ñòð. 5 |

0 ! F ! E ! G ! 0;

where F and G are rank 1 sheaves with degH (F ) = degH (G). We put

X

A(E) = F __ G__; l (F __ G__)=(F G) pp 2 X :

Z(E) = (k)

p2X

Claim: For E ; E 2 M we have (E ) = (E ) if and only if (A(E ); Z(E )) =

1 2 1 2 1 1

(A(E ); Z(E )). In other words: the sets (M) and N are equal.

2 2

116

We want to check the claim in a special case. Assume (E ) = (E ), 1 2

where E and E are -stable. Take D 2 jnHj general, then E jD =

E jD (otherwise, as L

m is very ample on MD , we can nd a section s 2

1 2 1

H (MD ; L

m ), such that 0 = s(E jD ) 6= s(E jD ). Then s(E ) = 0; s(E ) 6=

2 0

e e

0

1 2 1 2

0

0). The exact sequence

Hom(E __ ; E __ ) ! Hom(E jD ; E jD ) ! H (Hom(E __ ; E __ ( nH))) = 0 1

1 2

1 2 1 2

implies that Hom(E __ ; E __ ) 6= 0. The -stability of E ; E and therefore 1 2

also of E __ ; E __ then implies E __ = E __ . Now assume p 2 Z(E ) but p 62

1 2

1

Z(E ). Then we choose D 2 jnHj such that p 2 D and E jD is semistable.

1 2 1 2

Then E jD is not semistable and therefore we can nd s 2 H (MD ; L

m ),

2 2

0

1

such that s(E ) = 0 and s(E ) 6= 0.

0

e e

1 2

3.4 Donaldson invariants via algebraic geometry

X

Let again M := MH (C; c ) be the moduli space of Gieseker H-semistable

2

sheaves with Chern classes C and c . Assume that there is a universal sheaf2

E over X M. Write

d := 4c C 3(1 + pg (X)):

2

2

Let : H (X) ! H (M) be dened by 4

1

(a) := c (E) 4 c (E) =a; 2

2 1

(i.e. we write

c (E) 1 c (E) := i

i ; i 2 H (X; Q );

i 2 H (M; Q );

X

2

4

2 1

i

then X

hi ; ai

i ).

() =

i

Again is independent of the choice of a universal sheaf, and, if no universal

sheaf exists, can be dened without using it. We denote again by Ad (X)

the set of elements of degree d in Sym (H (X) H (X)), where the class p 0 2

of a point in H (X) is given weight 2 and a class in H (X) is given weight

0 2

1. For := a : : : ak 2 Ad (X), we dene

1

() := (a ) [ : : : [ (ak ) 2 H (M) 2d

1

117

Donaldson invariants in Algebraic Geometry

and Z

X;H

C;d () := ():

M

Theorem 3.7 [M],[Li] Under the conditions specied below we have

2 +KX C)=2 X;H

X;g(H) = ( 1)

:

(C

C;d C;d

Conditions:

1. Locally-free -stable sheaves are dense in M (otherwise replace M by

the closure of the locus of locally free sheaves).

2. Every L in Pic(S) n f0g with L C mod 2 and LH = 0 satises

L < (4c C ) (this means that H does not lie on a wall, see

2 2

2

below).

X

3. MH (C; c ) has dimension 4c c 3(OX ) and 2

2 2 1

X X

dim(MH (C; n)) + 2(c n) < dim(MH (C; c )); for all n < c :

2 2 2

4. If C 2 2H (X; Z) there is an extra condition; e.g. for 2 Symd (H (X)),

2

2

the condition is d > 2c C =2. 2

2

The point is that the classes () and () are related by : (()) =

(). Furthermore the fundamental classes of M and N are related by :

up to dierent sign convention ([M]) = [N]. Then the theorem follows

from the projection formula.

4 Flips of moduli spaces and wall-crossing for Don-

aldson invariants

In this and the next lecture we want to determine the dependence of the

Donaldson invariants on the metric in the case b = 1 when they indeed

+

depend on the metric. In this lecture we will restrict to the case of algebraic

surfaces. In this case the change of metric corresponds to a change of ample

X

divisor H. So we study how the moduli spaces MH (C; c ) vary with H. 2

We will nd out that under suitable assumptions, the variation is described

by an explicit series of blow ups and blow downs, with centers projective

bundles over Hilbert schemes of points. We use this to determine the change

118

of the Donaldson invariants as an explicit intersection number on a suitable

Hilbert scheme of points on X. Finally one can compute the leading terms of

this intersection number. We will follow mostly [E-G1]. A similar approach

can be found in [Fr-Q].

