ñòð. 2 |

')

1r-1 0)

( 010

001

in SL(3, Z) ˜ Aut T3. The Sp,q are again elliptic surfaces and, as was point˜,'

out by Kodaira [11 ], are homotopy equivalent to S if p + q is even. In particul˜:

b+(Sp,q) = 3 and, for k 2: 4, Theorem 1 gives an invariant qk,SPJf' which is'˜

multilinear function of degree d = 4k - 6. Friedman and Morgan have announced

partial evaluation of these invariants, for k > 4 using a description due to Fried -.

of the moduli spaces of stable holomorphic bundles over Sp,q, for a suitable Hod ˜

metric [12],[13]. Friedman's description of the moduli spaces starts from an analys

of the restriction of bundles on Sp,q to the fibres of the elliptic fibration. On

fibre the bundle is either decomposable into a connected sum, or an extension

the trivial bundle by itself. The first condition is open and for a bundle whi:,

decomposes on the generic fibre the choice of a factor in such a decompositi,

defines a double branched cover of the surface; then the bundle can be recover:

using the second construction mentioned above. Friedman also analyses the oth,

bundles by the first technique, using extensions, and these turn out to have few

moduli. In this way Friedman is able to obtain a very general and quite detail,'

˜

description of the moduli spaces.

To state the result of Friedman and Morgan we regard the invariants as polynom˜:

functions on the homology of the 4-manifolds. There are two basic such functio .

the intersection form Q - viewed as a quadratic polynomial, and the linear functio,

.....

˜p,q : H 2 (Sp,q) Z,

--+

given by pairing with the cohomology class -Cl(Sp,q). The Yang-Mills invari '

can be expressed as polynomials in Q and ˜p,q and have the form:

I

+'"" a 'Q[l-i] ˜2i .

qk , SP,f = (pq) Q[ij

(5) p,q

L.J I

;=1

Here 1= (d/2) = (l/4)dim Mk, and we have written Q[lJ for the "divided pow':

(l/l!)Q'. The formulae depend a little on ones choice of conventions for multipli

tion in the ring Symx,z: explicitly we have, for example:

21

Donaldson: Yang-Mills invariants of four-manifolds

lu formula (5) the ai are unknown integers. Friedman and Morgan deduce

Ulr

"t"t1 this partial calculation that the product (pq) is a differentiable invariant of

Ilua ;t lunnifold Sp,q. In particular there are infinitely many diffeomorphism types

,""UJ;i('cl within the one homotopy ( or homeomorphism) class. (Indeed Friedman

It.. t Mor˜an work much more generally, considering all simply connected elliptic

.\" rn.Â·.'H with b+ ˜ 3.)

AU"'."I'",1 question to ask is to what extent such information can be derived without

t.o algebraic geometry and explicit descriptions of the moduli spaces. One

"Ii'U'I'HC'

MhvhHIH approach to this is to think about the differentiable description of the

Ittl"dt.hluic transformation in terms of cutting and pasting along a 3-torus. One

"I" ..., t. n. good theoretical understanding of the effect on the Yang-Mills invariants

ttl ."u,1I (Â·utting and pasting operations across a homology 3-sphere using Floer's

",.ta"ll.oll homology groups. One would like to have an extension theory of this to

tRtu'r ".'Â·llC'ral 3-manifolds, like the 3-torus. The author has been told that analytical

,.."It.. iu t.his direction have been obtained by T. Mokwra.

A"uUu'" p;cneral problem is to find if there are any other simply connected 4-

Ittlu'l'olclH, beyond the connected sums and algebraic surfaces noted above. We

PIli nt.l,,"ution to a very interesting family of 4-manifolds, which provides at present

1tt1"IV c'u.ndidates for such examples. These candidates are obtained by starting

wUIt " Coolllplex algebraic surface X defined over the real numbers, so there is an

'flU I"tlcullorphic involution u : X ˜ X with fixed point set a real form X R of X -

â€¢ tf'1\1 u,lJ((\hraic surface. We let Y be the quotient space X/u, which naturally has

thtt "',I',.cÂ·f.l1re of a smooth manifold ( since the fixed point set has real codimension

,) If.\' is simply connected and XR is non-empty the quotient Y is also simply

MttuuÂ·ltl.r'd and its' classical numerical invariants can be found from the formulae

+ X(XR)

b+(Y) = pg(X) , 2X(Y) = X(X) Â·

U,fhâ€¢â€¢ hltckwards, the manifold X can be recovered as the double cover of Y,

It.aurl,,Â·c1 over the obvious copy of XR.

˜r'd. "oIlHtruction has been used by Finashin, Kreck and Viro in the case when X

,. " I )â€¢â€¢I,,;u.chev surface ( with Pg = 0 ). In this case the quotient Y does not give a

ItllW .Illlc'n'utiable structure - for a suitable choice of q it is diffeomorphic to the 4-

Itth"tf" Illst,ead they show that the branch surfaces give new exotic knottings in 8 4

˜ 1\ I f'f'k 's article in these Proceedings. A similar picture holds if we take X to be

It..˜ .. f 1.1 .., tnanifolds 8p ,q considered above. First, if (8,u) is a K3 surface with anti-

11 phie involution then S/u is one of the standard manifolds: S2 X 52 or X1,p.

h..h;f˜tll Yan's solution of the Calabi conjecture gives a u- invariant hyperkahler

ttttal tte' 4 H. S, compatible with a family of complex structures. It is easy to see that

"'" I, l4"fiP(\(˜t, to one such complex structure, J say, the map q is a holomorphic

ht".,lul.iotl of S. (This complex struct˜re is orthogonal to the original one present in

ntH f˜l(plicit. complex description of S.) Then J induces a complex structure on the

Donaldson: Yang-Mills invariants of four-manifolds

22

quotient space T = (5/u), such that the projection map is a holomorphic branch'

cover. But it is a simple fact from complex surface theory that if a K3 surface

a branched cover of a surface T then T is a rational surface; hence S/ u is ratio .

and so difFeomeorphic to 52 X 52 or some Xl ,fJ.

Now the argument of Finashin, Kreck and Viro shows that the quotient of a 10

1

rithmic transform 5 p ,q by an anti-holomorphic involution is again diffemorphic

one of these standard manifolds. By this means one can get "knotted compi

curves" in, for example, S2 x B2, i.e. embedded surfaces homologous to a compI.

curve of the same genus, but not isotopic to a complex curve.

While we do not obtain any new manifolds by this quotient construction in t

two cases considered above, in more general cases the problem of understanding t

diffeomorphism type of the quotient seems to be quite open. An attractive feat ,

of this class of manifolds is that one can still hope to get some explicit geometri ˜

information about the Yang-Mills solutions. The anti-holomorphic involution u,

X induces an anti-holomorphic involution 0- of the moduli spaces Mk,X. Recen

S-G.Wang has shown that the moduli space Mj,Y can essentially be identified wit'

component of the fixed-point set of q in M 2 j,x ( the "real" bundles over X ). On t;

other hand these real bundles can, in principle, be analysed algebro-geometric

SECTION THREE, TORSION INVARIANTS

(a) More cohomology classes. The theory outlined in Section Two can be

tended in a number of directions. In this Section we will consider one such extensi '

where we define additional invariants which exploit the torsion in the homology'

the space of connections. This extension was greatly stimulated by conversati

with R.Gompf during the Durham Symposium, and for aditional background \

refer again to Gompf's article in these Proceedings.

