precisely, the following formula holds:

Y Y X X

j mj m b X n i n

t q b X t q

j n i

G¨ottsche’s original proof uses the Weil Conjectures [14]. For a different approach see [3]. It is clear from this formula, that the Cohomology of the Hilbert schemes can be understood only if all Hilbert schemes are considered simultaneously. This

becomes even more apparent in Nakajima’s approach. We collect a few deﬁnitions:

Deﬁnition 2.2 — Let H denote the double graded vector space with

ni i n

H X Q X H Q

components H . Since is a point, . The unit in

X Q

H is called the ‘vacuum vector’ and denoted by .

ni ni

H H fH H

A linear map f is homogeneous of bidegree if

i f f EndH

for all n and .If are homogeneous linear maps of bidegree and

, respectively, their commutator is deﬁned by

f f f f f f

j jfj We use the notation j , etc. to denote the cohomological degree of homogeneous cohomology classes, homogeneous linear maps etc.

Setting

H X Q

for any deﬁnes a non-degenerate (anti)symmetric bilinear form

X Q H f H H

on H and hence on . For any homogeneous linear map its y

adjoint f is characterised by the relation

jfjjj y

f f

y y y

f g g f Clearly, . We will be mainly concerned with linear maps which are

deﬁned via correspondences.

Y Y u

Let and be smooth projective varieties, and let be a class in the Chow

A Y Y

group n . (We tacitly assume rational coefﬁcients. This will not always be necessary. On the other hand, we are not interested in integrality questions for

the moment, and hence will not pay attention to this problem). The image of u in

H Y Y u

n will be denoted by the same symbol. induces a homogeneous linear

map

i idim Y n

u H Y H Y y PD p u p y

PD H Y H Y

where is the Poincar´e duality map.

Y v A Y Y

Assume that is another smooth projective variety, and . Let

p Y Y Y Y Y

i j

ij be the projection from to the factors , and consider the element

w p p u p v A Y Y

nmdim Y

Then

w u v

See [13, Ch. 16] for details.

U Y Y V Y Y u

Suppose and are closed subschemes such that

v A V A U

and . Let

W p p U p V

w A W

Then the class deﬁned above is already deﬁned in .

T Y Y Y Y u

Let exchange the factors. Then a Chow cycle induces

two maps

u H Y H Y Tu H Y H Y

and

which are related by the formula

Z Z

Tu u

Y Y

Tu u This follows directly from the projection formula. Thus . The following operators were introduced by Nakajima [24]. The study of their properties is the major theme of this article. We take the liberty to change the notations

and sign conventions.

n n

n n Q

The fundamental classes of the -dimensional subvarieties

n n

X X

X (see 1.2) are cycles

n n n n

Q A X X X

n n

p p Let the projections to the factors be denoted by , and .

Deﬁnition 2.3 (Nakajima) — Deﬁne linear maps

q H X Q EndH Z

H X Q y H X Q

as follows: assume ﬁrst that .For and let

nn nn

q y Q y PD p Q p y

The operators for negative indices then are determined by the relation

q q

q jj

By deﬁnition, is a homogeneous linear map of bidegree .

q q

Moreover, , and if , the operator is induced by the subvarieties

n Q , . The following theorem is the main result of [24]. Similar results have been an-

nounced by Grojnowski [16].

Theorem 2.4 (Nakajima) — For any integers n and and cohomology classes

q q m

and , the operators n and satisfy the following ‘oscillator relations’:

q q n id

n m nm H

Here and in the following we adopt the convention that equals if and is

deg dim Z

zero else, and that any integral is zero if R . Z In [24] Nakajima only showed that the commutator relation holds with some uni-

versal nonzero constant instead of the coefﬁcient n. The correct value was computed

directly by Ellingsrud and Strømme [9]: up to a sign factor, which depends on our con-

n n

deg X X

vention, this number is the intersection number p for the subvarieties

n n

X X X

p . There is a different proof due to Grojnowski [16] and Nakajima [25] using ‘vertex operators’.

Consider the vector spaces

W H X Q tQ t W H X Q t Q t

and

W W W Deﬁne a non-degenerate skew-symmetric pairing on the vector space

n m

f t t g n

nm X

Note that we are taking the expression ‘skew-symmetric’ in a graded sense:

n m jjj j m n

f t t g f t t g W

The oscillator algebra is the quotient of the tensor algebra T by the two-sided ideal

v w fv wg v w W

I generated by the expressions with :

H T WI

H is the (restricted) tensor product of countably many copies of Clifford algebras

odd

X Q

arising from H and countably many copies of Weyl algebras arising from

ev en

H X Q W f g

.As is isotropic with respect to the skew-form , the subalgebra

H W S W Z

in generated by is the symmetric algebra (taken again in a -graded

sense). This becomes a double graded vector space if we deﬁne the bidegree of

n n jj t

as . With these notations, Nakajima’s Theorem says: sending

q EndH H H W

to n deﬁnes a representation of on .

9 W

The subspace of monomials of negative degree annihilates the vacuum vector

for obvious degree reasons. Hence there is an embedding

S W H HW H H

S W

It is not difﬁcult to check that the Poincar´e series of equals the right hand side

of G¨ottsche’s formula. This implies: H

Corollary 2.5 (Nakajima) — The action of H on induces a module isomorphism

S W H H

. In particular, is irreducible and generated by the vacuum vector.

3 The boundary operator

The key to our solution of the Chern class problem is the introduction of the boundary

EndH operator d . This is done in 3.2. We begin with the discussion of related topics and ingredients for later proofs.

3.1 Virasoro generators q

Starting from the basic generators n and the fundamental oscillator relations we will L

deﬁne the corresponding Virasoro generators n in analogy to the procedure in conformal ﬁeld theory. We will then give concrete geometric interpretations for these

generators.

H X H X X H X H X Let be the push-forward

map associated to the diagonal embedding. Equivalently, this is the linear map ad-

q q m

joint to the cup-product map. If , we will write n for

i i

q q

m n

i i

L H X Q EndH n Z

Deﬁnition 3.1 — Deﬁne operators n , , as follows:

L q q n

n if

and

L q q

Remark 3.2 — i) The sums that appear in the deﬁnition are formally inﬁnite. How-

ever, as operators on any ﬁxed vector in H , only ﬁnitely many of them are nonzero.