4.1 Walls and chambers

We start by reviewing general results about the dependence on the metric.

Let X be a compact simply-connected dierentiable 4-manifold.

In the case b (X) > 1 the Donaldson invariants X;g are independent of

C;d

+

the metric g (as long as it is generic). Now assume b (X) = 1. In this case

+

the Donaldson invariants will indeed depend on the metric g. Let H (X; R)

2 +

be the set of all 2 H (X; R) with > 0. In fact the Donaldson invariants

2 2

depend on g via a system of walls and chambers in H (X; R) . 2 +

We x C 2 H (X; Z) and d 2 Z0. The positive cone H (X; R) =R

2 2 + +

has two connected components

and

. A homology orientation (i.e.

+

the choice of an orientation on a maximal-dimensional linear subspace of

H (X; R) on which the intersection form is positive denite), which is needed

2

to dene an orientation on the moduli space of ASD-connections, is equiva-

lent to the choice of one of them, say

. +

Denition 4.1 Let g be a Riemannian metric on X. The period point

!(g) is the point in

dened by the one-dimensional subspace of g-self-

+

dual g-harmonic 2-forms. I.e these are the harmonic two forms 2

(X) 2

with g = . By the Hodge theorem this is a 1-dimensional subspace of

H (X; R). An element 2 H (X; Z) + C=2 is called of type (C; d) if

2 2

(d + 3)=4 + 2 Z0: 2

In this case

:= L 2

L=0

W

+

is called the corresponding wall of type (C; d). The chambers of type (C; d)

are the connected components of complement of the walls of type (C; d) in

.

+

It turns out that the Donaldson invariants with respect to the Fubini-

Study metric corresponding to H depend only on the chamber of the period

point of H.

119

Donaldson invariants in Algebraic Geometry

Theorem 4.2 [K-M]

1. X;g depends only on the chamber (of type (C; d)) of !(g).

C;d

2. For all of type (C; d) there exists a linear map ;d : Ad (X) ! C .

X

such that

X

X;g1 X;g2 = ( 1) C=4)C ;d :

X

(

C;d C;d

!(g2 2

)<0<!(g )

4.2 Interpretation of the walls in algebraic geometry

Now let X be a simply-connected algebraic surface with geometric genus

pg = 0 (this is equivalent to b (X) = (1 + 2pg (X)) = 1 ). Let H be an

+

ample divisor on X. Let C be the ample cone of X. We choose

as the +

connected component of H (X; R) =R , which contains C=R . Then the

2 + + +

period point of the Fubini-Study metric g(H) is !(g(H)) = R H 2 C=R . + +

Fix C 2 H (X; Z) and c 2 Z, such that d := 4c C 3 is a nonnegative

2 2

2 2

integer. By Section 3 we can compute X;g(H) on MH (C; c ). So we now

X

C;d 2

X

need to know how MH (C; c ) depends on H.

2

Let E be a torsion-free rank 2 sheaf on X with Chern classes C and c . 2

Let H and H be two ample line bundles on X, and assume that E is

+

Gieseker stable with respect to H , but Gieseker unstable with respect to

H . Then the Harder-Narasimhan ltration of E with respect to H gives

+ +

an exact sequence

0 ! IZ1 (F ) ! E ! IZ2 (G) ! 0;

where

1. The class := (F G)=2 satises

H < 0 < H : +

2. IZ1 and IZ2 are the ideal sheaves of 0-dimensional subschemes Z 2 1

X and Z 2 X and c (E) = FG + n + m or equivalently

[n] [m]

2 2

C =4 + = n + m 0:

c 2 2

2

This means that is a class of type (C; d) and there exists an ample line

bundle H between H and H with H = 0. In other words denes a wall

+

of type (C; d), such that W intersects C.

120

Denition 4.3 Let En;m be the set of all sheaves E lying in extensions

0 ! IZ1 (F ) ! E ! IZ2 (G) ! 0;

with := (F G)=2, Z 2 X , Z 2 X .