Our starting point is the following question : does the connected sum of a pair!:'

algebraic surfaces decompose into "elementary" factors? For example, can we s .

off an 52 X S2 summand ? The invariants we have defined so far are not at all us

for these problems, since they are trivial on such connected sums. So we will 11J'

look for finer invariants, which will not have such drastic "vanishing" propertÂ·,

These invariants use more subtle topological features. ,

As we explained in Section Two the rational cohomology of the space B* of equ

alence classes of irreducible connections on an SU(2) bundle over a compact,(

manifold X is very simple. The integral cohomology of B* the other hand, is m '

more complicated. For example, consider the case when X = 8 4 and, as in Sec

2, let' 8 be the space of "framed" connections - homot˜py equivalent to an SO'

bundle over Brtt. This basic example was discussed in detail by Atiyah and Jo'

[1]. The space 8 is homotopy equivalent to n3 5 3 - the third loop space of S3.

rational cohomology is trivial, but the cohomology with finite co-efficient gro˜p

very rich. Many non-zero homology classes are detected by a virtual bundle wh˜

corresponds, in the framework of connections, to the index of the family of coup)"

23

Donaldson: Yang-Mills invariants of four-manifolds

Ol"nc' operators parametrised by B. In general on an arbitrary spin 4-manifold X

WfI.'nll use the Dirac family to construct corresponding classes, as in [7]. One can

'h..u ˜() on to consider the problem of pushing these classes down to 8*. For our

Iltl,UcÂ·nt.ion below we want a certain class u E H 1 (8*; Z/2), or equivalently a real

It".. 1.'llulle ." over 8*. This is defined when the Chern class k of the bundle P over

Xw,Â· 'U'(˜ considering is even. We recall the construction from [7]. Over A there is

â€¢â€¢1˜t,.Â·nllinant line bundle Tj with fibres

= AmazokerDA Â® AmazokerD'A,

11A

WIHtl'C˜

DAis the Dirac operator coupled to A via the fundamental representation

.., ˜HJ(2), and regarded as a real operator. This admits a natural action of the

I,ulc˜ KI"OUP Q; and the element -1 in the centre of Q (which acts trivially on A)

I.th. tL'" (_l)indDA on the fibres of fi. On the other hand the numerical index of the

"uUI.I,4cl operator compares with that of the ordinary Dirac operator D by

= k + 2 ind D.

ind DA

(tl'll(' fa.ctor 2 appears here as the dimension of the fundamental representation.)

" râ€¢â€¢llt)ws that (-1) E Q acts trivially on ij precisely when k is even, and in this case

UtI' ItUlldlc descends to a line bundle 7J ˜ 8*. We then put u = Wl(7]).

(It) Additional invariants.

('''''Mld.'1" first a general case where we have a cohomology class 8 E H8(8*; R), for

"'Utti ('o-cfficient group R, and a Yang-Mills moduli space Mit: C 8* of dimension

It If WfÂ· (˜an construct a natural pairing between a fundamental class of the moduli

II'R."˜ ,u1d 8 we obtain a n\llllerical invariant of X. We recall that in Section 2 such

I I..,˜ri ..p; could be obtained, when 8 is a product of classes p(a), by extending

th" f'oluuIlology classes to the compactified space Mit: which carries a fundamental

1....lIolo.u:y class once k is large. It seems that this approach cannot be extended,

wtt.llout reservation, to all the cohomology of B* .(It is certainly not true that all

Ulf1 4 olu)(nology of 8* extends to Mk.) However, 88 we shall now show, it can be

t

.'RII tf'( 1t.hrough when the class 8 contains a large enough number of factors of the

tU11i1 11.( (\˜). Suppose then that the virtual dimension s of the moduli space under

t!.... t1idtÂ·rntion has the form s = 2d+r, where r = 1 or 2 and 8 is a cohomology class

ur t ltc' Hilape :

wllf'n˜ Â¢ E HT(8*; R). To construct a pairing between 8 and the moduli space we

tt" Wf'('d as follows. As in the second construction of Section 2 we let Vi, ... ,Vd be

t IlIlilUC'llsion 2 representatives for the.J.t(ai), based on surfaces E i in X, and chosen

Then the

!if" t.hnt. all multiple intersections are transverse to all moduli spaces.

lftftÂ·"N(˜(Â·tion :

24 Donaldson: Yang-Mills invariants of four-manifolds

is an r-dimensional, oriented, submanifold of Mk C 8*. If I is compact we

evaluate the remaining factor q, on I to obtain an invariant in the co-efficient gr ,

R.

The argument to show that I is compact, when k is large, is just the sam'e

that used in the basic case (when r = 0) considered in Section 1. In terms of ,

compactified space, we exploit here the fact that the lower strata have codimensÂ·

at least 4, and this is where the hypothesis r :5 2 enters. In fact, at this point

only need r :5 3. In detail; suppose that [Aal is an infinite sequence in I. '] â€¢

a subsequence we may assume that it converges to ([A], (Xl, ... ,Xl) in M ". Th

are at most 21 of the surfaces which contain one of the points Xi, so [A] must lie

at least d - 21 of the Vj. H I = k, so A is flat, [A] does not lie in any of the Vj'l

in this case we must have d:5 2k i.e.4k :5 3(1 + b+(X)) + r. So if we assume th'

4k > 3(1 + b+(X)) + r

this case does not occur. On the other hand if I < k the dimension of M,,_

2d + r - 81 and this must be at least 2(d - 21), since A lies in d - 21 of the Vj. He;

r ˜ 4/, and since r :5 3 we must have 1 = o. So A is a limit point of the seque:

in I. :

Now a similar argument involving families shows that for any two generic metÂ·

on X, or choices of Vj, the intersections are cobordant in 8*. This is where we n'

to use the assumption that r :5 2, since we introduce an extra parameter into

"dimension counting" .It follows then that the pairings are the same. Finally le˜

note that the group of orientation- preserving self- homotopy equivalences of X .

naturally on the cohomology of 8*. For simplicity we suppose that the class ˜

fixed by this action, we just call such a class an invariant class . Then to sum"

we obtain

7. Let X be a compact, smooth, oriented,and simply connecte

THEOREM

manifold with b+(X) > 1. Let q, be an invariant class in Hr(B*, R) for r

If 4k > 3(1 + b+(X) +r and the dimension s = 8k - 3(1 + b+(X) equals 2d + r t,

the map

H 2 (X : Z)

q",q"X : H 2 (X; Z) xÂ·Â·Â· R,

--+

X

given by q",q"x ([E I ], ... , [Ed]) =< 4> ,M" n VI n V2 nÂ· Â· Â·n Vd > defines an eleme ':,

Sym˜ R which is (up to sign) a differential-topological invariant of X, natural .

respedt to orientation preserving diffeomorphisms.