L L n

Hence the sums are locally ﬁnite and the operators n are well-deﬁned. is

n n jj homogeneous of bidegree

ii) Using the physicists’ normal order convention

q q n m m

n if

q q

n m

n m q q n m if

10 L

the operators n can be uniformly expressed as

L q q

n n

L q m Theorem 3.3 — The operators n and satisfy the following commutation rela-

tions:

L q m q

m nm

1. n

n n

L L n m L c X id

n m nm nm H

L n Z

Taking only the operators n , , we see that the Virasoro algebra acts on X

H with central charge equal to the Euler number of .

Proof. Assume ﬁrst that n . For any classes and with

i i i

we have

q q q q q q

m n m n

i i i i

j j jj

q q q

m n

i i

m q

nm nm

i i

j jjj

q m

nm m

i i

X i

If we sum up over all and , we get

L q q q q m q

n m n m nm

with

j jjj

pr pr pr pr

Similarly, for ,

q q q m q

m m m m

Thus summing up over all we ﬁnd again

L q m q

m m

This proves the ﬁrst part of the theorem.

As for the second part, assume ﬁrst that n . In order to avoid case considera-

q N

tions let us agree that N is zero if is odd. Then we may write:

L q q q

m m

By the ﬁrst part of the theorem we have

L q q q q mq q

n m n m nm

In the following calculation we suppress and up to the very end. Summing up

over all , we get:

m m m m

q q q q L L

n m

n n

X X

mq q q q

nm n m

m m

m m m m

q q q q

n n

X X

mq q n q q

nm nm

m m

Hence

m m m m

q q q L L n m q q q

n m nm

n n

mq q

n q q

mn m

Now split off the summands corresponding to the indices and

m from the sums. Substituting n for in the second sum on the right hand side,

we are left with the expression:

L L n mL mq q q q

n m nm nm

The right hand side is zero unless n . In this case, observe that the composition

H X H X H X H X

c X

is multiplication with . Hence we see that

L L n mL c X N

n m nm nm X

where N is the number

n n

N if is odd,

and

n n

N if is even.

n n

An easy computation shows that in both cases N equals .

n n n n

X X X Recall the deﬁnition of the varieties E in (1.2).

Deﬁnition 3.4 — Let be a nonnegative integer and let

e H X EndH

be the linear map

nn nn

e y E y PD p E p y

H X Q y H X Q for and .

The following theorem gives a ‘finite’ geometric interpretation of the infinite sums which define the Virasoro operators.

Theorem 3.5 — Let n be a nonnegative integer.

m mn m q

n if or

e q

n m else

e L q q

n n n

Proof. Ad 1: Assume ﬁrst that m . To simplify the notations we introduce the

short-hand

n n n n n n

k k

X X X X

Suppose , and consider the following diagram

p p

m nmm nmm

X X X

nm X

e q m

The product operator n is induced by the class

mnm m

z p p E p Q A Z

where

mnm m

Z p p E p Q

Z f xyj nx my x g

Here the notation should comprise the conditions: is a subscheme of ,

m y

and the ideal sheaf of in is of length and is supported at etc.

q e v A V

n m

Similarly, the operator is induced by a class with

V V f xyj mx ny y g

mn mn

X X

Moreover, if T exchanges the two copies

X e q z T v m

of in the middle, then the commutator n is induced by .

x y g X Z

Now observe that off the diagonal f the subsets

and T are equal. Moreover, there is only one component of (maximal possible)

n m

dimension . It is easy to see that this component has multiplicity both

T v

in z and : the intersection

mnm m

p E p Q

is transversal over a general point in this component of Z , and maps injectively into

z T v

Z . Thus the only contributions to may arise from the diagonal part. Now

V fx y g f xxj n mx x g

n m We have seen earlier (1.1) that this set has dimension and hence may be

disregarded. On the other hand

Z fx y g f xxj n mxg

Again using 1.1 we see that this set has only one component D of (maximal) dimension

n m

. Moreover, this component is the image of the embedding

nm nm

Q X x xx

H X Q y H X Q

Let and . Then we have

p D p p y

p Q p p y

p y p Q p

This shows that

e q q

n m nm

D z

for some integer . Hence it remains to compute the multiplicity of in . To this

D p E

end we pick a general point d and inspect the intersection of

Q p d

and p along the ﬁbre .

A general point in D is of the form

d with

X n m

where is a curvilinear subscheme of of length , supported in a single point

x which is disjoint from . Since is curvilinear, there is a unique subscheme

p d

of length m, and hence consists of the single point

d x x

mnm mnm

X X X

Near d the varieties and are lo-

mnm mnm

X E

cally isomorphic in theetale ´ topology; and similarly E to

X X X

and Q to . Thus we may split off the factors from the geomet-

ric picture. In the end this amounts to saying that we may assume without loss of generality that .

Moreover, the calculation is local (in theetale ´ topology) in X , so that we may

nm m

X A Sp ecC z w I w z I w z I

assume that and , and

nmm m n

w z d X A . Then has an afﬁne neighbourhood in with

coordinate functions

a a b b w z c c d d w z

nm nm m m

x y X

which parametrises quadruples of subschemes in given by the ideals

w g z f z w w z z w g z f z w w z z

where

nm nm

X X

i i nm

b z a z z g z f z

i i

and

m m

X X

i m i

f z c z z g z d z

i i

y X Supp fy g

Now belongs to , i.e. , if and only if

f z z z w g z

and (6)

nmm

x Q

And belongs to if and only if the following three conditions are

satisﬁed: , i.e.

g z g z f z hz f z f z k z

and (7)

h k n n I

with polynomials and of degree and , respectively; the ideal sheaf is

supported at x, i.e.

k z z z w g z

and (8)

and ﬁnally, x must be contained in , which imposes the condition

f z (9)

One easily checks that the equations (6) - (8) cut out a smooth subvariety which

Sp ec C z z b b

projects isomorphically to the afﬁne space . Moreover,

z z

in these coordinates the last condition (9) simply reads . Hence the m

multiplicity equals the exponent .