[n] [m]

1 2

Then we conclude

X

1. MH (C; c ) depends only on the chamber of type (C; d) of H. In par-

2

ticular the Donaldson invariants are constant on each chamber.

2.

X (C; c ) n M X (C; c ) a a E n;m :

MH H+

2 2

n;m

Here the sums run over all of type (C; d) such that H < 0 < H , +

and over all n; m 2 Z0 with n + m = c C =4 + . 2 2

2

X X

We would like to say that MH+ (C; c ) is obtained from MH (C; c ), by

removing the E and replacing them by the E m;n (for classes of type

n;m

2 2

(C; d) with H < 0 < H ). This however is not quite true. The problem

is that E and E l;r can intersect, i.e. we can have a diagram

n;m

+

0

?

?

y

IW1?(G)

&

?

y

! IZ1 (F ) ! ! IZ2 (G) ! 0

E

0 ?

& ?

y

IW2?(F )

?

y

0

To deal with this, we need a ner notion of stability. We use: Gieseker

stability is not invariant under tensorizing by a line bundle.

Assume H and H are separated by a unique wall W with H < 0 <

+

H . Let H lie between H and H with H = 0. If E is a torsion-free

+ +

H-semistable sheaf of rank 2 with Chern classes C and c , then 2

1. E is H -semistable, if and only if E(l(H H )) is H-semistable for

+

l 0.

121

Donaldson invariants in Algebraic Geometry

2. E is H -semistable, if and only if E(l(H H )) is H-semistable for

+ +

l 0.

This gives us a ner notion of stability. By using a parabolic structure of

length 1 (which essentially amounts to tensorizing with a fractional power

of H H ), we get moduli spaces

+

Ma ; a 2 [ 1; 1]; M = MH (C; c ); M = MH+ (C; c ):

X X

1 2 1 2

There are miniwalls ai 2 [ 1; 1] such that for all i and 0 < 1

Mai = (Mai n En;m ) q E m;n

+

for suitable n, m (for more details look at [E-G1]).

4.3 Flip construction

X

Now we want to see more precisely what happens to MH (C; c ) when H 2

crosses a wall. We want to see that its change can be described by a se-

quence of blow ups and blow downs. For this we need an additional assump-

tion which essentially guarantees that all the involved moduli spaces will be

smooth near the locus where they change.

Denition 4.4 A class of type (C; d) denes a good wall if W contains

ample divisors and 2 + KX and 2 + KX are not eective. In particular

if KX is eective, then all walls in the ample cone are good.

We want to describe the wall-crossing through a good wall dened by .

Let b be a miniwall, as above, and let

M := Mb ; M := Mb+ = (M n En;m) q E m;n :

+

We write

E := En;m ; E := E m;n :

+

Let T := X X , let Z ; Z S T be the two universal families i.e.

[n] [m]

1 2

f(x; Z; W) 2 X T j x 2 Zg

Z1 :=

:= f(x; Z; W) 2 X T j x 2 Wg:

Z2

122

Let q be the projection S T ! T. We write C := F + G, := F G=2

and write F := IZ1(F ), G := IZ2(G), (these are sheaves on X T and we

suppress the various pullbacks in the notation). Finally we write

A := Extq (F; G); A := Extq (G; F):

1 1

+

These are the relative Ext sheaves. Under our assumptions these sheaves

are locally free and the bers over a point (Z; W) 2 T are

A (Z; W) := Ext (IZ (F ); IW (G)); A (Z; W) := Ext (IW (G); IZ (F )):

1 1

+

We denote the tautological sub-line bundles on P(A ) and P(A ) by and

+

.

+

Lemma 4.5 1. A is locally free of rank (2 KS ) + n + m 1.

2. The tautological extension

0 ! F ! E ! G( ) ! 0

gives an isomorphism P(A ) ' E .

3. NE =M = A ( ). +

Proof. 1. follows from Riemann-Roch, because the condition of a good wall

implies Homq (F; G) = 0 and Extq (F; G) = 0.

2

2. is then easy.

For 3. we use that TM = Extq (E; E), where E is the universal sheaf on

1

X M . Then one has to do some diagram-chasing.