(c) Loss of compactness. Unfortunately, the author does not know any inter ˜

ing potential applications for the invariants of Theorem 7. So we now go furt

and see what can be done if we take r = 3 in the set-up above. For definiteness

25

Donaldson: Yang-Mills invariants of four-manifolds

nuw lix the class 4> E H 3 (8*; Z/2) to be u 3 , the cup-cube of the class u described

In (,,). Thus we should assume that X is spin and that the Chern class k is even,

' 'Ii clilllcnsion formula shows that we must then have b+(X) even. The essential

r.rl. ",hout this class u 3 is that it can detect the "glueing parameter" which appears

wh"11 W(˜ join together instantons over two different regions, after the fashion of our

˜IIIIIU'(Â·t.(˜d sum construction in 2(b). We shall use this fact twice below so we will

.".˜ .. II the main point now. Consider a pair of irreducible connections A 1 ,A2 on

hUllclI.'H Pl,P2 over spin manifolds X 1 ,X2 â€¢ Let the Chern classes of the bundles be

˜I t A"˜' with k1 + k2 even. Flattening the connections in small balls we construct a

p"u,uÂ·cÂ·t.ion Ao(p) for each gluing parameter p, and in this way we obtain a family

ttl ll."'''˜(˜ -equivalence classes of connections over the connected sum parametrised

lty S( J( ˜l). Up to homotopy this family is independent of the particular connections

A" ur t.he particular flattening procedure. We can restrict our determinant line

h,ul.II.˜ t.o this family, getting a real line bundle over 50(3). A simple application

ur t.luÂ· At.iyah-Singer "Excision Axiom" shows that this bundle is

wl.-Â·,,Â· eis the Hopfline bundle over 80(3), viewed as projective 3-space.(See [7].)

Nut.., t.hat there is no loss in symmetry in this formula, since k1 + k 2 is even. It

f..lluwH t.hen that the pairing of u 3 with the fundamental class of 80( 3) is (-1 )k 1 â€¢

WU.lt this fact at hand we will now go back to our discussion of invariants. Let

Ut" dill H˜nsion of the moduli space Mk(g) be 2d +3 and let 1(g) be the intersection

ur II ..Â· IlH)duli space with VI, ... , Yd. As we noted above, l(g) is still compact for

lfiIlC.tic- Inctrics 9 (so long as 4k > 6 + 3b+) and we can form the pairing of [1(g)]

wit.1I ,,:'. The argument to show that this is independent of the choices of surfaces

Mud c'odiluension-2 representatives Vi goes through just as before, and we obtain a

u,ult.ilillpar function

f3g : H 2 (X) xÂ·Â· Â· x H 2 (X) Z/2,

--+

=< u 3, 1(9) >.

I.v '˜.IU.illp; ,8g([˜l], ... ,[˜d])

1'1 ..\ IU'W feature that we encounter is that fig is not now independent of the generic

....,tilt' !I. The problem comes from the next stratum Mk,l = Mk-I X X in the

.....1I1.lIct.ified space. The moduli space M"-l has dimension 2(d - 2) - 1 so in a

t \'1., .. ,,1 I-parameter family of metrics 9t we should expect there to be some isolated

tlt""'1 when Mk-1(gt) meets d - 2 of the Vi, say Vi, ... ,Vd - 2 â€¢ IT A is a connection

... rllte"1I au intersection and x is a point in the intersection Vd-l n Vd then the pair

f 1.-l11 ,r) can lie in the closure of I == oN n VI Â· Â· Â· n Vd, and in that case I does not

lJ)"f' " cOlnpact cobordism from 1(go) to 1(gl).

AII III IIOt, lost, however, through this failure of compactness. The same analytical

I'll ic 11l(˜S used for connected sums allow one to model quite precisely the behaviour

I f"

". # .6

Donaldson: Yang-Mills invariants of four-manifolds

26

of the compactified moduli space around ([A], x), see [7]. The link L of the strat

Mk,l in the compactified space is a copy of 50(3), representing the gluing paramet I

which attachs a highly concentrated instanton to the background connection A.

.follows from the discussion in the previous paragraphs that the pairing of u 3 with.:

is 1. On the other hand a simple topological argument shows that a suitable trun

tion of the space I is a compact manifold-with-boundary, whose boundary has,')Â·

addition to 1(go) and 1(91), a component for each pair ([A], x), and this compon :.J

is a small perturbation of the link L. So the difference (P9o - ,891 ([˜l]'. Â· Â· ,[˜dl

is exactly the total number of pairs ([A], x). (Here we have, of course, to allow'

partitions of {I, ... ,d} of type (d - 2,2) when counting the pairs ([A], x) .)

To understand this better we consider briefly another kind of generalisation of t

Yang-Mills invariants. Suppose we Â·have a situation where the moduli space M j h

virtual dimension -1, and so is empty for generic metrics. We define an invari

for a path of metrics 9t by counting the number of points in the associated mod

space N. If b+ ˜ 3 this number depends on the path only through its' homoto

class, with fixed (generic) end points. This collection of invariants of paths give

class in Hl(1?,*), where 'R,* is the space of Riemarmian metrics on X with trivi

isometry group, modulo diffeomorphism. More generally, if we have a moduli sp .'

of dimension (2d - 1) we can take the intersection with subvarieties Vi to obtainj

multi-linear invariant of paths of metrics :

This is independent of the representatives Vi, and yields a homotopy invariant:

paths. It naturally defines a class in the twisted cohomology Hl(R,*;II), wh

IT is the local co-efficient system over 'R.* corresponding the representation of t

diffeomorphism group on the multilinear, Z/2 -valued functions in the homology::

x. (We can, of course, go further in this direction to define higher cohomolo'

classes over 1?,â€¢â€¢ )

Our analysis of the ends of the manifold I now leads immediately to the fonnul

(9)

where u is the (d-2) -linear invariant of paths defined by the moduli space Mk ;

as described above. In (9) we take any path from 90 to 91, and on the right h '

x

side we use the multiplication in the ring of multilinear functions Sym ,Z/2 .

the intersection form Q of X. In particular the functions 13gt are equal modulo t,

ideal generated by the intersection form, and we obtain an intrinsic invariant in t

quotient graded- ring

Sym X,Z/2/ < Q > Â·

( Notice that another consequence of (9) is that the product of the cohomolo

class defined by u with Q is zero in Hl('R.*;IT).)

To sum up then we have :

27

Donaldson: Yang-Mills invariants of four-manifolds

TII˜:.)I'IÂ·:M 10. Let X be a compact, simply connected, oriented, spin 4-manifold

.,.,. I, t ( ..X") > 1. Suppose k is even and is such that 8k - 3(1 +b+(X) = 2d +3 and

˜˜ .. :I( 1 +b+(X) + 3. Then tbe pairing

=< u 3 , MA; n VI n Â· Â·Â· n Vd >

Pk,x([E I ], â€¢â€¢â€¢ , [Ed])

fI"n'U'H lJ. differential- topological invariant Pk,X in Symi,Z/2/ < Q >.