e q m n

Next, we consider the case n with . There is nothing to prove

if m . Hence assume that . Dimension arguments similar to the ones above

v q e

m n show that the cycle which induces the commutator must be supported on

the closed subsets

V f x x j x n mxgX

n m dimV

The cycle v has degree , so that it sufﬁces to show that

n m

. This follows from Lemma 1.1.

e q m n m It remains to consider the case n with . A dimension check of the

set-theoretic support of the intersection cycle shows that we must have

e q q

n m nm

for some integer , independently of and . To determine , we proceed alge-

braically and take the commutator with m :

e q q q q n m id

n m mn nm mn H X On the other hand, combining the Jacobi identity, the oscillator relations and the ﬁrst

part of the proof yields

e q q e q q

n m mn n mn m

m nq q

m m

id mm n

It follows that .

y e L q q

n n n

Ad 2: Consider the difference . Com-

paring the expressions in 3.3 and part 1 of the theorem we see that y commutes with

q m Z H N y

all operators m , . Since is a simple -module, must be a scalar (in

n y

some algebraic extension of Q ), which is impossible: if , then has non-trivial

n n jj n y bidegree , and if , it is easy to see directly that .

Remark 3.6 — In particular, the operator has the following geometric inter-

X X

pretation: the universal family n induces a homomorphism

H X Q H X Q

and

L y y y H X Q

for all

L y n y y H X Q

If we insert X , we get for all . Thus

L X

is the ‘number’ operator, that counts with how many points we are dealing. L

This can, of course, also be deduced directly from the deﬁnition of .

3.2 The boundary of the Hilbert scheme

x n

i i s

For any partition of the tuples ,

n n n

x X S X S X X S X

i , form a locally closed subset in . Let . It follows

from Brianc¸on’s Theorem that X is irreducible and

dimX n s

The generic open stratum is X . It corresponds to the conﬁguration space of

unordered n-tuples of pairwise distinct points. Furthermore, there is precisely one

stratum of codimension 1, namely X

n X

If and s are partitions of , then is con-

tained in the closure of X if and only if there is a surjection

f sgf s g

such that j for all . It follows that

i j

n n

X X X

is an irreducible divisor in X . As it is the complement of the conﬁguration space in

n n X

X we might and will call it the boundary of .

We will need a different description of the divisor X in sheaf theoretic terms.

p X

Let n be the projection, and deﬁne sheaves

O p O CohX

n O n

As p is ﬂat and ﬁnite of degree , is locally free of rank .

X i

Lemma 3.7 —Wehave h

X c O

Moreover, let E be the exceptional divisor. Then

n n

p X p X E

n n

X X

Proof. is the branching divisor of the ﬁnite ﬂat morphism n . The

Y Y

assertion holds true in a more general setting: if Y is a smooth variety and

A O O

is a ﬁnite ﬂat map, so that Y is a locally free –sheaf, the branching O

divisor is given by the discriminant of the Y -bilinear form

A A AO

O Y

or, equivalently, by the determinant of the adjoint linear map AA , so that indeed

c A

the branching divisor is given by . p

Applying to the short exact sequence (5) we get an exact sequence

n n

O E p O p O

X X

from which one deduces the second assertion.

This proof was communicated to me by S. A. Strømme and replaces a slightly

longer one in an earlier version.

H H Deﬁnition 3.8 — Let d be the homogeneous linear map of bidegree

given by

h i

n n

x dxc O X x x H X

for all

EndH f d f

For any endomorphism f its derivative is . As usual, we write

n n

ad d f

f for the higher derivatives.

It follows directly from the Jacobi identity that f is a derivation, i.e. for any

b EndH

two operators a the ‘Leibniz rule’ holds:

ab a b ab a b a ba b

and

H X H X jf j

Moreover, if f is a homogeneous linear map, then

fj f f

j , so that and have the same parity. Furthermore,

y y

f f

Indeed, this follows formally from the obvious fact that d .

n n

n X

Let n be nonnegative integers, and consider the incidence variety

n n

I X X n

. Recall the deﬁnition of the ideal sheaf n and the exact sequence

p O I p O

n n

X n X

n n

p I n n X

Then n is a locally free sheaf of rank on .

n n

u H X Q H X Q

Lemma 3.9 — Let be the induced linear map

n n

u A X

associated to a class . Then

u c p I u

n n

H X Q

Proof. Let y . Then

u y du y u dy

c p O PD p u p y

PD p u p c p O y

u p y PD p p c p O p c p O

v y

v p c p O p c p O u

with , and

p c p O p c p O c p p O c p p O

n n

X X

n x

c p I

n n

3.3 The derivative of n

In order to understand the intersection behaviour of the boundary X we need to

d q

know how the operator commutes with the basic operators n , in other words: we q

need to compute the derivative of n . q

The following theorem describes the derivative of the operator n in two ways: By its action on any of the other basic operators, and as a polynomial expression in the

basic operators. This is the main technical result of the paper. X

Let K denote the canonical class of the surface .

Z H X Q

Main Theorem 3.10 — For all n m and the following holds:

n o

jnj

q nm q q K id

m nm nm

1. H

njnj

q K q n L n

2. n

d q H X H

Corollary 3.11 — The operators and , , sufﬁce to generate from the vacuum .

Proof of the theorem. The second assertion is an immediate consequence of the

ﬁrst: by Nakajima’s relations 2.4 and the relations 3.3 we see that

njnj

n L q K q

n n m

n jnj

K id nm q

H nm nm

Hence the difference of q and the expression on the right hand side in the theorem

q m Z H N

commutes with all operators m , . Since is an irreducible -module, it

19 follows from Schur’s Lemma that this difference is given by multiplication with a

scalar (say, after passage to some algebraic closure of Q ). But this is impossible for

n n jj n

degree reasons: the bidegree of q is . (The case being trivial n anyhow.)

