<< ñòð. 4(âñåãî 12)ÑÎÄÅÐÆÀÍÈÅ >>
We sum this up over all  and i, to obtain
 
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 de nition the ber F [n] (W ) of F [n] over a point W 2 X [n] is naturally
identi ed 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] de nes 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 de nition 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 identi cations are under the assumption that L is suciently
ample. It is e.g. sucient, but not necessary that L is the n-th tensor power
of a very ample line bundle on X. The identi cations 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 suciently 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 de ned 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] ].
De nition 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 de ned 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 justi cation 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 di erence 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 di erence is the multiplication
by a scalar. This scalar must be zero, because the di erence 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 dicult to keep track of the indices. The most dicult 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 de nition 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) suce to generate H (X)
from 1.
12. Chern classes of tautological sheaves
We de ne 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] ).
De nition 12.1. Let u 2 K(X). We de ne 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 de nition

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:
n0
Proof. We note that
q1 [X]n 1 = 1 :
X
n! [n]

Therefore
X
c(u[n]) = c[u] exp(q1 [X])1

n0
= 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
;k0
()
(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 
;k0
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
n0 m1
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
n0 m1
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) :=
n0
The second equality is by Proposition 12.2. Therefore U satis es the di er-
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
m1
98

we want to show that S(t)1 satis es the same di erential equation. By
de nition
d S(t) = S(t)  X ( 1)m q [c(L)]tm :
m+1
dt m0
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 :
m1
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 :
m0

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
de nitions 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 suciently 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)
n0
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. Astrisque, vol 100.
e
n C fx; yg. Inventiones Math. 41, 45-89 (1977).
[2] J. Brianon, 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. Strmme, On the homology of the Hilbert scheme of points
in the plane. Invent. Math. 87 (1987), 343{352.
[5] G. Ellingsrud and S. A. Strmme, 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. Gttsche, The Betti numbers of the Hilbert scheme of points on a smooth pro-
o
jective surface. Math. Ann. 286 (1990), 193{207.
[9] L. Gttsche 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 ane 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 Gttsche
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 di erentiable invariants of smooth compact 4-manifolds X, de ned
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 identi ed 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 de nition 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 De nition 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 compacti cation . . . . . . . . . . . . . . . . . . . 109
2.5 De nition of the invariants . . . . . . . . . . . . . . . . . . . 110
2.6 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . 111
3 Algebro-geometric de nition of Donaldson invariants 112
3.1 Determinant line bundles . . . . . . . . . . . . . . . . . . . . 113
Construction of sections of L (nH) . . . . . .
3.2 . . . . . . . . . 114
1

3.3 Uhlenbeck compacti cation . . . . . . . . . . . . . . . . . . . 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
classi cation of compact di erentiable 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 di erentiable classi cation, ones needs ad-
ditional invariants. The Donaldson invariants are de ned via gauge theory
in terms of moduli spaces of anti-self-dual connections on di erentiable bun-
dles on X. If X is an algebraic surface, then these moduli spaces can be
identi ed 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 de nite intersection form. An
example where this happens is when X is the blow up of Y in a point.
2. The di erentiable classi cation of elliptic surfaces.
3. The Kodaira dimension of an algebraic surface is a di erentiable 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 di erentiable 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 De nition and properties of the Donaldson in-
variants
In this lecture we de ne 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
de ned via intersection theory on a moduli space of anti-self-dual connections
on P.
SU(2)-bundles on X are classi ed by their second Chern class c (P ), 2

and SO(3)-bundles on X are classi ed 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 identi ed 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 di erentiable 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 .
De nition 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 de nite.
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 Khler 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 de nes 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
de ne via moduli spaces of ASD-connections) and moduli of vector bundles.

2.4 Uhlenbeck compacti cation
We want to de ne 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 compacti cation:
110

Theorem 2.5 There exists a topology on
a
n)  X
Ng (c ; c (n)
1 2

n0
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 e ective 0-cycles on X of degree n.
2.5 De nition 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 DjXfDg =
D, then we can de ne 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 de ne  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 compacti cation
N g (E). For generic metric g, N g (E) is a strati ed 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 de ne 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
X;g := X X;g : A (X) ! Q :
c1 c1 ;d 
d0
By de nition 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 di erentiable 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
n0
De nition 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
(aa)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 de nition 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 compacti cation N := Ng(H) (C; c ). 2

Here g(H) is the Fubini-Study metric associated to H. As the Donaldson
invariants are de ned in terms of the Uhlenbeck compacti cation, 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!

De nition 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 (EjDM ))

det(EjfagM )
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 de ne
L := det(p G)

det(GjfagMD )
(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 de ne L (nH). The de nition 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 de ned
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 de ne 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 de ned on U(D). By de nition 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 compacti cation
L (nH)
m de nes a morphism
1

: M ! P(H (M; L (nH)
m )_ ):
0
1

Theorem 3.6 (M) is homeomorphic to the Uhlenbeck compacti cation
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 e ective
0-cycle of length k := c (E) c (E __ ) on X, i.e. a point in the
2 2

symmetric power X . (k)

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