Part 3. of this lemma lets us hope that the blow up of M along E and

the blow up of M along E might be the same. So let M be the blow up

f

+ +

of M along E and M the blow up of M along E . Let D ' P(A ) T

f + + +

P(A ) be the exceptional divisor. We see that O(D)jD = OD ( + ).

+ +

Theorem 4.6 M = M .

f f +

Proof. Let E be the universal family on X M (and denote by the

same symbol its pullback to M ). Let E be the kernel of the composition

f +

E ! E jD ! G( )D , where G( )D is the pullback of G() from S T

to S D. So we dene E via elementary transform along the exceptional

+

divisor D. Then we check that E is a b + stable family for 1 > 0.

+

123

Donaldson invariants in Algebraic Geometry

Thus E denes a morphism M ! M . It is not dicult to check that it

f

+ +

is the blow up along E . +

X X

So we see that MH+ (C; c ) = M is obtained from MH (C; c ) = M via

n;m

2 1 2 1

a sequence of blow ups along smooth subvarieties of the form E followed

by a blow up of the exceptional divisor in another direction to E m;n .

4.4 Computation of the wall-crossing

X

Now we want to compute the wall-crossing terms ;d . For simplicity we

restrict to Symd (H (X)). Let a 2 H (X). Let b run through the miniwalls

2 2

corresponding to and write Mb for the blow up Mb = Mb+ . From the

f f f

denitions we see that

!

Z Z XZ

;d (ad ) =

X (a)d (a)d = ( (a)d (a)d ):

X X +

MH+ MH Mb

f

2 2 b

(C;c ) (C;c )

Here (a) := (c (E ) c (E ) =4)=a, and similarly for . (The E and

2

2 1 +

E and therefore the and also depend on b.)

+ +

Let us again put ourselves in the situation of the previous section: M is

f

the blow up of M along E and M the blow up of M along E , and D

f + + +

is the exceptional divisor.

Lemma 4.7 1. (a) (a) = h; aiD.

+

R

2. M ( (a)d (a)d ) is the evaluation of a suitable (explicitly com-

f +

putable) cohomology class on X X . [n] [m]

Proof. 1. Is an easy application of Riemann-Roch without denominators see

[Fu] (which tells how to compute the Chern classes of sheaves supported on

subvarieties).

2.

(a)d (a)d = ( (a) (a))( (a)d + : : : + (a)d ) 1 1

+ + +

is by 1. divisible by D, so we can view it as a class on D. We can then push

the class from D down to T.

Putting all this together and summing over all the miniwalls corresponding

to a given wall we obtain the following: Note that the various X X [n] [m]

`

with m + n = l can be collected to (X t X) = n+m=l X X ). [l] [n] [m]

124

Theorem 4.8

d

2b d h; aid b

X

;d (ad ) =

X

b

b=0

Z

b (Extq (IZ1; IZ2

(O( 2) O( 2 + KX )))):

b s 1

[l]

2l

tX)

(X

Here p and q are the projections of X (X t X) to X and (X t X) [l] [n]

respectively and l := c C =4 + . Z and Z X (X t X) are the

2 2 [n]

2 1 2

universal families

f(x; Z; W) 2 X (X t X) j x 2 Zg;

Z1 := [l]

:= f(x; Z; W) 2 X (X t X) j x 2 Wg:

Z2 [l]

si(E) denotes the i-th Segre class, dened by

1 + s (E) + s (E) + : : : = 1=(1 + c (E) + c (E) + : : : )

1 2 1 2

and := q (p ([Z ] + [Z ])): So we are reduced to a (very complicated)

1 2

intersection computation on the Hilbert scheme of points on X. The inter-

section theory of X is in general not understood. It gets harder very fast

[n]

as n grows. So in our case the diculty of the computation depends on the

number l := c C =4+ . The intersection number above can be computed

2 2

2

for l not too large, say l 3. For l = 0 we get for instance

;d (ad ) = h; aid :

X

There is an alternative way of carrying out the nal step of the compu-

tation, i.e. the computation in the cohomology ring of the Hilbert scheme

of points. Assume X is a blow up of P . On P we have actions of C with

2 2

nitely many xpoints. We can do the blow up in such a way that X still

carries an action of C with nitely many xpoints (at each step it is enough

to only blow up xpoints). This action lifts to an action on X , which [n]

still has only nitely many xpoints. All the intersection numbers we have

to compute for the wall-crossing are indeed intersection numbers of Chern

classes of equivariant bundles for this action.

