ñòð. 4 |

2 Ln [u]; qm [w] = ( m)qn+m [Z];

90

with Z Z

X X

Z= si ti w + ti wui :

X X

i i

Each of the sums on the right-hand side equals uw. This shows part 1. Part

2. can easily be reduced to part 1.

10. Tautological sheaves

We can, in a natural way, associate a tautological sheaf F [n] on X [n] to

a vector bundle F on X. These sheaves are very important in geometric

applications of the Hilbert scheme X [n] . Let again

Zn := (W; x) 2 X [n] X x 2 W

be the universal family with the projections p : Zn ! X [n] , q : Zn ! X.

Then the tautological sheaf

F [n] := p q (F )

is a locally free sheaf of rank rn on X [n], where r is the rank of F. (This

is because p : Zn ! X [n] is

at of degree n.) In particular F [1] = F.

By denition the ber F [n] (W ) of F [n] over a point W 2 X [n] is naturally

identied with H 0 (W; FjW ).

If 0 ! F ! E ! G ! 0 is an exact sequence of locally free sheaves, then

so is

0 ! F [n] ! E [n] ! G[n] ! 0:

Therefore ( )[n] : F 7! F [n] denes a homomorphism from the Grothendieck

group K(X) of locally free sheaves on X to K(X [n]).

The Chern classes of the tautological sheaves have interesting geometric

interpretations.

1. Let L be a line bundle on X. Then cn (L[n] ) 2 H n (X [n] ) is the Poincar

e

[n] = C (n) , where C 2 jLj is a smooth curve.

dual of the class of C

2. More generally cn l (L[n]) is the Poincar dual of the class of all W 2

e

[n] with W Ct for Ct a curve in a general l-dimensional linear

X

subsystem of jLj.

3. The top Segre class s2n(L[n] ) is by denition just the top Chern class

c2n ( L[n]) (here ( L[n]) is the negative of L[n] in the K(X [n])). In

other words that means that s2n(L[n] ) is the part of degree 2n of

1=(1 + c1 (L[n]) + c2 (L[n] ) + : : : ):

Hilbert schemes and Heisenberg algebras 91

The degree of s2n (L[n] ) is the number of all W 2 X [n], which do not im-

pose independent conditions on curves in a general (3n 2)-dimensional

sub-linear system of jLj.

All these identications are under the assumption that L is suciently

ample. It is e.g. sucient, but not necessary that L is the n-th tensor power

of a very ample line bundle on X. The identications are proven by using

the Thom-Porteous formula ([7] Theorem 4.4), which gives the class of the

degeneracy locus of a map of vector bundles in terms of their Chern classes.

There is a natural evaluation map

evn : H 0(X; L)

OX ! L[n] ; (s; W) 7! sjW 2 H 0 (Z; LjW ) = L[n](W )

[n]

from the trivial bundle with ber H 0 (X; L) to L[n]. The assumption that

L is suciently ample ensures that evn is surjective. In this situation the

Thom-Porteous formula says that cn l (L[n]) is the class of the locus where

the restriction of evn to a trivial vector subbundle of rank l+1 is not injective.

Such a vector subbundle corresponds to an l-dimensional linear subsystem

M of jLj, and the locus where the map is not injective is easily seen to be

the locus of W 2 X [n] with W 2 C for a curve C 2 M. This shows parts 1.

and 2.

Part 3 is similar. In this case we look at the dual map

(evn )_ : (L[n])_ ! H 0 (X; L)

OX ; [n]

and the locus we are looking for is the locus where (evn )_ is not injective.

So we get in particular

Z

s2 (L) = # base points in a pencil of jLj = c1 (L)2 :

X

R

is the number of double points of the map X ! P4 given by a

[2]

X [2] s4 (L )

general 4-dimensional linear subsystem of jLj. The numbers s2n(L[n] ) are,

for instance, interesting from the point of view of Donaldson invariants.

11. Geometric interpretation of the Virasoro operators

Our aim is to give a more geometric interpretation of the action of the

Virasoro algebra, which was dened in section 9. We shall see that they are

related to the "boundary" of X [n] , i.e. the locus of subschemes of X with

support less then n points. If we write @ for the operation of multiplying by

the cohomology class of the boundary, then Ln will turn out to be essentially

the commutator qn@ @qn . In order to be able to prove this result we have

92

to give another description of the class @, which relates it to tautological

sheaves. This is done by looking at the incidence scheme

n o

[n;n+1] := (Z; W) 2 X [n] X [n+1] Z W :

X

As a special case of diagram (5) we have the diagram

pn+1

! X [n+1]

X [n;n+1]

X ?

pn ?

(15) y

X [n]:

In particular there is a morphism := (pn ; ) : X [n;n+1] ! X [n] X, which

sends a pair (Z; W) of subschemes of X to Z and the residual point. It is

evident that is an isomorphism over the open subset of all (Z; x) 2 X [n] X

with x 6= Z, i.e. over the complement of the universal family

n o

Zn := (Z; x) 2 X [n] X x 2 Z :

More precisely we have the following theorem:

Theorem 11.1. [5] X [n;n+1] is the blowup of X [n] X along the universal

family Zn .