(l'lvn.rinnts with this kind of ambiguity have appeared in a slightly different con-

IHt, lâ€¢â€¢ the works of Kotschick and Mong. The identification of the precise correction

fatAl. u' ",rising from the failure of compactness has been discussed, in this other con-

".t. hy Kotschick. )

(1I)llIvnriants for connected sums. We suppose now that the manifold X ap-

˜'I""1l: in Theorem 9 is a smooth, oriented, connected sum Xl˜X2 and that each of

t+( 1\' I), b+(X2 ) is odd. We shall use the analytical techniques described in Section

I (t.) I.eÂ» l)artially calculate the new invariant of X in terms of the factors in the SUIn.

tit.. tl"˜ltments involved are very similar to those in the Vanishing theorem for the

"'' tllnl invariants - but we shall see by contrast that these torsion invariants for X

".....t Hot, be trivial, due to the fact that they detect the glueing parameter which

Itt,I",.uÂ·('d in our description of the moduli space. The discussion here is very similar

˜" Uud. in [7] for the complementary problem of the existence of 4-manifolds: it

,. "htH very similar to Furuta's use of such torsion classes in his generalisation of

rl.NtI'" coohomology groups; see Furuta's article in these proceedings.

˜ '''altlyse the invariant Pk,X we fix a partition d = d l + d2 and homology classes

"tJ, .. 0' [˜dl] in Xl, represented by surfaces Ej in t˜e obvious way, and classes

fJi\). .. , [˜d2] in X 2 â€¢ Recall that 8k - 3(1 + b+(XÂ» = 2d +3, where

4k > 6 + 3b+(X).

=,8k,x([E I ], â€¢.â€¢ , [Ed

.....1a".11 (˜valuate the pairing (J [E˜, ... , [Ed,]) assuming that

W l ],

(II )

Â·l'lu\ point of this condition is that if we define ki by

t...tl. of k I , k2 are in the range where the polynomial invariants ql:.,x, developed

h. ˜t.Â·.Â·tioll 2(a) are defined. Let us write qI,q2 E Z for the evaluation of these

Itt\'''' "Ult.S on the classes [E j ], [Ejl, in H2 (X 1 ),H2 (X2 ) respectively.

\\,'1 IIC)W proceed in the familiar fashion, considering a family of metrics g( A) on

1( wltb t.he neck diameter O(A 1 / 2 ), and "converging" to given, sufficiently generic,

I

Donaldson: Yang-Mills invariants of four-manifolds

28

metrics 91,g2 on X t ,X2 â€¢ We let l(A) C Mk,X(g(AÂ» be the intersection of mod'

space with all of the 'Vi and Vj. We will show that, for small A, leA) is a disjoÂ·

union of copies of SO(3). In one direction, suppose that At is a connection over

which represents a point of

and similarly that A 2 is a connection over X 2 which represents a point of the

tersection [2 of Mk 2 ,X2 with the VJ. Then the glueing theory sketched in 2(b) sh '

that, for small enough A, there is a family of ASD connections over X parametri '

by a the product of a copy of SO(3) ( the gluing parameter ), and neighbourh "1:

of the points [Ail in their respective moduli spaces. Taking the intersection f.

the 'Vi and Vj is effectively the same as removing these two latter sets of paramet

in the family; so we obtain a copy I([A t ], [A 2 ]) of SO(3) in the intersection, w

clearly forms a complete connected component of I(A).

Now, under the condition (10) the sets I t ,I2 are finite, so for small A we ,

111 1.112 1 copies of 80(3) in I(A). We will now show that these make up all of Ie

Again the argument takes a familiar form : suppose we have a sequence An ;'

and connections An in leAn). After taking a subsequence we can suppose that,

connections converge to limits B 1 ,B2 over the complement of sets of sizes 11, 1\

the two punctured manifolds; where Bi is an anti-self-dual connection on a bu

with Chern class "'i over Xi. We have an "energy" inequality

(12)

Now the argument is the usual dimension counting. First note that at least,

of the Ki must be strictly positive, by (11). Suppose next that K2, say, is zero'"

B 2 is the product connection. Then each surface ˜j must contain one of th

exceptional points in X 2 , so:

d2 2 1â€¢

:::; 2

'Vi

Over the other piece: at least d1 - 2 11 of the must meet the moduli spac .

Chern class Kl, so :

+ 8Kl - + b+(X1 ).

411 3(1

2d1 :::;

Combining these inequalities with (11) we obtain a contradiction. Similarl:

the case when neither "'1 nor ˜2 is zero one deduces that in fact 11 = 1 = 02

k = "'1 + "'2. It follows then that for large n the point [A(n)] lies in [([B 1 ], [ :

and hence that I( A) is indeed the union of these components, for small A. :

We can now use the relation (8) between the class u and the gluing construe'

to evaluate (3. The copies [([AI], [A 2 ]) of 80(3) are small perturbations of t \

29

Donaldson: Yang-Mills invariants of four-manifolds

..I,t.n,iued by flattening the connections, so the cohomoloogy classes restrict in just

'ht' tuune way. We obtain the formula

f3 = 0 if k1 , k 2 even

= ql Â·q2 if kl , k2 odd.

w., (˜nn sum up in the following theorem

'aUI:ouEM 13. Let X be a simply connected, spin, 4-manifold with b+(X) even

.,,,1 k he even with 4k > 6 + 3b+(X). If X can be written as a connected sum

,V - .\I UX2 , with each of b+(Xi) odd, the invariant I3k,x has the form:

L )+ â‚¬1+ â‚¬2 mod 2,

(3k,X=( Qk t ,XtÂ·Qk 2 ,X2

k1,k 2 odd,k t +k 2 =k

+b+(Xi)) in H 2 (Xi).

wl,,',.(˜ â‚¬i contains terms of degree at most (3/2)(1

WfJ t.hen that the torsion invariants are more sensitive than those defined by

14(1('

'Itp IItt.ioual cohomology: the latter are killed by connected sums, since the glueing

ttl. nt.u't.er is rationally trivial, but the torsion classes can detect the gluing parame-

,.... nud p;ive potentially non-trivial invariants. Moreover, if Xl and X 2 are complex

.1.fil.l'",i(˜surfaces we can hope to calculate some components of the new invariant

ftt" t.I.., (˜onected sum, using Theorem 13, and hence show, for example, that the

ftUu.irold does not split off an 8 2 x S2 summand. In this direction, one can use a

",˜..n'1I1 of Wall [20], which tells us that if X = YU(8 2 X 8 2 ) and b+(Y) ˜ 1 then

t'tf' t1l1l.olIlorphism group of X realises all symmetries of the intersection form. The

tt\V"dnllf.H for such a manifold must be preserved by the automorphism group, and

Iltht p"iVC'H strong restrictions. The author has, however, not yet found any examples

whf.'lf˜ t.his scheme can be applied: in a few simple examples various arithmetical

"t!f,˜n˜ He'em to conspire against the success of the method. Perhaps more elaborate

""Iuple':-I will be successful, or perhaps there is some deeper phenomenon in play

Whlt'h Illakes these torsion invariants also vanish on connected sums.

N˜.t ''J'I()N 4, REMARKS ON HOLOMORPHIC AND PSEUDO-HOLOMORPHIC CURVES

II) Â«:urves, Line bundles and Linear systems.