The proof of the ﬁrst assertion has two parts of quite different nature: We need to

m n m

distinguish the cases n and and deal with them separately.

q q nm q n m

Proposition 3.12 — m for any two integers

m H X

with n and cohomology classes . m Proof. Step 1: Assume that n and are positive. We proceed as in the proof of

Theorem 3.5. Let be nonnegative, and consider the diagram

p p

nmm nmm m

X X X

Let

m mnm

Q A V Q p v p

mnm m

V p Q p Q

q q

According to Lemma 3.9, the operator m is induced by the class

w p p c I v A W W p V

mnm mn

V W W x y

Let V and denote the open subsets of those tuples and

x y x y x y

, respectively, where either or but x is curvilinear. Certainly,

W V p p V W

, but in fact we even have that is an isomorphism:

for the conditions imposed on V imply that is already determined by the remaining

x y

data .

n m

Claim: V is irreducible of dimension .

nfx y g

For it follows from Brianc¸on’s Theorem that the open part V is ir-

n m

reducible of dimension , and tuples of the second kind, i.e.

x x

with x curvilinear, are easily seen to deform into this open subset.

W n W m n W

Claim: dim . In particular, the complement of in w

W cannot support any contribution to .

f x x j n mxg T

Indeed, the set T has a stratiﬁcation

T T length

i x

i , where the stratum is the locally closed set of all tuples with

i T T x

. Let be the closed subset that consists of tuples where is not curvilinear.

W n W T T T T n m

Then .Now is irreducible of dimension ,

and T is a proper closed subset and therefore has strictly smaller dimension. The

assertion now follows from Lemma 1.1.

mn mm

Q p Q

Claim: The intersection of p and is transversal at

general points of V .

In fact, the intersection is transversal at all points with x and curvilinear.

V r r

We conclude, that the intersection cycle v equals , where is a cycle

W n W

supported on p and therefore irrelevant for our further computations for

dimension reasons. Let us return to the deﬁnition of the cycle w .

W W

Identifying V and we see that the variety parametrises three families

Z W X

of subschemes in X . In terms of these we can summarise the discussion above by

q q

stating that m is induced by the cycle

c p I W A W

Having reached this point we pause to reﬂect what changes in this picture if we

q q T m

exchange the order of the operators n and . Up to the usual twist that ﬂips the

X W W factors X in , not a iota is changed in . Indeed, parametrises

not only three but rather four families of subschemes

s xyZ s

where and are characterised by the property that at a point s

the subschemes s are the unique ones with

s s

Z mx ny

s s

and

mx Z ny

s s

q q

This means: the commutator m is induced by the cycle

c p I c p I W A X

The ideal sheaves corresponding to the various inclusions between the families

Z , , and are related by the following commutative diagram of short exact

sequences

I I I

Z Z

I I I

Z Z

The homomorphism

p p I p I

x y g W

is an isomorphism off the diagonal f . On the other hand the clo-

fx y g Q

sure of W equals the image of the ‘diagonal’ embedding

X . It follows that

c p I c p I W Q

p Q

where is the length of coker at the generic point of the variety . This

proves

q q q

m nm n

and it remains to show that

x y Q x x

A general point d of is of the form where

x and is a curvilinear subscheme supported at . As the computation

is local in X we may apply the same reduction process as in the proof of Theorem

X A Sp ecC z w x

3.5: we may assume that , that , and

I w z d W

. Then there is an open neighbourhood of this point in which

A Sp ecC a a st

isomorphic to such that the families

and are given by the ideals

n m m

I w f z z t z s I w f z z s

and

I w f z z t

f z a a z a z

where . We ﬁnd

p O stz z t C a

and

m m n

p I z s C a stz z s z t

The cokernel of

m m n n

p z s C a stz z s z t C a stz z t

a st

is isomorphic to the C -module

m n m n

C a stz z s z t C a s tz s z tz s z t

s tg This module is supported along the diagonal f (as we expected), and its stalk at

the generic point of the diagonal has length nm (as we had to prove).

m n

Step 2: Assume that m is positive and . First one shows as above

q q A X

that the commutator is induced by cycles in

Q for each , which are supported on the diagonally embedded varieties ,

so that

q q c q

m nm nm n

22 c

for certain constants nm . In order to determine these constants we apply the commu-

tator . Then the oscillator relations yield for the right hand side

c n m id

nm H X

On the other hand

q q q q q q

m nm nm m

n n

Now

y y

q q q q

nm n n

m y

q q

m y

nn mq nn mq

which by Step 1 equals m . Hence

q q q nn mq q

m nm m m

nn mm id

c nm

Choose classes with . It follows that nm . X

Step 3: The general case can now be reduced formally to the cases already treated.

The assertion is certainly trivial if either n or . If the assertion is known to

n m y

be true for some pair , we may apply the operation to both sides and ﬁnd:

nm y y

q q q q

n n m

y y nm

q q

m n

nm y nm

q q nm q

nm q

This and the identity

jjj j

q q q q

m n

n m

allow us to reduce anything to cases checked in Step 1 and Step 2.

m In order to prove part 1 of Theorem 3.10, it remains to treat the case n

. This will be done in two steps. First, we prove a qualitative statement about the

structure of the ‘correction term’, and afterwards we determine the precise value of the K

‘coefﬁcient’ n :

K PicX Q n Z

Proposition 3.13 — There exist rational divisors n , , with

K K K

n n

and and such that

q q n K id

n H

n (10)

H X for all .

Proof. There is nothing to prove for n . Moreover,

jjj j

q q q q

n n

K n

It follows that if there is a divisor n so that (10) holds for , then (10) also holds for

n K K

n n with the choice . Hence it sufﬁces to prove the proposition for positive

integers n.

Let be a nonnegative integer and consider the diagram

p p

n n n

X X X

Let

n n

v p Q p Q A V

n n

V p Q p Q

q q

According to Lemma 3.9, the operator n is induced by the class

w p p c I v A W W p V

T W fx y g

Consider the diagonal part ﬁrst. It is contained in i , where

T f x x j ig T X

x x

i . The closure of is the diagonal

X i

X and is therefore irreducible of dimension . Whereas for ,

i i

T X X X X

the set i embeds into the irreducible variety of dimension

ii i

fx y g n n

The off-diagonal part W is empty if .If it has precisely

one irreducible component W of maximal dimension : it contains as a dense

subset the image of the embedding

n n

f X X g W X

j and are pairwise disjoint

x y n W

Since the function x is semicontinuous and is at least on ,

T W

it follows that is contained in n . In particular, this intersection has

dimension . As we want to compute a cycle of degree , we may restrict our

attention to the open part W and may disregard the complement of in its closure.

p p W W W

is an isomorphism, which we use to identify and the

W X

off-diagonal part of V .Now parametrises four ﬂat families of subschemes on :

Z Z

besides the families and of ﬁbrewise length , these are the families and

Z n n W w

of ﬁbrewise length and . The contribution of to is the class

c p I W A W

q q u

Reversing the order of the operators and n shows that the part of the cycle

q q W

inducing the commutator n , that is supported on , is the class

c p I c p I W

Z Z

I I

Z Z Since the ideal sheaves and are isomorphic, this class is zero.