Proof. Let : Y ! X [n] X be the blowup along Zn with exceptional

divisor E. On X [n] X X, let Wn be the pull-back of Zn from the rst

and third factor. On Y X, let

:= ( 1X ) 1 ; Wn := ( 1X ) 1 Wn :

e f

Then the projection pY j : ! Y is an isomorphism, which maps \ Wn

e ef

e

isomorphically onto the exceptional divisor E. Therefore Zn+1 := [ Wn

e ef

is a

at family of degree (n + 1) over Y , and on Y X we have a sequence

0 !O ( E) ! OZn+1 ! OWn ! 0:

(16) e e f

The

at family Zn+1 induces a morphism Y ! X [n+1] , which together with

e

the projection Y ! X [n] gives a morphism Y ! X [n] X [n+1] with im-

age X [n;n+1]. One checks that the induced morphism Y ! X [n;n+1] is an

isomorphism.

Let E be the exceptional divisor of the blowup X [n;n+1] ! X [n] X. Then

E can be described as

n o

E = (Z; W) 2 X [n;n+1] supp(Z) = supp(W) :

Hilbert schemes and Heisenberg algebras 93

Let F be a vector bundle on X. Then tensoring the sequence (16) with p F X

[n;n+1] gives the exact sequence

and pushing down to Y = X

0 ! F( E) ! p F [n+1] ! p F [n] ! 0

(17) n+1 n

which relates the tautological bundles F [n] and F [n+1] . This makes it pos-

sible to try to treat the tautological bundles via an inductive argument. In

particular we get

( E) = p O[n+1] p O[n]

OX n+1 n

[n;n+1]

in the Grothendieck group K(X [n;n+1]).

[n]

Let @X [n] be the closure of the stratum X(2;1;::: ;1) , i.e. the locus in X [n] ,

where the subscheme does not consist of n distinct points. The class of

[@X [n] ] (i.e. the class Poincar dual to it) is related to the rst Chern class

e

of the tautological sheaves.

[n]

Lemma 11.2. [@X [n]] = 2c1 (OX ).

Proof. @X [n] is the branch divisor of the projection p : Zn ! X [n] , therefore

2c1 (p OZn ) = [@X [n] ].

Denition 11.3. Let d : H (X) ! H (X) be the operator of multiplying

by c1 (OX ), i.e. for y 2 H (X [n] ) we have dy = c1 (OX ) y. For f 2

[n] [n]

End(H (X)) the derivative f 0 of f is dened to be

f 0 := [d; f]:

It is easy to check that

(fg)0 = f 0 g + fg0 ; [f; g]0 = [f 0 ; g] + [f; g0 ];

which gives some justication for calling it derivative.

We have the following geometric interpretation of the derivative in terms

of tautological sheaves. Let X [n;m] X [n] X [m] be the incidence variety of

pairs of subschemes (Z; W) with Z W (in particular n < m). Let pn and

be pm be the projections of X [n;m] to X [n] and X [m] .

Then taking the derivative of f 2 End(H (X)) amounts to multiplying

[m] [n]

with c1 (p OX ) c1 (p OX ).

m n

Proposition 11.4. Let f : H (X [n]) ! H (X [m] ) be a homomorphism

which is given by f() := pm (p \ u), for a suitable u 2 H (X [n;m] ).

n

Then

0 () = pm p () c1 (p O[m] ) c1 (p O[n]) \ u :

f n mX nX

94

In particular, in case m = n + 1, we get

f 0() = pn+1 p () ( E) \ u :

n

Proof.

f 0() = df() fd

c1 (O[n]) \ u

[m] ) p (p () \ u) p

= c1 (OX m pn

m n X

Now we apply the projection formula.

X [n;n+m] X carries two universal families Zn Zm+n . The above result

can also be reinterpreted as saying that we multiply by the rst Chern class

of the push-forward to X [n;n+m] of the ideal sheaf IZn =Zn+m .

Now we come to the most important technical result of Lehns paper. It

gives a geometric interpretation of the Virasoro operators Ln .

Theorem 11.5. 1. !!

Z

jnj 1

0

KX uw 1 :

qn[u]; qm [w] = nm qn+m [uw] + 2 n+m

X

2.

qn [u] = nLn[u] + n(jnj2 1) qn[KX u]:

0

Part 2. Says that the Virasoro generators Ln [u] are essentially the deriva-

tives of the qn[u].

Proof. We show that 1. implies 2. By the Heisenberg relations for the qn

and from the formula Ln[u]; qm [w] = mqn+m[uw] from Theorem 9.2, we

get

n(jnj 1) q [K u]; q [w]i

h

nLn[u] + nX m

2 !

Z

2 (jnj 1)

= nmqn+m[uw] + n 2 KX uw 1:

n+m

X

Therefore the dierence between the right-hand side and the left-hand side

in 2. commutes with all the qm [u]. Since H (X) is an irreducible Heisenberg

module, it follows by Schurs lemma that the dierence is the multiplication

by a scalar. This scalar must be zero, because the dierence has weight n

(i.e. sends H (X [l] ) to H (X [l+n] )).

The proof of part 1. requires a complicated geometric argument, and it

is also dicult to keep track of the indices. The most dicult part is the

Hilbert schemes and Heisenberg algebras 95

case n = m (when the Theorem also has an extra term). We will sketch

the proof of 0

q1 [X]; qn [u] 1 = nqn+1[u]1;

which illustrates some of the geometric ideas, without running into any of

the technicalities. In the application to Chern classes of tautological sheaves,

0

we mostly use q1 [X].