We can therefore apply the Bott residue formula. This allows us to com-

pute the intersection numbers by looking at the weights of the action on the

bers of the equivariant bundles over the xpoints. This gives an algorithm

125

Donaldson invariants in Algebraic Geometry

for computing the wall-crossing for rational surfaces. We used this in [E-G2]

to compute the Donaldson invariants of P of degree 50.

2

The fact that we can compute the Donaldson invariants of P , where there

2

are no walls might seem surprising. We use the blow up formulas (see the

next lecture) to relate the Donaldson invariants of P to those of the blow

2

up of P in a point. On this blow up we can then apply the wall-crossing in

2

order to do the computation.

5 Wall-crossing and modular forms

Let X be a simply-connected 4-manifold with b (X) = 1. In this case the

+

Donaldson invariants were rst studied in [K]. In this lecture I want to give

X

a generating function for the wall-crossing terms ;d . We will see that such

a generating function can be found in terms of modular forms. This is the

contents of the paper [G]. The strategy will be to compare the wall-crossing

an X and on the connected sum of X with P with the opposite orientation.

2

X

This will give us recursion formulas for the ;d .

5.1 Ingredients

There are several ingredients which have to be put together in order to

compute the generating function.

(1) Kotschik-Morgan conjecture. In their paper [K-M], where they

X;g

show that the Donaldson invariants C;d depend only on the chamber of the

period point of the metric g, Kotschik and Morgan also make a conjecture

about the structure of the wall-crossing terms ;d . For a class 2 H (X)

X 2

and a 2 H (X), we denote by h; ai the pairing of H (X) with H (X) and

2

2 2

by (a a) the intersection form on the middle homology H (X). 2

Conjecture 5.1 [K-M] ;d(ad ) is for a 2 H (X) a polynomial in h; ai and

X 2

(a a), whose coecients depend only on , d and the homotopy type of X.

2

In a series of papers Fehan and Leness are working on a proof of this con-

jecture [Fe-Le1],[Fe-Le2-4].

(2) Blow up formulas. The blow up formulas relate the Donaldson

invariants of a 4-manifold X with those of the connected sum X := X#P of

b 2

X with P with the opposite orientation. In the case that X is an algebraic

2

126

surface, we can take X to be the blow up of X in a point. In the case

b

b (X) = 1, when the Donaldson invariants depend on the choice of a metric,

+

we need to choose the metric on X to be very close to the pullback of a metric

b

on X, in order to make the blow up formulas applicable. Let E be the class

of the exceptional divisor, then H (X; R) = H (X; R)R E. We will identify

b 2 2

H (X; R) with the classes in H (X; R) orthogonal to E.

b

2 2

If L 2 H (X; R) is (a representative of) the period point of the metric

2 +

g, we write

X;L := X;g :

C;d C;d

For C 2 H (X; Z), H 2 H (X; R) , we write

b 2 2 +

X;H := X;H E

b b

C;d C;d

for 0 < 1. (This will be independent of for suciently small > 0.)

Theorem 5.2 Let C 2 H (X; Z), a 2 H (X), H 2 H (X; R) . We write

2 2 +

2

e 2 H (X; Z) for the Poincar dual of E. Then

e

b2

1. X;H (ad ) = X;H (ad ).

b

C;d C;d

2. X;H (e ad ) = X;H (ad ).

b

C+E;d+1 C;d

3. X;H (e ad ) = 0.

b 2 2

C;d

This result holds also if b > 1. In fact this is the case in which it

+

was originally proved. In the case b > 1 the Donaldson invariants are

+

independent of the metric, so one does not have to worry about which metric

to choose on X for a given metric on X.

b

More generally Fintushel and Stern [F-S] found generating functions for

the blow up formulas: Let p 2 H (X) be the class of a point. Then there

0

are power series

X

B(x; t) = Bk (x)tk ;

k

X

Sk (x)tk ;

S(x; t) =

k

such that

X;H (ad ek ) = X;H (ad Bk (p));

b

C C

X;H (ad ek ) = X;H (ad Sk (p)):

b

C+E C

127

Donaldson invariants in Algebraic Geometry

B(x; t) and S(x; t) can be expressed in terms of elliptic functions, e.g. S(x; t) =

e t2 x=6 (t), where is the Weierstrass function.