Thus we may fully concentrate on the contribution of the diagonal part . (Also

T q q

note that for the reversed order n any diagonal parts must be contained in

and are therefore too small and irrelevant.)

T X X n

The complement of the open subset in has codimension

p X X T

. Locally near there are isomorphisms between and

n n

X Q X X w A X

, and similarly between and . Hence if is the

intersection cycle for the special case , then the general cycle is simply given by

w X w A X X

. But that was all we had to prove: a cycle of this

form induces the linear map

y y H X Q y H

Corollary 3.14 — For all positive integers n one has

q nL nq K

n n n n

Proof. Use the same argument as in the ﬁrst paragraph of the proof of the main

theorem after Corollary 3.11.

To ﬁnish the proof of Theorem 3.10 it remains to show:

Proposition 3.15 — For all positive integers n the rational divisor deﬁned by Propo-

sition 3.13 is given by

K K

X where K is the canonical class of the surface . This will be done in the next section.

3.4 The vertex operator, completion of the proof

H X

Deﬁnition 3.16 — Let be an element which is of even degree though not

t S

necessarily homogeneous, and let be a formal parameter. Deﬁne operators m ,

m ,by

X X

n m

S t q t S t exp

n m

q q n Since is of even degree by assumption, any two operators and n commute in the ordinary, i.e. ‘ungraded’ sense. In particular, there is no ambiguity in

the meaning of the expression on the right hand side in the deﬁnition. S

The geometric meaning of the operators m is explained by the following theorem:

n n

X S C C

let C be a smooth curve in . There is an induced closed embedding

n n n

X C H X C H X . Let and be the corresponding cohomology classes, i.e., the Poincar´e dual classes of the fundamental classes of these varieties.

Theorem 3.17 (Nakajima, Grojnowski) — The following relation holds for all non-

negative integers n:

C S C n

For proofs see [25] and [16].

H X

Lemma 3.18 — Let be an element of even degree. Then

n n

S tS t t L q K

n n n

a a Proof. Assume ﬁrst that a is an operator of even degree, and that commutes

with a. Then

X X X

i ni

a a a

n n

X X

n n

na a a n i a a

n n

X X

a a n

a a a

n n

a a expa a

Next, let be a family of commuting operators of even degree such that any

a a a

commutes with every . Then it follows from Step 1 and

a exp a expa a a

that

X X X X

exp a exp a a a a

a q t

Now apply this formula to the family and use our previous

a a t L q K a t q

results n and . One

tS t

gets S with

X X

n n

t q L q K t

n n n

n n

t L q K N

n n n n

N n

where n is the number of pairs of positive integers and that add up to , i.e.,

N n

n .

n n

X X C

Let C be a smooth projective curve. The boundary intersects

n n C

generically transversely in the boundary C of , i.e. in the set of all tuples with

n C

multiple points. The subvarieties X and have complementary dimensions

n X

n and in and we may compute the intersection number

I X C

n X We will do this ﬁrst using our algorithmic language, and afterwards using a geometric

argument. The comparison of the two results will lead to the identiﬁcation of the K

divisors n .

q X C S C X

Lemma 3.19 — n and .

Proof. The ﬁrst assertion follows from the deﬁnition of the operators n .By

n n

S C C X

Nakajima’s Theorem, n is the class of the submanifold , and

hence according to Lemma 3.7:

n n n

S X C C C d S C

Lemma 3.20 —

Z Z

q S C nK C C

n X n

X X

Proof. Indeed,

Z Z

q S q S C C

n X n X

n n

X X

q S C

n X

q q q s i q

n X n i j

since .Now commutes with any product i if ,

i n S

and j . Thus the only summand in that contributes to the commutator

q q C K C n

n n n

with is . Hence

q S C n C id C K

n X H n

27 This proves the lemma.

Next, we give the geometric computation of I :

Lemma 3.21 —

X C nn C C K

n n n

X C X X C X

Proof.Wehave . The intersection of

n n

C C

and C is transversal and is equal to the image of the closed immersion

C n c

sending a point c to the unique subscheme of of length that is supported in . Thus

n n

I degO X j degO C j

n n

C C

X C

C C

The embedding factors through the diagonal embedding and the quotient

n n n

C C pr C C

map . Moreover, if denotes the projection to the

ij j

product of the i-th and -th factor,

O O C pr

C C C

C ij

From this we conclude:

I deg O C deg O j

C C C

nn C C K

Proof of Proposition 3.15. From Lemma 3.19 and Lemma 3.20 we conclude

I C nK C

K K

Comparison with Lemma 3.21 shows that n .

This ﬁnishes the proof of Theorem 3.10.

4 Towards the ring structure of H

4.1 Tautological sheaves

There is a natural way to associate to a given vector bundle on X a series of tauto-

n logical’ vector bundles on the Hilbert schemes X , . The Chern classes of the

tautological bundles may be grouped together to form operators on H .

Consider the standard diagram

X X X

X n

Let F be a locally free sheaf on . For each the associated tautological bundle

on X is deﬁned as

F p O q F

n F

Since p is a ﬂat ﬁnite morphism of degree , is locally free with

rkF n rkF

F F

Note that F and .

F F F

Furthermore, if is a short exact sequence of locally free

n n

F F F

sheaves on X , the corresponding sequence is again

F F F exact. Hence sending the class of a locally free sheaf to gives a group

homomorphism

n n

K X K X

K X

Deﬁnition 4.1 — Let u be a class in . Deﬁne operators

u EndH chu EndH

c and

H X Q

as follows: For each n , the action on is given by multiplication with

n n

u chu the total Chern class c and the Chern character , respectively.