Let U X be the submanifold represented by u. Let

n o

Mn (U) := Z 2 X [n] supp(Z) is one point of U ;

n o

(Z; W) 2 X [n;n+1] supp(Z) = supp(W)

Mn;n+1(U) := is one point of U :

By denition and by Proposition 11.4 we obtain

q1[X]qn[u]1 = q1 [X][Mn (U)] = pn+1 (( E) \ p [Mn (U)]):

0 0

n

We recall that n o

E = (Z; W) 2 X [n;n+1] supp(Z) = supp(W) :

Therefore, set-theoretically Mn;n+1 = Mn X E, but the map E ! X [n]

[n]

has degree n, and the map Mn;n+1 ! Mn has degree 1. Therefore

pn+1 (( E) \ p [Mn (U)]) = pn+1 (n[Mn;n+1(U)])

n

= npn+1 [Mn+1 (U)]

= nqn+1[U]1:

0

On the other hand q1 [X]1 = 0:

Corollary 11.6. d and the q1[u] for u 2 H (X) suce to generate H (X)

from 1.

12. Chern classes of tautological sheaves

We dene operators on H (X) of multiplying by the Chern classes of the

tautological sheaves F [n] on X [n]. If we can understand how these commute

with the qn , this allows us to compute the Chern numbers of all tautological

sheaves, and to partially understand the ring structure of the H (X [n] ).

Denition 12.1. Let u 2 K(X). We dene operators c[u] 2 End(H (X))

by

c[u]y = c(u[n] ) y for y 2 H (X [n]):

So if u is the class of a vector bundle on X, then c[u] just multiplies for each n

a class on X [n] with the total Chern class of the corresponding tautological

96

sheaf F [n]. We also write ck [u]y for ck (u[n] ) y. Note that by denition

d = c1 [OX ]. Obviously the c[u] commute among each other (and therefore

they also commute with d). We put

C[u] := c[u]q1 [X][u] 1 :

c

We can use the operator C[u] to write down the total Chern classes of the

tautological sheaves in a compact way.

Proposition 12.2.

X

c(u[n]) = exp(C[u])1:

n0

Proof. We note that

q1 [X]n 1 = 1 :

X

n! [n]

Therefore

X

c(u[n]) = c[u] exp(q1 [X])1

n0

= c[u] exp(q1 [X])[u] 1 1

c

= exp([u]q1 [X][u] 1 )1:

c c

Now we express C[u] in terms of the derivatives of the Heisenberg operator

q1 applied to the Chern classes of u. This establishes a relation between the

Chern classes of the tautological sheaves and the Heisenberg generators.

Theorem 12.3.

X r

k q() [c (u)];

C[u] = 1k

;k0

()

(here q1 [ck (u)] is the -th derivative of q1 [ck (u)]).

Proof. Let F be a locally free sheaf on X. Recall the incidence variety

pn+1

! X [n+1]

X [n;n+1]

X ?

pn ?

y

X [n]

and the exact sequence

0 ! F( E) ! p F [n+1] ! p F [n] ! 0:

n+1 n

Hilbert schemes and Heisenberg algebras 97

This gives

k( E) c (F ):

[n+1] ) = p F [n] X r

p

n+1c(F

(18) k

n

;k0

y 2 H (X [n] ), we get

So, for

C[F]y = c(F [n+1] ) pn+1 p (y c(F [n] ) 1 )

n

= pn+1 p c(F [n+1] ) p c(F [n] ) 1 p y :

n+1 n n

We insert (18) into this formula and apply Proposition 11.4, which says that

multiplying by ( E) corresponds to taking derivatives.

At least in the case of a line bundle L on X, the results obtained so far

are enough for nding an elegant formula for the Chern classes of L[n].

Theorem 12.4.

!

( 1)m 1 q [c(L)] 1:

X X

c(L[n] ) = exp mm

n0 m1

Remark 12.5. Note that for the top Chern classes this gives the following.

Let D 2 jLj be a smooth curve, then cn (L[n] ) = [D[n] ] = [D(n) ]. Then the

theorem gives

!

1)m 1

(

X X

qm [c1 (L)] 1:

[D(n) ] = exp m

n0 m1

This is Theorem 7.5, which was used to determine the constant in the Heisen-

berg relations.

Proof. Let X

c(L[n])tn = exp(C[L]t)1:

U(t) :=

n0

The second equality is by Proposition 12.2. Therefore U satises the dier-

ential equation

d U(t) = C[L]U(t); U(0) = 1:

dt

Now let !

X ( 1)m 1

q [c(L)]tm ;

S(t) := exp m

m

m1

98

we want to show that S(t)1 satises the same dierential equation. By

denition

d S(t) = S(t) X ( 1)m q [c(L)]tm :

m+1

dt m0

By the Lehns Main Theorem 11.5, we have

0

q1 [X]; qm [c(L)] = mqm+1[c(L)]:

As this commutes with qm [c(L)], we get

ni n1

0 [X]; qm [c(L)] = qm [c(L)] ( m)qm+1 [c(L)]:

h

q1 n! (n 1)!