WI' will.IOW change tack and make make two remarks on the relation between holo-

tt.u. phico nnd pseudo-holomorphic curves. These remarks can be motivated by the

..."l"p,ic'H with Yang-Mills theory on 4-manifolds described in Section 1. We have

Ifif'1t III Sc'etion 2 that, at present, the utility of the Yang-Mills invariants is derived

hUJl.'11y frOID the link with algebraic geometry in the case when the 4-manifold is

ft â€¢ lIluplc'x algebraic surface. One might hope that, in a similar way, the space

Itt IUff'lulo-holomorphic curves in a general almost Kahler manifold captures infor-

'U"f â€¢â€¢ -II whi<:h depends only on the symplectic structure and which reduces, in the

Donaldson: Yang-Mills invariants of four-manifolds

30

special case of Kahler manifolds, to well-known facts about complex curves. This i

certainly true to some extent: for example Gromov proved in [15] that a symplecti˜'

4- manifold which has the homotopy type of Cp2 and contains a complex line 0,

self-intersection 1 (for a suitable, compatible, almost complex structure) is sym'

plectomorpbic to the standard CP2. Gromov's argument reduces in the integrabl'

case to classical geometry, effectively a step in the Enriques-Castuelnuevo theore

[14]. For many more general results on these lines see the contribution of Mac D ,

to the Proceedings. Observe, by the way, that an embedded (real) surface E in :

symplectic manifold (V,w) is pseudo-holomorphic with respect to some compatibl˜

'almost complex structure if and only if it is a symplectic submanifold ,i.e. if '

restricts to a symplectic form on E. So Gromov's Theorem asserts that a symplec'

tic homotopy Cp2 in which a generator of the homology can be represented by

symplectic 2-sphere is standard.

With this in mind we consider what can be said about moduli spaces of holo

morphic and pseudo-holomoxphic curves. In the integrable case one can of cours

apply a great deal of existing algebro-geometric theory. First, in a general wa˜

the moduli spaces of holomorphic curves will be quasi-projective varieties - an'

the coresponding hyperplane class has a simple toploogical description, much as i '

the Yang-Mills case. While it is a difficult problem to find holomoxphic curves i

general there is one class of examples which are easy to find and describe - th"

curves given by complete intersections of hypersurlaces in a complex manifold. W'

recall that in a hypersurlace W in a complex manifold V can be identified with'

eover e.

line bundle V and a holomoxphic section s of If V is simply connecte

eis

then the line bundle in turn specified uniquely by its' first Chern class -

ethe

integral class of type (1,1). Having fixed cOrresponding hypersurfaces for '

a linear system, parametrised by the projective space P(r(e)Â·). Thus the stud

of complete intersection curves, and in particular of all holomomorphic curves i '

a complex surface, reduces to questions about line bundles and their holomorphi .'

sections.

This familiar theory has a number of simple consequences. We will concentrat '

on the case of symplectic 4-manifolds and complex surfaces, although some of ou.

remarks apply in higher dimensions. First, the existence of any holomorphic curv .

at all in some complex manifolds is a very unstable phenomenon. Take for exampl'

a generic (Kahler) metric on a K3 surface. The integer lattice in H2 only meet

the subspace of (1,1) classes in the origin, so there are no non-trivial holomorphi'

curves. (Note that the ideas here are very close to those we encountered wh :

discussing how to avoid reducible instantons ). The same phenomenon applies mor'

generally, and we shall now see how it can be understood in the framework of th,

local deformation theory of solutions to the holomorphic equation, and cohomology;'

Let us now go back for a while to review some of the general theory of pseud "

holomorphic curves, of a given genus 9 and a given homotopy class, in an almos

complex manifold V. We can define two moduli spaces ME and M ; the first bein

31

Donaldson: Yang-Mills invariants of four-manifolds

'h" Npace of holomorphic maps from a fixed Riemann surface Ii, and the second

'HI'IIK the space of all pseudo holomorphic curves of the given topological type, in

wlll.oll the induced complex structure is allowed to vary. (Thus ME is a fibre of the

""t.urn.l map from M to the moduli space of Riemann surfaces of genus g.) The

U""ltrisation of the equation defining M about a given solution f : ˜ --+ V (which

Wfl t.11.ke, for simplicity, to be an embedding) is given by a linear elliptic operator

,'t t\(˜ting on sections of the normal bundle. The Fredholm index s of 6/ is easily

f'f\lcÂ·,.lnted to be

s = 2(E.E + 1 - g)

.... 1 t.his index is the virtual dimension of the moduli space M of all the pseudo-

huloillorphic curves. To describe ME locally we introduce a similar operator 6/,0'

".'t.iup; on the pull back of the tangent bundle of V. The Fredholm index of fJ/,o is

â€¢ (Gg - 6), and this is the virtual dimension of ME.

Nc)w for a generic almost-Kahler structure on V one can show that the operators

˜ I tU I( 16/,0 are surjective, for all embedded pseudo-holomorphic curves (the analogue

IIr til(' Freed-Uhlenbeck result in the Yang-Mills case), see [15]. This means that

ll,,Â· Illoduli spaces M, ME are smooth manifolds whose dimension agrees with

UuÂ·'" virtual dimension. We will now see that the picture for Kahler metrics is quite

tUlrfÂ·l'C˜llt. Consider a holomorphic curve f : ˜ -+ V in a complex Kahler surface

V ; fc)l' simplicity we assume f is an embedding. The cokernels of ˜/ and 6/,0 can

It" lclc'utified with the sheaf cohomology groups Hl(E; v), Hl(E, TVIE) respectively,

wl,,'n' v is the normal bundle of E in V. We have then:

14.

PUOPOSITION

If V is a compact complex surface with Pg(V) > 0 and E is an embedded curve

I" V then Hl(E; v) and HI(E; TV(E) are both non-zero, except for the cases

( I) Pg(V) = 1 and ˜ ( or some multiple thereof) is cut out by the section of

Kv.

(:˜) E is an exceptional curve in V (i.e. an embedded 2-sphere with self-

intersection -1 ).

(:1) V is an elliptic surface, and E is a multiple fibre in V - a 2-torus whose

normal bundle is a holomorphic root of the trivial bundle.

'1'0 prove this it suffices to consider the normal bundle, since the holomorphic

eis

,,,up from TVh˜ to v induces a surjection on HI. Now if the line bundle over

eto

V C"C HTcsponding to E the normal bundle v is the restiction of E and we have

hit f'xnct sequence

'I'his induces a long exact cohomology sequence, the relevant part of which is:

Donaldson: Yang-Mills invariants of four-manifolds

32

The space H 2 (V; 0) has dimension Pg(V) which is positive by hypothesis. ':D:'

show that HI (v) is non-zero it suffices to show that the map to H2 (V; e) is no

injective. By Serre duality this is equivalent to showing that the map between th "

duals

m.,: HO(V;e* Â®Kv) ˜ HO(V;Kv)

ecutting out E. S',

is not surjective. Here m s is multiplication by the section s of

m s is surjective if and only if all sections of the canonical line bundle K V vanish 0

E. Thus we have established that Hi (E; v) is non-zero if ˜ is not a fixed componen

oflKvlÂ·

To complete the proof we examine the case when E is a fixed component of IKvJ-

Thus we can write Kv = [E + Cj, where C is another curve in V. Now if E is , ˜

exceptional curve in V then certainly all sections of K v must vanish on E, so we g

a fixed component this way, as allowed for in case (i) of the Proposition. Conversel .

leaving aside exceptional curves, we may as well replace V by its minimal mode

So we assume now that V is itself minimal. We now appeal to the classification .

surfaces, as on p.188 of [2]. The only cases that can occur are when V is an minim

elliptic surface or a minimal surface of general type. In the first case the curve .

must be a fibre of the elliptic fibration. If it is an ordinary fibre the normal bund \,

is trivial and H1(v) is non-zero, so the only curves that occur in this way are t ˜

multiple fibres allowed for in part(3) of the Proposition. In the second case, wh "

V is of general type, we can assume C is non-empty, otherwise we fall into catego '

(i) of the proposition. Then we must have E.G> 0, since IKvi is connected ( [2

page 218). On the other hand some routine manipulation using the adjunctio'

formula shows that the holomorphic Euler characteristic X(v) is given by -˜.G, .