Let

X X

cu ch u c u ch u

k and

k k

k be the decompositions into homogeneous components of bidegree . Since all

of these operators are of even degree and only act ‘vertically’ on H by multiplication,

they commute with each other and in particular with the previously deﬁned boundary

d c O X operator .

Moreover, we have

u v cu cv ch u v chuch v

c and

K X

for all u v .

K X r H X

Theorem 4.2 — Let u be a class in of rank and let . Then

ch u q expad dq chu

or, more explicitly,

q ch u ch u q

Similarly,

r k

cu q cu q c u

k F

Proof. We may assume that u is the class of a locally free sheaf . Recall the

standard diagram for the incidence variety X :

X X X

X The variety X parametrises two families of subschemes of . Their structure

sheaves ﬁt into an exact sequence

E O p O O O

X X X

X X X E

where p is the projection and is the exceptional divisor.

p q F

Applying the functor to this exact sequence yields

F O E F F

(11)

c O E

Let . Then

chF chF chF exp

and

r k

cF cF c F

H X Q

It follows for any x :

ch F q x chF PD X x

PD X chF x

q chF x q ch F x

Here we used Lemma 3.9 which says that the cycle induces the operator

q . This is the equation for the Chern character. The equation for the total Chern

class is proved analogously.

K X Cu

Corollary 4.3 — For any u let be the operator

rku k

Cucu q cu q c u

Then

cu expCu

Note that the right hand side can be explicitly expressed in terms of the basic q

operators n by applying Theorem 3.10.

Proof.Wehave

X X

cu cu

n n

cu exp q

cu exp q cu

expcuq cu

expCu

Remark 4.4 — The sequence (11) was used by Ellingsrud in a recursive method to

determine Chern classes and Segre classes of tautological bundles (unpublished, but

see [28],[5]). He expresses the classes in terms of the Segre classes of the

X X

n universal family . Thus one needs to control the behaviour of these Segre classes under the induction procedure. This method yields qualitative results on the structure of certain classes and integrals, but all attempts to get numbers have

ended so far in unsurmountable combinatorial difﬁculties.

Remark 4.5 — The results of the present and the previous section provide an algo-

rithmic description of the multiplicative action of the subalgebra A which is

generated by the Chern classes of all tautological bundles: The elements i

H Q q s

i generate as a -vector space. By Corollary 3.11, each such element

can be written as a linear combination of expression w , where is a word in an

d q H X Q

alphabet consisting of and operators , . By Theorem 4.2 the

F commutator of ch with any of these is again a word in this alphabet. And ﬁnally,

Theorem 3.10 shows how such a word can be expressed in terms of the basic oper- q

ators n . Admittedly, without a further understanding of the algebraic structure this

X Q description is useful for computations in H only for small values of or if one implements it in some computer algebra system. The following sections deal with special situations where one can say more.

31 4.2 The line bundle case

The results of the previous section sufﬁce to compute the Chern classes of the tauto-

n L

logical bundles L associated to a line bundle in terms of the basic operators. X

Theorem 4.6 — Let L be a line bundle on . Then

X X

cL exp q cL

n m

Remark 4.7 — Expanding the term on the right hand side, one realises that the coho-

X Q n

mological degree of any summand contained in H is , and, moreover,

n q

the maximal degree can only be attained if the arguments of all operators in- volved have degree 2. In other words, considering elements of top degree only, the

equation of the theorem specialises to

X X

c L exp q c L

n (12)

n m

C X L O C

This is Nakajima’s result 3.17: for suppose is a smooth curve and X .

X O O C C X If , the natural homomorphism vanishes if and only if .

Hence the vanishing locus of the global vector bundle homomorphism

n n

O O C L

n n n

C C c L

is the subvariety . Therefore n . Inserting this into (12), we recover

Nakajima’s formula 3.17

X X

C exp q C

n m

Based on this observation, the theorem was conjectured by L. G¨ottsche in a letter to G. Ellingsrud and the author.

Proof of the theorem. We shall give two variants of the proof which differ slightly

in ﬂavour. We have seen that the left hand side in the theorem equals exp ,

where in this case because of r we have

CLq cL q

Variant 1. Expanding the right hand side of

q cL q expCL

q cL q X

yields summands which are words in the two symbols and .Mov-

ing all factors X within a given word as far to the right as possible using the

commutation relations of the main theorem we can write

q cL q A B q A

X X

where A is a sum of expressions of the form

q cL q cL

jj i

Let denote a partition and let i , and

. We get

X Y X

q cL

N q cL q

(13)

n jj i

i n N

where the natural number counts how often the operator

q cL

q q cL jj

arises from a word in X and of length . It is not difﬁcult to see that

N jj

equals the number of possibilities to partition a set of elements into subsets in

such a way that there are i subsets of cardinality . Hence

Inserting this into equation (13) above one gets

X Y X

q cL

q cL q

n i

i n

Y X

q cL

exp

q cL exp

In fact, being a little more careful, one gets

expCL exp q cL expq

i X

Variant 2. Starting again from the sequence

cL q CL cL

n n

q t n X

we multiply by from the right and sum up over all :

X X

n n n n

cL cL q t q t

X X

dt n n

n n

q t cL CL

This means that the series

n n

cL t cL expq t

satisﬁes the linear differential equation

CL X

X (14) dt

with initial condition

X (15)

On the other hand, consider the operator

S cLtexp q cLt

We ﬁnd

m m

S cLtS cLt q cLt

and

h i

fq c L q gScLt

X X

h i

S cLt q q cL t

X m

m m

S cLt q cLt

This shows

fq c L q gS cLt

X X

m m

q cLt S cLt

S cLt q cL

m m

S cLt q cLt

cLt

Hence S satisﬁes the system (14) and (15) as well and therefore equals

cL expq t X . This proves the theorem.