(3) Vanishing results. The last ingredient is that in certain cases (for

rational ruled surfaces) the Donaldson invariants vanish. This will give a

starting point for the calculations.

Lemma 5.3 Let X be a rational ruled surface. Let F be the class of

a ber and assume CF = 1. Let H be an ample divisor on X. Then

MF+H (C; c ) = ; for 0 < 1. In particular X;F+H = 0 for 0 < 1.

X

C;d

2

More generally the following holds: Let f : X ! C be a surjective mor-

phism of an algebraic surface to a curve. Let F be the class of a ber and

let H be ample on X. Then a vector bundle E over X is semistable with

respect to F +H if and only if the restriction of E to the generic ber of f is

semistable. This fact is also e.g. used by Friedman to study the Donaldson

invariants of elliptic surfaces.

5.2 The result

Our aim is to show:

Theorem 5.4 Let a 2 H (X) and let t be a variable. Then

2

h; ait (a a)G()t

X (exp(at)) = Coe 0 f()R()()(X) q 2 =2 exp

2

:

q

f() f()

2

Here (X) is the signature of X. For the rest of the notations I brie

y

review modular forms.

Review of modular forms: Let H := 2 =() > 0 be the

C

complex upper half plane. For in we denote q := e . The group

H 2i

SL (Z) acts on H by

2

a b = a + b :

cd c + d

A function g : H ! C is called a modular form of weight k for SL (Z), if 2

a+ b = (c + d)k f(); for all a b 2 SL (Z);

g c + d cd 2

128

and furthermore g has a q-development

1

X

anqn:

f() =

n=0

One can associate an elliptic curve E := C =(Z + Z ) to 2 H , and E

and E 0 are isomorphic if and only if and 0 are related by an element of

SL (Z). Therefore modular forms are related to moduli of elliptic curves.

2

One can also talk about modular forms for subgroups of nite index of

SL (Z). In this case one requires the transformation behavior only for the

2

elements in and the requirement on the q-development has to be modied.

All the functions appearing in the theorem are (related to) modular forms.

1

Y

(1 qk )

() := q 24

k=1

is the discriminant, a modular form for SL (Z). 2

() = () 1=24

is the Dirichlet eta-function.

X2

() := qn =2

n2Z

is the theta function for Z.

1

1 + X X dqn

G () := 24

2

n=1 djn

is an Eisenstein series and

1

1 + X X d q n ;

e () := 12

3

n=1 djn;d odd

the value of the Weierstrass }-function at one of the two-division points. We

put

() 2

R() := (=2)(2) ;

i=4 ()

3

f() := e () ;

G() := G () + e ()=2:

2 3

129

Donaldson invariants in Algebraic Geometry

As a corollary to this result we can compute all the Donaldson invariants of

the projective plane P . The projective plane is in some respects the simplest

2

algebraic surface. Therefore, if one wants to understand the Donaldson

invariants of algebraic surfaces, one should at least be able to compute them

for P .

2

Let H 2 H (P ; Z) be the hyperplane class, and let h be its Poincar dual.

e

2

2

Corollary 5.5

"

P2;H ( exp(ht)) = Coeq0 f()R()

H

#

G()t

(n + )t

X 1 1 ((aa) 1 )2 ) 1

( 1)n+ 4 q 2

2

:

2 exp

(n 2

f() f()2

an>0

There is a similar formula for P2;H . 0

Proof. The blow up X of P at a point is a ruled surface, the class of the

ber is F = H E. So we get X;F+H = 0 by the vanishing result above.

2

H

On the other hand the blow up formulas give that P2;H (hd ) = X;H (hd );

H H

X

and the last can be computed by adding all the wall-crossing terms ;d for

all classes of type (H; d) with H > 0 > F:

5.3 Proof of the theorem

Now I want to sketch the proof of the theorem. The idea is as follows: We

want to relate the wall-crossing on X and its blow up X. So x C 2 H (X; Z)

b 2

and let dene the only wall of type (C; d) on X between H and H . +

Instead of directly applying the wall-crossing formula for the wall W , we

can also rst apply the blow up formulas, then cross all the walls between

H and H on X and then apply the blow up formula again to get back to

b

+

X

X. This gives us two dierent ways to compute the wall-crossing term ,

which will give us recursion formulas.