HI (v) must be non-zero.

We see then that for most purposes the Kahler metrics are quite unlike the gene .,.

almost-Kahler metrics as far as the pseudo-holomorphic curves which they defin

are concerned. Again, one should contrast this discussion with that for holomorp .,'

bundles and instantons where the key result, obtained from the estimate (4),

the fact that the Kahler metrics behave quite like the generic metrics.

A partial remedy for the degeneracy we have noted above can be achieved

allo,ving the symplectic form to vary. Fix a conformal structure on V such th'

the symplectic form w is self-dual. Then w is an element of the space H.+ of se '

dual harmonic forms, which has dimension b+(V). There is an open set U C 11"

containing w such that any Wi E U is a non-degenerate 2-form, defining a symplect

structure on V. Also there is a unique metric in the conformal class which is almosÂ·

Kahler with metric form Wi. Thus we have a natural family of almost- Kahl

structures on V parametrised by U. Fixing the volume of V we get a b+ -

dimensional family parametrised by the subset S(U) of the sphere 8(11.+). We c '

then consider an enlarged moduli space M+ whose points consist of pairs (w',f'

where Wi is in S(U) and f : ˜ ˜ V is pseudo-holomorphic with respect to t

33

Donaldson: Yang-Mills invariants of four-manifolds

structure. Thus the space M considered before is a fibre of the

P-"It'r'˜p()nding

".t,1I1'1l1 Inap from M+ to S(U).

Tit., point of this construction is that the space M+ is in many respects the more

_",u'upriate generalisation of the moduli space of holomorphic curves in a Kahler

IUt fU4Â·(t. The Hodge decomposition of the cohomology shows that if the original

..,.... i. is Kahler then M = M+ - i.e. there are no pseudo-holomorphic curves, in

o

'h" ".iV(˜ll homology class, for the perturbed structures. Moreover the dimension of

AI til. t.he Kahler case is then typically equal to the virtual dimension s+b+(V) -1.

ft,nt) t.llis point of view the degeneracy detected by Proposition 14 in the Kahler case

'Itl,,'n.rH a.s the degeneracy of the map from M+ to M. The great drawback with

tltllll hÂ·I)1 )I'oach, as far as applications go, is the fact that in general we will lose the

˜AlIlf' C(uupactness properties in the manifold M+, since the symplectic structure

"'ftlf will break down at the boundary of U.(In the special case of a K3 surface,

"Uti It. hyperkahler metric, the set S(U) is the whole 2-sphere and this breakdown

tit*,. 11()f, occur: the importance of considering all the complex structures in this

"'f! WitH pointed out to the writer by Mario Micallef.)

(tt) Ilnrlnonic theory on almost Kahler manifolds.

˜f4 will focus on a specific question - the existence of symplectic submanifolds

BOW

1ft " p,f'IH˜ral almost Kahler manifold. To be quite precise we will consider the

fHUt.will.L!; problem:

' 'U' I.EM. Let V be a compact 2n-dimensional manifold and w a symplectic form

fltt \' \vil,}) integral periods, i.e. [w] E H2(V; R) is the reduction of a class a

itt III ( ˜\ ; Z). Is there a positive integer k such that the Poincare dual of ka is

PI,JlrHIÂ·III,ed by a symplectic submanifold of V ?

I Wf1 III\.V(˜ mentioned in (i) that, when n = 2, such a submanifold would be pseudo-

1w1"'llc"'I)hic for a suitable almost-Kahler structure on V.)

˜h" â€¢C'It.HOll for phrasing the problem in this way is that there is a simple, famil-

,â€¢â€¢.â€¢ nllHW(˜r in the case when the form is compatible with an integrable complex

e-+ V with Cl(e) = a, and

.Itllt".un'. There is then a holomorphic line bundle

ehaving eis

curvature form -21riw. Thus

â€¢â€¢<.tlBlu,.foible unitary connection on a

'tt.,""r eis

hne bundle and, according to the Kodaira embdding theorem, ample,

e

tit til" H('ctions of some power , k Â» 0, define a projective embedding of V.

k

("Vii" this we obtain many holomorphic curves, Poincare dual to ka, as hyperplane

1fI.:UUllti in the projective space, or equivalently as the zero sets of holomorphic sec-

th..," of ˜k. These holomorphic curves are a fortiori symplectic submanifolds, so we

fU'I1IW,'. the problem affirmatively in the "classical" case.

h ftt "tlIH that there is esentially only one proof of the Kodaira theorem known; using

1

˜'H"Jllaill.u; theorems and harmonic theory over Kahler manifolds ( see for example

II ˜I ' 121]). Thus it is natural to ask whether this kind of proof can be adapted

It. ntltlw.'r the problem in the general, non-integrable case. In the remainder of this

'-tit t " -. 1 WC˜ will make some first moves in this direction. Let us consider then an

34 Donaldson: Yang-Mills invariants of four-manifolds

almost- Kahler manifold V, of any dimension and mimic, as far as possible, th:

usual differential geometric theory from the Kahler case. .

First, we can decompose the differential forms on V into hi-type (since this is

purely algebraic operation) and define operators :

a: at;q --. nt:q+

lJ : at;q at,+l,q , 1

--+

by taking the relevant components of the exterior derivative d. In general we d˜'

a

- -2

not have d = () + {J, and is not zero. Instead we have,

and

82 = N 0 a:a-f;q nt;q+2,

--+

where N is the Nijenhius ten80r in Tl,O Â® AO,2, which defines bundle maps fro':;'

AP+l,q to AP,q+2 (see [4]). I

We will now go on to consider vector bundles over V. Let E --. V be a compi

Hermitian bundle with a connection having covariant derivative V E and curvat

FE. We decompose VE, much as in Section 2(c), to write:

We can extend these operators to E- valued differential forms, getting operata'

oE,8E with:

) dE=8E+8E+N+N,

,BE = V'/i; on se˜tions of E. The important case for us will

such that OE = V E

e e

when E is a complex line bundle k â€¢ Let s be a smooth section of k , with zero s

Z C V. It is a simple exercise in linear algebra to show that Z will be a symplect

submanifold of V if the section s satisfies the condition:

(15)

This condition is thus a natural generalisation, from the point of view of t "

zero set, of the notion of a holomorphic section in the integrable case. We c .'

interpret the problem of finding a symplectic submanifold as the problem of fin Â·

"approximately holomorphic" sections in this sense. To see why it is plausib,

that such sections should exist we will go on to consider the interaction betw

curvature and the coupled a-operator, in the almost-Kahler case.