4.3 Top Segre classes

The following problem was posed by Donaldson in connection with the computation

jH j

of instanton invariants: let n be an integer , and consider a linear system of

n X P

dimension inducing a map . A zero-dimensional subscheme

X jH j does not impose independent conditions on the linear system if the

natural homomorphism

H P O H O H

fails to be surjective. The subscheme of all such has virtual dimension zero,

c W W n

and its class is given by , where is the virtual vector bundle

n n

H P O H O OH

P X

Thus the number of those that impose dependent conditions is given by

Z Z

cO H c O H N

n n

X X

N N

More explicitly, is the degree of the linear system, is the number of double

N P N

points, is the number of trisecants to a surface in and is the number of

quadruples of points on a surface in P that span a plane.

N X

Problem: Express n in terms of intrinsic invariants of such as the degree

HH HK KK

d , the intersection and and the topological Euler

e c X

characteristic . Note that even the fact that such an expression in terms of the given invariants exists is not evident a priori. This has been proved by Tikhomirov [28]. It also follows immediately from our approach.

Using our algorithm, we can attack this problem as follows. Theorem 4.2 yields

O H r

for F and the formula:

CO H q c H

X X

q H

q H H

cO H exp q t X It follows as in the proof of Theorem 4.6 that satisﬁes

the following differential equation and initial value condition:

CO H X X

X and dt As long as no explicit generating function is available we must be content with the

following semi-explicit solution to the problem:

N CO H

n n

X N

Example 4.8 — As a special case, let us compute . This is the number of secant

x P

lines to an embedded surface in P that pass through a ﬁxed but general point .

Hence we should ﬁnd Severi’s double point formula [26] (see also [2]). Let

H H

. Then

q N CO H CO H

with

n CO H

Since q for all and for all parameters, we have

. Moreover, for degree reasons the inﬁnite sum reduces to

CO H q q q q q q

q xq y q xy

Using this becomes

CO H q q q q q q

For the higher derivatives q , , there is the following recursion formula:

q x q x q Kx

n n

q c X xq Kx

X H X H X H X (Recall that the composite map H is the multiplication with the self intersection of the diagonal, i.e. the second Chern class

c X X Ke K

of .) Using this formula repeatedly and keeping in mind that

K K e e

e and , , we ﬁnally arrive at

CO H q q K K eq K K e

Only the ﬁrst two summands contribute to the integral. Hence

Z Z Z

K K e CO H

X X X

d d e

n N

For higher , the practical calculation of n quickly becomes rather difﬁcult. Al- N

ready the case of surpassed my personal calculation skills. Using MAPLE, I com-

N n

puted n for . One obtains for example:

N d d d d e

N d d d e

d e e e e

N d d d e

d e d e

e e e

e e e e

These calculations verify LeBarz’ trisecant formula for [21, Th´eor`eme 8] and

N N

the computation of by Tikhomirov and Troshina [29]. The formula for seems

N N

to be new. I omit the presentation of and : the information is contained in the

X P O H O m

X following analysis of these numerical data. For and P these

tally with the polynomials computed by Ellingsrud and Strømme using a torus action on P and the Bott formula [8].

Taking the logarithm of the generating function, we may write:

X X

m n

d z N z exp

m n

d d H HK

where the coefﬁcients m a priori are rational polynomials in , ,

K e and . One can show that these polynomials are in fact linear (cf. [5]). The

explicit calculation yields

d d

d d e

From this one can attempt to guess the generating functions. Let

k z z z Q z

be the inverse power series of the rational function

k k k

k k

This is a solution of the differential equation

dz dk

z k k k k k

Conjecture 4.9 — Using the notations above the following formula holds:

a b

k k

N z

k k

a HK K b H K O c H H K O X

with , X , and . We thank Don Zagier for pointing out to us the existence of Sloane’s ‘Encyclopedia of Integer Sequences’ [27]. Intensive use of the on-line version of the Encyclopedia, numerous numerological experiments and some inspiring help from Don Zagier al- lowed me to guess the generating functions. He also found a simple substitution to turn my still awkward version of the generating function into the smooth form pre-

sented above.

4.4 The cohomology ring of

A In this section we will describe an identiﬁcation of the cohomology ring of with the ring of certain explicitly given differential operators on the polynomial ring in countably many variables. After writing this section I learnt from a letter by N. Fakhruddin [10] of his description of the cohomology ring using a different approach.

Of course, the afﬁne plane A is not projective, so that we cannot directly apply

the methods of the previous sections. On the other hand, in [24] Nakajima does work

q n

with non-projective surfaces, the only difference being that the operators n , , must be modelled on cohomology classes with compact support rather than ordinary

cohomology classes. The reason for this is that, in the notations of Deﬁnition 2.3, the

p p morphism is proper, so that push-forward is deﬁned, whereas is proper only if

the variety X is proper. With this modiﬁcation Nakajima’s main theorem holds for the

afﬁne plane as well.

H A Q Q q q

m m

As , we simplify notations by putting A . Then

i n

Q q q H H A Q

, the polynomial ring in countably inﬁnitely

q m H H A Q n

many variables, and if m is given degree , then is the ho-

n A

mogeneous component of H of degree . As any vector bundle on is trivial, there

n n

A ch H H

is essentially only one tautological bundle O on . Let be the

components of the associated Chern character operator, and let d as before. The

n n

P A P

inclusion A induces an open embedding which in turn gives

n n

P Q H A Q

rise to an epimorphism of rings H . This implies that

q ch H

all commutation relations for the m and hold in as well. In fact they become

c P c P

much simpler as the pull-back both of and is zero. To describe these

relations in the given special setting, let m .

q m

Theorem 4.10 — The Chern character of the tautological bundle acts on H as follows:

ch q

n n n n

n n

n H Q ch O For each , the cohomology ring n is generated as a –algebra by , and the

relations between these generators are those of the restriction of the given differential H

operators to n . H

That n is generated by the chern classes of the tautological bundle had earlier been proved by Ellingsrud and Strømme [7]. In order to prove the theorem we consider a larger class of differential operators

on H deﬁned by

X Y

q D

n n

n D q n

for nonnegative integers and , with the usual conventions n for

D d D

and . The key observation is that . This follows directly from

A Q Q Theorem 3.10 and the fact that in the present situation H . It is easy to check by explicit calculation that these operators satisfy the following commutation

relations

D D m n D

n m nm

q D D n D

n n

In particular, , or more generally, by induction:

q n D

n n

We can now easily generalise Theorem 4.2:

m D ch q

mn m

For m the assertion follows from the basic relation (Theorem 4.2)

q D ch q

n n

m mq q q

and for we deduce it by induction using m as well as

q D ch q ch q

and .

n n

Proof of the theorem. We must ﬁrst show that

Observe that by the commutation rules for the operators we have

n n n

D q D D m D

n m n m mn

n n n

ch D

Thus and show the same commutation behaviour with all generators

q H

m of and clearly act trivially on the vacuum. Hence they are equal. H

It remains to check that the Chern classes of the tautological bundle generate n .