Therefore we obtain

X

0

q1 [X]; S(t) = S(t) ( 1)m qm+1 [c(L)]tm :

m1

We recall from Theorem 12.3 that

0

C(L) = q1 [c(L)] + q1 [X]:

So we nally get by putting everything together

0

C(L)S(t)1 = q1 [X]; S(t) 1 + q1 [c(L)]S(t)

X

= S(t) ( 1)m qm+1 [c(L)]tm :

m0

Let L again be a line bundle on X. We want to compute the top Segre

classes Z

s2n (L[n])

Nn :=

X

[n]

as polynomials in the intersection numbers L2 , LKX , KX , c2 (X) on X. A

2

priory it is not clear that this should be possible. We rewrite

Z Z

C[ L]n 1:

[n] ) =

Nn = c2n (( L) n!

X [n] X [n]

By Theorem 12.3 we get

L] = X( 1) q1 [c( L)+1 ]:

()

C[

0

By the main theorem 11.5 we can express the derivatives of q1 in terms of

the Virasoro generators Ln and the Heisenberg generators qn . Applying the

denitions 9.1 of the Virasoro generators, we can express this in terms of

the Heisenberg generators. We can do all these computations explicitly to

Hilbert schemes and Heisenberg algebras 99

compute the Nn for suciently small n. The calculation shows that the

following conjecture is true until n = 7.

Conjecture 12.6. (Lehn) Let k be the inverse power series to

k(1 k)(1 2k)4 :

t = (1 6k + 6k2 )

Then

LK 2KX (1 2k)(L KX ) +3(OX )

n = (1 k) X

X 2 2

Nnt :

(1 6k + 6k2 )(L)

n0

2

(Here (L) = L(L KX )=2 + (KX + c2 (X))=12 is the holomorphic Euler

characteristic of L.)

100

References

[1] A. Beilinson, J.N. Bernstein, P. Deligne, Faisceaux pervers. Astrisque, vol 100.

e

n C fx; yg. Inventiones Math. 41, 45-89 (1977).

[2] J. Brianon, Description de Hilb

c

[3] J. Cheah, Cellular decompositions for various nested Hilbert schemes of points.

Pac. J. Math., 183 (1998), 39{90.

[4] G. Ellingsrud and S. A. Strmme, On the homology of the Hilbert scheme of points

in the plane. Invent. Math. 87 (1987), 343{352.

[5] G. Ellingsrud and S. A. Strmme, An intersection number for the punctual Hilbert

scheme of a surface. Trans. Amer. Math. Soc. 350 (1998), 2547{2552.

[6] J. Fogarty, Algebraic Families on an Algebraic Surface. Am. J. Math. 10 (1968),

511{521.

[7] W. Fulton, Intersection Theory. Ergebnisse 3. Folge, Band 2, Springer 1984.

[8] L. Gttsche, The Betti numbers of the Hilbert scheme of points on a smooth pro-

o

jective surface. Math. Ann. 286 (1990), 193{207.

[9] L. Gttsche and W. Soergel, Perverse sheaves and the cohomology of the Hilbert

o

schemes of smooth algebraic surfaces. Math. Ann. 296 (1993), 235{245.

[10] I. Grojnowski, Instantons and ane algebras. I. The Hilbert scheme and vertex

operators. Math. Res. Letters 3 (1996) 275-291.

[11] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on

surfaces. Inventiones Math. 136, 157-207 (1999).

[12] I. G. Macdonald, The Poincar Polynomial of a Symmetric Product. Proc. Cam-

e

bridge Phil. Soc. 58 (1962), 563-568.

[13] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective sur-

faces. Ann. Math. 145 (1997), 379-388.

[14] H. Nakajima, Lectures on Hilbert schemes of points on surfaces. University Lecture

Series, 18. American Mathematical Society, Providence, RI, 1999.

Donaldson invariants in Algebraic Geometry

Lothar Gttsche

o

The Abdus Salam International Centre for Theoretical Physics,

Trieste, Italy.

Lecture given at the

School on Algebraic Geometry

Trieste, 26 July { 13 August 1999

LNS001003

gottsche@ictp.trieste.it

Abstract

In these lectures I want to give an introduction to the relation of

Donaldson invariants with algebraic geometry: Donaldson invariants

are dierentiable invariants of smooth compact 4-manifolds X, dened

via moduli spaces of anti-self-dual connections. If X is an algebraic

surface, then these moduli spaces can for a suitable choice of the metric

be identied with moduli spaces of stable vector bundles on X. This

can be used to compute Donaldson invariants via methods of algebraic

geometry and has lead to a lot of activity on moduli spaces of vector

bundles and coherent sheaves on algebraic surfaces.

We will rst recall the denition of the Donaldson invariants via

gauge theory. Then we will show the relation between moduli spaces of

anti-self-dual connections and moduli spaces of vector bundles on alge-

braic surfaces, and how this makes it possible to compute Donaldson

invariants via algebraic geometry methods. Finally we concentrate on

the case that the number b+ of positive eigenvalues of the intersection

form on the second homology of the 4-manifold is 1. In this case the

Donaldson invariants depend on the metric (or in the algebraic geomet-

ric case on the polarization) via a system of walls and chambers. We

will study the change of the invariants under wall-crossing, and use this

in particular to compute the Donaldson invariants of rational algebraic

surfaces.

Keywords: Donaldson invariants, moduli spaces of sheaves.