35

Donaldson: Yang-Mills invariants of four-manifolds

rh'Nt., turning back to the general differential geometric theory for the bundle E

ttWSI" V, we have:

F˜,q is

wla"I'(' the (p, q) component of the curvature. We wish to combine these

˜""uuln.e with the "Kahler identities". To state these we introduce the algebraic

'URI-

.1.1.'11 is the adjoint of multiplication by the metric form w, with respect to the

.huacln.rd inner product on the forms. The basic Kahler identities extend to the

alâ€¢â€¢â€¢â€¢â€¢,..t.-Kahler case, in that we have:

"lte Â»POSITION 16. On any almost Kahler manifold the formal adjoints of the oper-

- \

.tUI'N D,8 are ::

8* = i[A,8] , {)* = -i[A,8].

'l'ht'HC identities calf be verified by checking that the usual proof for the Kahler

2

for example in [21], does not use the equations 8 = 8 2 = o. Consider for

'."", II..';

"'.\tuple the formula for the operator F on forms of type (0, q). This is very easy

'It prove. We need to show that any (J E Oo,q, Â¢> E OO,q-l with compact supports

â€¢â€¢u˜ry :

< 8,84> >=< iA89, 4> >=< i89, w " 4> > .

'˜llht follows from the algebraic identity:

It,,- /. ( We have:

QO,r.

J861\" J861\"

"j, 1\ w n - r = r

"j, 1\ w n - ,

St,okes' theorem and the fact that w is closed. Now the same algebraic identity

'''tlll'';

.l.uWH t.hat the last expression is just < iA88,w 1\ 4> >, as required. The Kahler

hlt1t1t.it.ies extend to bundle-valued forms ( since the connection is trivial to first

ttl tI., .. ). We apply this first to the two Laplacians on sections of E to get:

(V˜)*V˜ - (V'E)*V'E = iA(F˜,l + NN + NN).

( I'l)

Donaldson: Yang-Mills invariants of four-manifolds

36

Turning now to the E -valued (0, q) forms, we can consider three Laplace- type

operators. First we have the "8 -Laplaci˜ "

:

Second we have the a-Laplacian on n˜9

The Kahler identities give :

= 0' + iA(F1,1 + NN + NN).

(18) 6.

On the other hand there is a unique connection on the bundle A0,9 Â® E compatible

with the metric and having aE for its (0, 1) component. We write the covariant;

derivative of this connection as V = V' + V" , so V' = BE. We can form the thir ;

o˜r˜˜ :

= V"Â· V" .

0"

A0,9 we get :

Applying (17) to the bundle E @

iA˜,

(19) 0' - 0" =

where ˜ is the (1,1) part of the curvature of the connection V. Combining (18,

with (19) we obtain:

˜ = 0" + iAF1,1 +iA(NN +NN˜).

(20)

e ehas

Now suppose that E is the line bundle k , where curvature 21riw. Then it i'

easy to see that the operator iAF˜,l is multiplication by 21t'k( n - q) on (0, q) form .

e

The curvature ˜ has two components - one from k and one from the bundle Ao,˜˜

The former gives a contribution 21rkn to iA'l> and the latter is independent of'

Similarly the term N N +N N does not vary with k. In sum then we get the tw

formulae

(21)

and

(22)

where 0 1 , C2 are tensors which depend only on the geometry of the base manifo!

The most important of these formulae for us is the first, which shows that, for lars

k, ˜ is a very positive operator on nO,q<e k for all q > O.

),

37

Donaldson: Yang-Mills invariants of four-manifolds

Nc)w it is easy to see, in various ways, that when k is large the operator D = 8E+a˜

lAking no,evenCek) to nO,odd(e k ) has a large kernel. The quickest route is to use the

At.iyah-Singer index theorem, which shows that the index of this operator is given

'.yt,he familiar "Riemann-Roch " formula

index D =< ch (el:) Td (V), [V] >,

wlac'''(˜

the Todd class is defined in the usual way by the almost complex structure

.... V. This formula represents a polynomial of degree n in k and the leading term

I" A˜n Yolume(V), thus we have:

= kRYolume(V) + O(k n - 1 ).

dim ker D ˜ indexD

OUIiHider now an element of the kernel of D, which we write in the form (s + q),

.Ia(˜rc˜ s is a section of {k and u contains the terms in nO,2T(e k) for r ˜ 1. We write:

o= D* D(s + u) = 6(8 + u) + ([)2 + (8*)2)(S + u).

e , using the

simplicity, written 8, '8* for the operators coupled to k

we have, for

Mfll'f'

1.".1 (˜()nnection. We obtain then:

+ {)* N*s + N8s

(In) 6u = 8*N*u

It..1

(I˜) 6.8 = {)*N*u.

ta˜'IIK the inner product of equation (23) with u and using the formula (21) above

'f' â€¢.˜

I

t.:

IIV"uIl2 + 21rkllull2 =< Ct(f,U > -2 < {)*N*u,u > - < 8*N*,s > .

1If1,r n.ll11orms are L2. The key observation now is that the term ()*N*u is hounded

,..lut,wiHe by a multiple of IV"ul+ lui, where the multiple is independent of k. This is

,la"1 wh(\n one expands out the terms by the Leibnitz rule, in local co-ordinates, and

..t""it VPH that {)* only involves the derivatives in the "antiholomorphic " directions.

W˜ 'tt*t t.hen, for large enough k, an estimate of the form:

IIV"uIl 2 + (21rk - C)lIull 2 :5 constÂ·(lIuli + IIsll)(lIuli + IIsll)Â·

It 1ft .Â·I,ollientary to deduce from this that

lIull :5 const. k- 1 / 2 I1 s l1 , IIsll,

IIV"ull:5 const.

38 Donaldson: Yang-Mills invariants of four-manifolds

with constants independent of k.Now use (19), and the fact that the curvature is of

order k, to obtain

which gives a bound on the "holomorphic" derivative:

IIV'o-li $ IIsli.

const.

Using the first order relation 8s = -8*0-, we deduce that

IIsli.

˜ const.

lIa811

On the other hand, returning to the equation (24), and using (22), we get:

so

from which we deduce that

We see then that each element of the kernel of D gives rise, when k is large, t!

e

an "approximately holomorphic" section s of k , in that the L2 norm of 8s is mu

less than that of 8s. Precisely, we have a bound

˜ 118sllÂ·

const. k- 1 / 2

118811

We sum up in the following proposition.

Let ebe a line bundle with a Hermitian connection over

PROPOSITION 25.

almost- Kahler manifold (V, w) whose curvature is 21riw There is a linear space

e

of sections of k , with

dim Hk "'" Volume V k n ,

and a constant C, independent of k, such that

for all s in Hk.

Here we have identified the space H k , as defined above, with a space of sectio

e

of k by taking the {lO(e k) component.

39

Donaldson: Yang-Mills invariants of four-manifolds

'I'lais L 2 result falls, of course, a long way short of proving the existence of a section

IlIn.t.isfying the pointwise inequality (15) which would give a symplectic submanifold,

1.'If, it seems possible that a more sophisticated analysis of the kernel of D, for large

k, would show that suitable elements of this kernel do indeed satisfy (15). (One can

1".lllpare here the work of Demailly [5] and Bismut [3] in the integrable case.) In

e

"' HiInilar vein, one can show that for large k the sections Hk generate k , so they

cl,˜fille a smooth map:

j" : V --+ P(HZ).