P Q

n n kk q i q

Let be a partition of , i.e. and let

i i i

q kk n Q

be the associated monomial. The monomials with form a -basis of

H q q

. Let us say that if in the lexicographical order. We want to

H ch m n

show that the subring H in generated by the action of , ,

q q q n

on contains the monomial for all partitions of . As this is

true for the smallest possible monomial , we proceed by induction. Given

q H q q n a

we assume that for all .As , let be the smallest

a a index such that a , i.e. Consider now the

partition

a a

q q H

Then and hence is contained in and

q ch q

where stands for a linear combination of smaller monomials. This ﬁnishes the

induction.

References

C f x y g [1] J. Brianc¸on, Description de Hilb . Inventiones math. 41, 45-89 (1977).

[2] F. Catanese, On Severi’s proof of the double point formula. Comm. in Algebra 7 (1979), 763-773.

[3] J. Cheah, On the Cohomology of Hilbert Schemes of Points. J. Alg. Geom. 5 (1996), 479-511.

[4] J. Cheah, Cellular decompositions for various nested Hilbert schemes of points. Pac. J. Math., 183 (1998), 39-90.

[5] G. Ellingsrud, L. G¨ottsche, M. Lehn, On the cobordism class of Hilbert schemes of Points on Surfaces. In preparation.

[6] G. Ellingsrud, A. Strømme, On the homology of the Hilbert schemes on points in the plane. Inv. Math. 87 (1987), 343-352.

[7] G. Ellingsrud, A. Strømme, Towards the Chow ring of the Hilbert scheme

of P . J. reine angew. Math. 441 (1993), 33-44.

[8] G. Ellingsrud, A. Strømme, Bott’s Formula and Enumerative Geometry, J. Am. MathSoc.,˙ 9 (1996), 175-193.

[9] G. Ellingsrud, A. Strømme, An intersection number for the punctual

Hilbert scheme of a surface. Trans. Amer. Math. Soc. to appear.

[10] N. Fakhruddin, On the Chow ring of A . Letter to the author.

[11] B. Fantechi, L. G¨ottsche, The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety. J. reine angew. Math. 439 (1993), 147-158,

[12] J. Fogarty, Algebraic Families on an Algebraic Surface. Am. J. Math. 10, 511-521 (1968).

[13] W. Fulton, Intersection Theory. Erg. Math. (3. Folge) Band 2, Springer Verlag 1984.

[14] L. G¨ottsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286 (1990), 193-207.

[15] L. G¨ottsche, W. S¨orgel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296 (1993), 235-245.

[16] I. Grojnowski, Instantons and Afﬁne Algebras I: The Hilbert Scheme and Vertex Operators. Math. Res. Letters 3 (1996), 275-291.

41 [17] A. Grothendieck, Techniques de construction et theor´ emes` d’existence en geom´ etrie´ algebrique´ IV: Les schemas´ de Hilbert.S´eminaire Bourbaki, 1960/61, no. 221.

[18] A. Grothendieck, Sur quelques points d’algebre` homologique.Tˆohoku Math. J. 9 (1957), 119-221.

[19] A. Iarrobino, Punctual Hilbert Schemes. Memoirs of the AMS, Volume 10, Number 188, 1977.

[20] B. Iversen, Linear determinants with applications to the Picard scheme of a family of algebraic curves. Lect. Notes Math. 174, Springer Verlag, Berlin (1970).

[21] P. LeBarz, Formules pour les trisecantes´ des surfaces algebriques´ . L’Ens. Math. 33 (1987), 1-66.

[22] M. Lehn, Chern classes of Tautological Sheaves on Hilbert Schemes of Points on Surfaces. Habilitationsschrift, Fakultät für Mathematik, Georg- August-Universität Göttingen, Oktober 1997.

[23] I. G. Macdonald, The Poincare´ Polynomial of a Symmetric Product. Proc. Cambridge Phil. Soc. 58 (1962), 563-568.

[24] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. 145 (1997), 379-388.

[25] H. Nakajima, Lectures on Hilbert schemes of points on surfaces. Preprint, University of Tokyo, 1996.

[26] F. Severi, Intorno ai punti doppi impropri di una superﬁcie generale dello spazio a quattro dimensioni e ai suoi punti triple apparenti. Rend. Circ. Mat. di Palermo, 15 (1901), 33-51. Also in: Memorie Scelte, I, Zufﬁ Bologna 1950.

[27] N. J. A. Sloane, S. Plouffe, The Encyclopedia of Integer Sequences. Aca- demic Press, San Diego, 1995. On-line version:

http://www.research.att.com/njas/sequences.

[28] A. S. Tikhomirov, Standard bundles on a Hilbert scheme of points on a surface. In: Algebraic geometry and its applications, Yaroslavl’,1992. Aspects of Mathematics, Vol. E25. Vieweg Verlag, 1994.

[29] A. S. Tikhomirov, T. L. Troshina, Top Segre class of a standard vector

Hilb S S

bundle E on the Hilbert scheme of a surface . In: Algebraic D geometry and its applications, Yaroslavl’,1992. Aspects of Mathematics, Vol. E25. Vieweg Verlag, 1994.

42 [30] A. S. Tikhomirov, The variety of complete pairs of zero-dimensional subschemes of an algebraic surface. Izvestiya RAN: Ser. Mat. 61 (1997), 153-180. = Izvestiya: Mathematics, 61 (1997), 1265-1291.

[31] C. Vafa, E. Witten, A strong coupling test of S -duality. Nucl. Phys. 431 (1994), 3-77.

Manfred Lehn Mathematisches Institut der Georg-August-Universit¨at Bunsenstraße 3-5, D-37073 G¨ottingen, Germany e-mail: [email protected]