Contents

1 Introduction 105

2 Denition and properties of the Donaldson invariants 106

2.1 Moduli spaces of connections . . . . . . . . . . . . . . . . . . 106

2.2 ASD-connections . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.3 Relations to holomorphic vector bundles . . . . . . . . . . . . 108

2.4 Uhlenbeck compactication . . . . . . . . . . . . . . . . . . . 109

2.5 Denition of the invariants . . . . . . . . . . . . . . . . . . . 110

2.6 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . 111

3 Algebro-geometric denition of Donaldson invariants 112

3.1 Determinant line bundles . . . . . . . . . . . . . . . . . . . . 113

Construction of sections of L (nH) . . . . . .

3.2 . . . . . . . . . 114

1

3.3 Uhlenbeck compactication . . . . . . . . . . . . . . . . . . . 115

3.4 Donaldson invariants via algebraic geometry . . . . . . . . . . 116

4 Flips of moduli spaces and wall-crossing for Donaldson in-

variants 117

4.1 Walls and chambers . . . . . . . . . . . . . . . . . . . . . . . 118

4.2 Interpretation of the walls in algebraic geometry . . . . . . . 119

4.3 Flip construction . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4 Computation of the wall-crossing . . . . . . . . . . . . . . . . 123

5 Wall-crossing and modular forms 125

5.1 Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 131

References 133

105

Donaldson invariants in Algebraic Geometry

1 Introduction

Donaldson invariants [D1] have played an important role in the study and

classication of compact dierentiable 4-manifolds X. The most compre-

hensive introduction to Donaldson invariants is [D-Kr]. Discrete invariants

of 4-manifolds are the fundamental group (X) and the intersection form

1

on H (X; Z). If X is simply-connected, then the homotopy type of X is

2

essentially determined by the intersection form. Friedman showed that in

this case X is determined up to homeomorphism by its homotopy type.

In order to attempt to make a dierentiable classication, ones needs ad-

ditional invariants. The Donaldson invariants are dened via gauge theory

in terms of moduli spaces of anti-self-dual connections on dierentiable bun-

dles on X. If X is an algebraic surface, then these moduli spaces can be

identied with moduli spaces of stable vector bundles on X. This makes it

possible to apply methods of algebraic geometry to compute the Donaldson

invariants. In fact, because of this, for a long time most of the calculations of

Donaldson invariants have been carried out in the case of algebraic surfaces.

On the other hand the Donaldson invariants have provided a lot of interest

for the study of moduli spaces of vector bundles and coherent sheaves on

algebraic surfaces.

Some results obtained with Donaldson invariants are:

1. Algebraic surfaces are essentially indecomposable: If a simply-connected

algebraic surface X is the connected sum X = Y #Z of two 4-manifolds,

then either Y or Z must have a negative denite intersection form. An

example where this happens is when X is the blow up of Y in a point.

2. The dierentiable classication of elliptic surfaces.

3. The Kodaira dimension of an algebraic surface is a dierentiable in-

variant.

Recently the Seiberg-Witten invariants have appeared, which are also de-

ned via gauge theory, but are often easier to compute [W],[D2]. A number

of conjectures from Donaldson theory were immediately proved, e.g.

1. The plurigenera of algebraic surfaces are dierentiable invariants.

2. The generalized Thom conjecture: Let X be an algebraic surface, then

each smooth algebraic curve in X minimizes the genus of embedded

2-manifolds in its homology class.

106

Conjecturally the Donaldson- and Seiberg-Witten invariants are very closely

related and in particular the Donaldson invariants can be computed in terms

of the Seiberg-Witten invariants.

Since the appearance of the Seiberg-Witten invariants the interest in the

Donaldson invariants has become a bit less, but there is still a large number

of interesting open questions.

2 Denition and properties of the Donaldson in-

variants

In this lecture we dene the Donaldson invariants via gauge theory and

state some of their most important properties. We prefer here to formulate

everything in terms of vector bundles, which should be more familiar to the

audience, instead of principal bundles, which would be more natural.

2.1 Moduli spaces of connections

Let X be a smooth simply-connected compact oriented 4-manifold. Let P

be a principal SU(2)- or SO(3)-bundle on X. The Donaldson invariants are

dened via intersection theory on a moduli space of anti-self-dual connections

on P.

SU(2)-bundles on X are classied by their second Chern class c (P ), 2

and SO(3)-bundles on X are classied by their second Stiefel-Whitney class

w (P ) 2 H (X; Z=2) and their rst Pontrjagin class p (P ) 2 H (X; Z).

2 4

2 1

In the SU(2)-case the moduli space of anti-self-dual connections on P

can be identied with the moduli space of anti-self-dual connections on the

associated complex vector bundle on E with rst Chern class c = 0. In 1

the SO(3)-case (after choosing a lift c 2 H (X; Z) of w (P )) it corresponds

2

1 2

to a moduli space of Hermitian Yang-Mills connections on the associated

complex vector bundle with Chern classes c ; c (with c 4c = p (P )).

2

1 2 2 1

1

For simplicity, in the following we will concentrate on the SU(2)-case.

Let E be a rank 2 complex dierentiable vector bundle on X. We x a

hermitian metric h on E. (That is for each x 2 X we have a hermitian inner

product on the ber E(x), varying smoothly with x.) We denote by

i (E)

the space of C 1 -sections of E

i TX .