By denition we see that the classes of type (C; d) on X with H <

b

0 < H are precisely the

+

= + nE; n 2 Z; n (d + 3)=4 + ;

2 2

130

and the classes of type (C + E; d + 1) are precisely the

= + (n + 1=2)E; n 2 Z; (n + 1=2) (d + 4)=4 + : 2 2

We write X

X X

;d :

:=

d0

Then together with the above discussion the blow up formulas give:

X (ad ) = X X (ad );

b

(5.0.1)

+nE

n2Z

X (ad ) = X( 1)n X d

b

+(n+1=2)E (e a ); (5.0.2)

1

n2Z

Xb

0 = +nE (e ad ):

X (5.0.3)

2 2

n2Z

Now we use the Kotschick-Morgan conjecture. Let X(b) be the blow up

of X in b points. The Kotschick-Morgan conjecture allows us to write

X(b) (ad =d!) = X h; ail (a a)k P(l; k; b; );

2

l! k!

l+2k=d

for universal constants P(l; k; b; w) for l; k; b 2 Z, w 2 Z=4. Then the rela-

tions (5.0.1), (5.0.2), (5.0.3) imply in turn

X

P(l; k; b; w) = P(l; k; b + 1; w n ); (5.0.4)

2

n2Z

X

( 1)n (n + 1=2)P (l; k; b + 1; w (n + 1=2) );

P(l; k; b; w) = 2

n2Z

(5.0.5)

X X

n P(l; k; b; w n ) = 2 P(l; k; b; w n ): (5.0.6)

2 2 2

n2Z n2Z

Now we put

l kb

w=2 L Q t ;

X

P(l; k; b; w)q

X := l!k!b!

l;k;b;w

for variables q; L; Q; t. We see that we have encoded all the information

about the wall-crossing formulas into the generating function X . So our

task is to determine X explicitly.

131

Donaldson invariants in Algebraic Geometry

The formulas (5.0.4), (5.0.5), (5.0.6) for the P(l; k; b; w) translate into the

following dierential equations for X .

@

() @t X = X ;

@@

() @L @t X = X ;

3

@ = q @ () @ : 2

2() @Q X @L X

@q 2

These dierential equations are trivial to solve: Writing

X (q) := X (0; 0; 0; q)

we get

L Q G() + t (q):

X = exp X

f() f() 2

Finally we need to determine X (q). It is enough to do this in the case

X = P P : For every simply-connected 4-manifold with b = 1, the

1 1 +

blow up Y of X in two points is homotopy-equivalent to the blow up of

P in a number of points. The Kotschick-Morgan conjecture says that

P1 1

the wall-crossing terms only depend on the homotopy type of X.

Let F and G be the bers of the two projections of P P onto its factors.

1 1

By the vanishing result we get

P1P1;F+G = P1P1;G+F = 0:

F+G;d F+G;d

Therefore the sum of all the wall-crossing terms for all the walls between

F and G has to vanish. This is enough to determine all the coecients of

P1P1(q). This part of the calculation is slightly more dicult and involves

some tricks with modular forms.

5.4 Further results

This result has later been used in [G-Z] to prove structure theorems (like

those of Kronheimer and Mrowka in the b > 1 case) also for manifolds

+

with b = 1. These results work when one takes the limit of the Donaldson

invariants X;H as H tends to a class F with F = 0. We write X;F for

+

2

C C

this limit.

132

We get for instance the following: Let X be a rational elliptic surface (i.e.

the blow up of P in the 9 points of intersection of two smooth cubics). Let

2

F be the class of a ber. Then for all a 2 H (X) we get

2

2 =2

e

X;F (eat (1 + p=2)) =

(aa)t

cosh(hF; ait) :

H

To prove such results, one has to sum over all walls between two classes F, G

with F = G = 0. These sums organize themselves into theta functions, and

2 2

somewhat complicated arguments with modular forms will give the result.

133

Donaldson invariants in Algebraic Geometry

References

[D1] S.K. Donaldson, Polynomial invariants for smooth four-manifolds,

Topology 29 (1990), 257{315.