A hermitian connection on E is a connection D :

(E) !

(E), which

0 1

is compatible with h. (That D is a connection means that it is a linear map

107

Donaldson invariants in Algebraic Geometry

satisfying the Leibniz rule

D(f s) = d(f

s) + f D(S)

and that D is compatible with the metric means furthermore that

d(h(s ; s )) = h(D(s ); s ) + h(s ; D(s )) .)

1 2 1 2 1 2

D is called reducible if E is the direct sum L L of two line bundles, and

1 2

D = D D with Di a connection on Li .

1 2

We write A(E) for the space of hermitian connections on E, (which are

trivial on det(E) in the case c (E) = 0 and equal to a xed connection on

1

det(E) otherwise). A (E) A(E) is the subspace of irreducible connec-

tions. The gauge group G is the set of C 1 -automorphisms of E which are

compatible with h and act as the identity on det(E). G acts on A(E) and

A(E) via D

!

?

(E)

?

(E)

0 1

? ?

y y

(E) (D)

(E):

!

0 1

Let B(E) := A(E)=G, B (E) := A (E)=G.

2.2 ASD-connections

We assume in this part that c (E) = 0. Now x a Riemannian metric g on

1

X. It gives rise to a Hodge star operator

g : TX ! TX ; g = 1:

2 2 2

We write for the (+1)-eigenbundle and for the ( 1)-eigenbundle of

+

g .

Denition 2.1 For D 2 A(E), let F(D) = D D 2

(End(E)) be it's 2

curvature. F is called anti-self-dual (ASD), if

F(D) = F(D):

In other words, writing F(D) := F (D) + F (D), with F (D) a section of

+

(End(E)) (and similarly for F (D)), the condition is F (D) = 0. We

+ +

write Ng (E) for the moduli space

Ng (E) := fASD-connections on Eg=G B(E):

108

In the case c (E) 6= 0 we have instead to take the moduli space of Hermitian

1

Yang-Mills connections on E, because only these correspond to the moduli

space of ASD-connections on the corresponding principal bundle. The dif-

ferentiable type of E is determined by its Chern classes c (E), and c (E).1 2

Therefore we also write Ng (c ; c ) for Ng (E).

1 2

If D is an ASD-connection (or Hermitian Yang-Mills in case c (E) 6= 0) 1

on E, then by Chern-Weil theory

1 Z tr(F(D)^F(D)) = 1 Z kF (D)k > 0:

4c (E) c (E) = p (E) = 4

2 2

4 X

2 1

X

1 2 2

Let b (X) be the number of positive eigenvalues of the intersection form on

+

H (X; R). We write

2

k := (c c =4)(E): 2

2 1

Then we have the following generic smoothness result:

Theorem 2.2 If g is generic, then Ng (E) is a smooth manifold of dimension

2d = 8k 3(1 + b (X)): +

For a generic path gt of metrics, the corresponding parameterized moduli

space is smooth.

Furthermore NE (g) is orientable. The orientation depends on the choice

of an orientation of a maximal-dimensional subspace H (X; R) of H (X; R)

2 + 2

on which the intersection form is positive denite.

2.3 Relations to holomorphic vector bundles

Assume here, that c (E) = 0. Let X be a projective algebraic surface. Let

1

H be an ample divisor on X. Let g(H) be the corresponding Hodge metric

and ! the Khler form. We write p;q for the bundle of (p; q) forms. Then

a

we get

= <( ) R!

2;0 0;2

+

= !? in : 1;1

We can write D := @D +@ D , where @D :

(E) !

(E) and @ D :

(E) !

0 1;0 0

(E). So we get

0;1

F(D) = @D + (@D @ D + @ D @D ) + @ D : 2

2

D is ASD if

109

Donaldson invariants in Algebraic Geometry

1. @ D = 0.

2

2. @D = 0 and F(D) ^ ! = 0.

2

1. means that @ D denes a holomorphic structure on E. 2. implies after

some work that (E; @ D ) is -stable with respect to H.

Recall that a vector bundle E of rank 2 on an algebraic surface X is called

-stable (slope stable) with respect to an ample divisor H, if

Hc (L) < Hc 2(E) 1

1

for all locally free subsheaves L of rank 1 of E. We denote by MH (c ; c )

X 1 2

the moduli space of -stable rank 2 bundles on X with Chern classes c and 1

c . We have motivated (at least in the case c = 0) that there is a map

2 1

: Ng(H) (c ; c ) ! MH (c ; c ) :

X

1 2 1 2

In fact this map exists for any c , and furthermore we get:

1

Theorem 2.3 (Donaldson) is a homeomorphism.

This will give a relation between the Donaldson invariants (which we will

dene via moduli spaces of ASD-connections) and moduli of vector bundles.

2.4 Uhlenbeck compactication

We want to dene the Donaldson invariants as intersection numbers on

Ng (E) which is usually not compact. We have therefore to compactify.

Theorem 2.4 Let (Ai )i be a sequence in Ng (E). After passing to a sub-

sequence we obtain: There is a nite collection of points p ; : : : pl 2 X 1

with multiplicities n ; : : : ; nl > 0, such that up to gauge transformation

1

AijXnfp1 ;::: ;plg converges to an ASD-connection A1 . A1 can be extended to

an ASD-connection on a vector bundle E 0 with

l

0 ) + X ni :

det(E 0 ) = det(E); c (E) = c (E

2 2

i=1

This leads to the Uhlenbeck compactication:

110

Theorem 2.5 There exists a topology on

a

n) X

Ng (c ; c (n)

1 2

n0

such that the closure N g (c ; c ) is compact.