[D2] S.K. Donaldson, The Seiberg-Witten equations and 4-manifold

topology, Bull. Amer. Math. Soc. 33 (1996), 45-70.

[D-Kr] S.K. Donaldson and P. Kronheimer, The Geometry of four-

manifolds, Oxford Mathematical Monographs, Oxford University

Press, Oxford, 1990.

[D-N] J.-M. Drezet, M.S. Narasimhan, Groupe de Picard des varits de

ee

modules de brs semistables sur les courbes algbriques, Invent.

e e

Math 97 (1989), 53{94.

[E-G1] G. Ellingsrud and L. Gttsche, Variation of moduli spaces and

o

Donaldson invariants under change of polarization, J. reine angew.

Math. 467 (1995), 1{49.

[E-G2] G. Ellingsrud and L. Gttsche, Wall-crossing formulas, Bott

o

residue formula and the Donaldson invariants of rational surfaces,

Quart,J. Oxford 49 (1998), 307{329.

[Fe-Le1] P. Feehan and T. Leness, Donaldson invariants and wall crossing

formulas, I: Continuity of gluing maps, preprint math.DG/9812060

1999.

[Fe-Le2-4] P. Feehan and T. Leness, Homotopy invariance and Donaldson

invariants, II, III, IV, in preparation.

[F-S] R. Fintushel and R.J. Stern, The blow up formula for Donaldson

invariants, Annals of Math. 143 (1996), 529{546.

[Fr-Q] R. Friedman und Z. Qin, Flips of moduli spaces and transition

formulas for Donaldson polynomial invariants of rational surfaces,

Communications in analysis and geometry 3 (1995), 11-83.

[Fu] W. Fulton, Intersection Theory, Springer Verlag 1984.

[G] L. Gttsche, Modular forms and Donaldson invariants for 4-

o

manifolds with b = 1, J. Amer. Math. Soc. 9 (1996), 827-843.

+

134

[G-Z] L. Gttsche und D. Zagier, Jacobi forms and the structure of Don-

o

aldson invariants for 4-manifolds with b = 1, Selecta Math. 4

+

(1998), 69{115.

[H-L] D. Huybrechts and M. Lehn, The geometry of moduli spaces of

sheaves. Aspects of Mathematics E 31, Viehweg, 1997.

D. Kotschick, SO(3)-invariants for 4-manifolds with b = 1, Proc.

[K] +

London Math. Soc. 63 (1991), 426{448.

D. Kotschick und J. Morgan, SO(3)-invariants for 4-manifolds with

[K-M]

b = 1 II, J. Di. Geom. 39 (1994), 433{456.

+

[Kr-Mr] P. Kronheimer und T. Mrowka, Embedded surfaces and the struc-

ture of Donaldson's polynomial invariants, J. Di. Geom. 33 (1995),

573-734.

[LP] J. Le Potier, Fibr determinant et courbes de saut sur les surfaces

e

algbriques, Complex Projective Geometry, London Mathematical

e

Society: Bergen (1989), 213-240.

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invariants, J. Di. Geom. 37 (1993), 417{465.

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488.

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1 (1994), 769{796.

Holomorphic bundles over elliptic manifolds

John W. Morgan

Department of Mathematics, Columbia University,

2990 Broadway, 509 Mathematics Building, New York NY 10027, USA

Lecture given at the

School on Algebraic Geometry

Trieste, 26 July { 13 August 1999

LNS001004

jm@math.columbia.edu

Contents

1 Lie Groups and Holomorphic Principal GC-bundles 139

1.1 Generalities on roots, the Weyl group, etc. . . . . . . . . . . . 140

1.2 Classication of simple groups . . . . . . . . . . . . . . . . . . 142

Groups of Cn-type . . . . . . . . . . . . . .

1.3 . . . . . . . . . . 145

Principal holomorphic GC -bundles . . . . .

1.4 . . . . . . . . . . 148

Principal G-bundles over S 1 . . . . . . . . .

1.5 . . . . . . . . . . 150

Flat G-bundles over T 2 . . . . . . . . . . .

1.6 . . . . . . . . . . 152

1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2 Semi-Stable GC-Bundles over Elliptic Curves 155

2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

2.2 Line bundles of degree zero over an elliptic curve . . . . . . . 156

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