1 2

Here X = X n =S(n) denotes the n-th symmetric power of X, the quo-

(n)

tient of the n-th power of X by the action of the symmetric group S(n) via

permuting the factors. It parameterizes eective 0-cycles on X of degree n.

2.5 Denition of the invariants

We write H (X) := H (X; Q ) and H (X) = H (X; Q ). If on X Ng (E)

there exists a universal bundle E with a universal connection D with DjXfDg =

D, then we can dene the -map as follows.

1

: H (X) ! H (Ng (E)); () = 4 p (E)=: 1

Here p (E) = (c (E) c (E) )=4, and the slant product p (E)= means:

1 1

2

1 2 1 1

4 4

write

1 p (E) = X

; 2 H (X);

2 H (N (E)):

i i i i g

4 1

i

Then

1 p (E)= = Xh ; i

:

i i

4 1

i

If the universal bundle does not exist, its endomorphism bundle End(E)

will still exist, and we can dene by replacing (c (E) c (E))=4 with 2

2 1

c (End(E))=4.

2

It can be shown that () extends over the Uhlenbeck compactication

N g (E). For generic metric g, N g (E) is a stratied space with smooth strata,

and the submaximal stratum has codimension at least 4. Therefore N g (E)

has a fundamental class.

Now let

3

d := 4c c 2 (1 + b (X)) 2

2 +

1

and write d = l + 2m. Let ; : : : l 2 H (X) and let p 2 H (X) be the class

1 2 0

of a point. Then we dene the Donaldson invariant

Z

X;g ( : : : l pm) := ( ) [ : : : [ (l ) [ (p)m :

c1;d 1 1

g

[N (E)]

111

Donaldson invariants in Algebraic Geometry

More generally let

A (X) := Sym (H (X) H (X)):

2 0

This is graded by giving degree (2 i=2) to elements in Hi (X). We denote

by Ad (X) the part of degree d. By linear extension we get a map X;g :

c1;d

Ad (X) ! Q and

X;g := X X;g : A (X) ! Q :

c1 c1 ;d

d0

By denition the Donaldson invariants depend on the choice of the metric

g, because the ASD-equation uses the Hodge operator, which depends on

g. We have however

1. If b (X) > 1, then X;g is independent of the generic

Theorem 2.6 C;d

+

metric g.

2. If b (X) = 1, then X;g depends only on the chamber of the period

C;d

+

point of g.

We will discuss walls and chambers later. The result means that the

Donaldson invariants are really invariants of the dierentiable structure of

X. In the case b (X) > 1, we can therefore drop the g from our notation.

+

The argument for showing the theorem is that one connects two generic

metrics by a generic path in order to make a cobordism. Reducible connec-

tions occur in codimension b (X), so they make no problem for b (X) > 1,

+ +

but can disconnect the path for b (X) = 1.

+

2.6 Structure theorems

It is often useful to look at generating functions for the Donaldson invariants.

For a 2 H (X) and 2 A (X) and a variable z we write

2

X

X (eaz ) := X (an =n!)z n :

C C

n0

Denition 2.7 A 4-manifold X is of simple type if

X ((p 4)) = 0

C

2

for all 2 A (X) and all C 2 H (X; Z).

2

112

Many 4-manifolds like K3 surfaces and complete intersections are known to

be of simple type, and it is possible that all simply-connected 4-manifolds are

of simple type. The famous structure theorem of Kronheimer and Mrowka

[Kr-Mr] says that all the Donaldson invariants of a manifold of simple type

organize themselves in a nice generating function, which depends only on a

nite amount of data: a nite number of cohomology classes in H (X; Z) 2

(the basic classes) and rational multiplicities associated to these numbers.

Theorem 2.8 Let X be a simply-connected 4-manifold of simple type.

Then there exist so-called basic classes K ; : : : ; Kl 2 H (X; Z) and rational

2

1

numbers (C); : : : l (C), such that for all a 2 H (X)

1 2

l

2 =2 X

i (C)ehKi ;ait :

X (eat (1 + p=2)) = e

C

(aa)t

i=1

(Here (a a) denotes the intersection form on H (X), and hKi ; ai the dual

2

pairing between cohomology and homology.)

3 Algebro-geometric denition of Donaldson in-

variants

Let X be a simply-connected algebraic surface, and let H be an ample divisor

on X. For a P F and a line bundle L on X we denote F(nL) := F

L

n .

sheaf

Let F) = i ( 1)i dimH i (X; F) be the holomorphic Euler characteristic

of F. Recall that a torsion-free coherent sheaf F on X is -stable with

respect to H if (c (G) H)=rk(G) < (c (F ) H)=rk(F) for all non-zero

1 1

strict subsheaves of F. F is called (Gieseker) H-semistable if (G(nH))

(F(nH) for all nonzero strict subsheaves G of F.

X

We denote by M := MH (C; c ) the moduli space of (Gieseker) H-semistable

2

rank 2 torsion-free coherent sheaves F on X with c (F ) = C and c (F ) = c :

1 2 2

We want to relate M to the Uhlenbeck compactication N := Ng(H) (C; c ). 2

Here g(H) is the Fubini-Study metric associated to H. As the Donaldson

invariants are dened in terms of the Uhlenbeck compactication, this allows

us to compute them on the moduli space M of sheaves.

The steps of the argument are as follows:

1. Introduce the determinant bundles L (nH) on M for n 0.

1

113

Donaldson invariants in Algebraic Geometry

2. Construct sections of L (nH)

m for n; m 0 and show that the

1

corresponding linear system is base-point free, thus giving a morphism

: M ! P(H (M; L (nH)

m )_ ):

0

1

3. Show that Im( ) is homeomorphic to N.

4. Apply this to the computation of the Donaldson invariants.

3.1 Determinant line bundles

We will assume for simplicity that there is a universal sheaf E over X M.

For instance, this is the case if H is general and either C is not divisible by

2 or c C =4 is odd.

2

2

For a coherent sheaf F on X M, let

0 ! Gl ! : : : ! Gs ! 0

be a nite complex of locally free sheaves which is quasi-isomorphic to

Rp (F). Then we put

2

O j

2 Pic(M):

det(p (F)) := det(Gj )

( 1)

2!

Denition 3.1 Let D 2 jnHj be a smooth curve. For a general E 2 M let

:= (EjD ). Let a 2 X be a point. Then we put

1

L (nH) := det(p (EjDM ))

det(EjfagM )

1 : 2

1 2!

Let MD be the moduli space of semistable rank 2 vector bundles on D of

degree D C. Assume for simplicity that also on D MD there is a universal

sheaf G. Let G be any element in MD . Then we dene

L := det(p G)

det(GjfagMD )

(G) :

2

0 2!

Remark 3.2 L (nH) is independent of the choice of E (and also of D and

1

a). Any other choice of a universal sheaf F can be written as F = E

p for

a line bundle on M. Then the projection formula implies that Rp (E

2

2

p) = Rp (E)

, and therefore

2

2

det(p (E

p )) =

(E)

det(p (E)):

2! 2!

2

114

So L (nH) stays unchanged if we replace E by E

. In fact we do not need

1

the existence of E in order to dene L (nH). The denition is part of a more

1

general formalism of determinant sheaves as was explained in the lectures of

Huybrechts and Lehn (see [LP], [H-L] where these line bundles are dened

via descent from the corresponding Quot scheme).

In the same way we see that L is independent of the choice of F and

0

indeed we do not need the existence of G to dene L . 0

3.2 Construction of sections of L (nH ) 1

We have the following theorem

Theorem 3.3 [D-N] L is ample on MD .

0

Let U(D) M be the open subset of all sheaves E such that EjD is

semistable. Thus for E 2 U(D), we get that EjD 2 MD . We obtain therefore

a rational map

j : M ! MD ;

which is dened on U(D). By denition we see that

j (L ) = L (nH) on U(D):

0 1

Fix an integer m 0. As L is ample, L

m will have many sections. So we

want to extend the pullbacks j (s) of sections s 2 H (MD ; L

m ) to sections

0 0

0

s 2 H (M; L (nH))

m . By Bogomolovs theorem ([H-L] p. 174) we have

0

e 0

1

the following: For n 0 and all E 2 M the restriction EjD is semistable,

unless E is not locally free over D. For c 0 the general element in M

2

is locally free. If E 2 M is not locally free, then its singularities occur

in codimension 2. Therefore the condition that EjD is not locally free has

codimension 1 in the locus of not locally free sheaves. So, putting things

together, we see that the complement M n U(D) has codimension 2 in

M. Furthermore M is normal. Therefore every j (s) for s 2 H (MD ; L

m )

0

extends to s 2 H (M; L (nH))

m .

0

e 0

1

More precisely one can show the following ([Li], Prop. 2.5).

Lemma 3.4 For every s 2 H (MD ; L

m ) the pullback j (s) extends to

0

s 2 H (M; L(nH)

m ). Furthermore the vanishing locus of s is

0

e e

0

Z(e) = E 2 M EjD is semistable and s(EjD ) = 0

s

or EjD is not semistable :

115

Donaldson invariants in Algebraic Geometry

Now choose m; n 0.

Proposition 3.5 H (M; L (nH)

m) is base-point free.

0

1

Proof. Let E 2 M. By the theorem of Mehta and Ramanathan (see

[H-L] Theorem. 7.2.1), we can nd a smooth curve D 2 jnHj such that

EjD is semistable. Choose s 2 H (MD ; L

m ), such that s(EjD ) = 0. Then

6

0

6

0

s(E) = 0.

e

3.3 Uhlenbeck compactication

L (nH)

m denes a morphism

1

: M ! P(H (M; L (nH)

m )_ ):

0

1

Theorem 3.6 (M) is homeomorphic to the Uhlenbeck compactication

N.

We want to give a brief sketch of the proof of this theorem.

For E 2 M, we introduce the pair (A(E); Z(E)), where

1. If E is -stable, then

X

A(E) = E __ ; l(E __ =E)p p:

Z(E) =

p2X

l(E __ =E)p is the length of the sheaf E __ =E at p. Z(E) is an eective

0-cycle of length k := c (E) c (E __ ) on X, i.e. a point in the

2 2

symmetric power X . (k)

ñòð. 4 |