It. iH interesting to investigate in what sense jk is, for large k, an "approximately

lu.l(unorphic" map.

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Phys. 61 (1978), 97 -118.

'/.. W.Barth, C.Peters and A. Van de Ven, "Compact Complex Surfaces," Springer, Berlin, 1984.

:I. J .M.Bismut, Demailly's Asymptotic Morse Inequalities: A Heat equation proof, Jour. of

I'unctional Analysis 72 (1987), 263-278.

4. S.S.Chern, "Complex manifolds without potential theory," Springer, Berlin, 1979.

rI .â€¢J.P. Demailly, Champs magnetiques et inegalites de Morse pour la d"-cohomologie, Ann.

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fl. S.K. Donaldson, Anti-self-dual Yang-Mills connections on complez algebraic surjaces and

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ff S.K.Donaldson, Connections, cohomology and the intersection forms oj4-manifolds, Jour.

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tn A.l4 'loer, An instanton invariant for 3-manijolds, Commun. Math. Phys. 118 (1989),215-240.

t I U.S.Freed and K.K.Uhlenbeck, "Instantons and Four-Manifolds," Springer, New York, 1984.

I˜ II .14)-iedman, Rank two vector bundles over regular elliptic surfaces, Invent. Math. 96 (1989),

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'4 P.< :rifliths and J .Harris, "Principles of Algebraic Geometry," John Wiley, New York, 1978.

Itt M.<.romov, Pseudo-holomorphic curves in Symplectic manifolds, Invent. Math. 82 (1885),

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Ifa M.(. romov, Soft and Hard Symplectic Geometry, in "Proc. International Congress of Math-

'˜lIu,ticians, Berkeley,U.S.A., 1986," American Mathematical society, Providence, Rhode Is-

Il\lId, 1987, pp. 81-98.

tf 1\ .I{odaira, On homotopy K3 surfaces, in "Essays on topology and related topics (dedicated

I... (;. de Rham)," Springer, New York, 1970, pp. 58-96.

Ht (',II. Taubes, Self-dual Yang-Mills connections o1Jer non-self-dual 4-manifolds, Jour. Differ-

'˜IIf.in.l Geometry 17 (1982), 139-170.

ICI\. Uhlenbeck and S-T.Yau, On the ezistence of Hermitian Yang-Mills connections on

IU

df&/Jle bundles over compact Kahler manifolds, Commun. Pure and Applied Maths. 39 (1986),

'It.7-293.

(. '1'.(˜.Wall, Diffeomorphisms of 4-manifolds, Jour. Lond. Math. Soc. 39 (1964), 131-140.

;In

Donaldson: Yang-Mills invariants of four-manifolds

40

21. R.O.Wells,Jr., "Differential analysis on Complex manifolds," Springer, New York, 1980.

22. E. Witten, Topological Quantum Field Theory, Commun.Math.Phys. 117 (1988), 353-386

On the topology of algebraic surfaces

I()DERT E. GOMPF

'I'lac' University of Texas at Austin

INTRODUCTION

A major focus of current research in topology is the classification problem for

closed, simply connected 4-manifolds. While we are still a long way from

Mtllf)()th,

....Â·oluplete solution of this problem, progress is now being made, driven mainly

tty t.he new gauge-theoretic tools introduced by Donaldson. In this article, we

will break the problem into several smaller problems, and discuss what is known

"houtJ each.

The most basic approach to a classification problem is to list examples.

'Ill .., ˜dmplest example of a smooth, closed, simply connected 4-manifold is the

rÂ£ f;4phere 8 4 â€¢ After this, we may think of Cp2, S2 X 8 2 and perhaps the famous

= 22 and signature

˜

1\':1 Httrface. The latter example has second Betti number

,,˜ -16, and it is the simplest known example which is spin and has nonzero

.'''''IH.t,ure. We see that except for 8 4 , these examples are all algebraic surfaces,

.It ;J luanifolds obtained as zero loci in cpn of collections of homogeneous poly-

"u'lIi",}s. In fact, they are hypersurfaces in Cp3, cut out by a single polynomial.

˜\.IÂ· c'n.ch d ˜ 1, there is a unique diffeomorphism type of smooth hypersurface

,â€¢â€¢ c: />:1 obtained from a degree d polynomial. For d = 1,2,4 we obtain Cp2,

and K3. The degree 3 hypersurface is the connected sum Cp2 Cp2

,')2 #6

.... l l(

â€¢â€¢r Nt'V('1l Cp2,S, six of which have reversed orientation. (Algebraically, it is CP2

"lowll up at 6 points.) For each d 2:: 5 we obtain a new example of a smooth,

.I"lif'd, simply connected 4-manifold. In general, the simply connected algebraic

"'" fncc's provide a rich class of examples.

N<'xt, we ask what simply connected 4-manifolds are not algebraic surfaces.

tIIU'Ct,cd sums of algebraic surfaces provide many examples. For example,

C l ..

Cp2 # Cp2 is not algebraic, since its tangent bundle admits no complex

˜I

Gompf: On the topology of algebraic smfaces

42

=

structure. (If it did, we would have the characteristic class identity cÂ¥[M]

(2C2 +Pl)[M] = 2X(M) +30'(M) = 2Â·4 +3Â· 2 = 14. But it is easily checked that

no element of H2(M; Z) has square 14.) In fact, Donaldson's invariants show

that simply connected algebraic surfaces can never split as connected sums of

pieces with bt > 0 [6], so we obtain many new examples this way.

What simply connected 4-manifolds are not connected sums of algebraic

surfaces? Here we are stymied. Although many attempts have been made to

construct such examples, none has been proven successful. In fact, many con-

structions have yielded only known examples such as connected sums # Â±Cp2 of I

Cp2,s with both orientations. It is possible that all simply connected 4-manifolds

are sums of algebraic surfaces.

˜he

We now see that classification problem for smooth, closed, simply con-

nected 4-manifolds splits into three subproblems:

Problem A. Classify simply connected algebraic surfaces up to diffeomorphism.

Problem B. Understand how algebraic surfaces behave under connected sum.

Problem C. Is every smooth, closed, simply connected 4-manifold a connected .˜

...˜

sum of algebraic surfaces?

As previously indicated, Problem C is a major open problem, about which noth-

ing is known. We will address Problems A and B in the next two sections.

2 ALGEBRAIC SURFACES

There is a partial classification of algebraic surfaces (or, more generally, ˜

compact, complex surfaces) due to Kodaira [19]. (See also [3].) In the sim.. ',

ply connected setting, algebraic surfaces (up to diffeomorphism) fall into three' ˜

types: rational, elliptic, and general type. The rational surfaces include CP2

and its blow-ups Cp2 #n Cp2, 88 well as 8 2 X 52. This provides a complete list

of diffeomorphism types of rational surfaces, although there are many algebraic

structures on these manifolds. The elliptic surfaces represent many more diffeo-

morphism types, including the K3 surface, and will be described in detail below.

The simply connected surfaces of general type include everything which is nei-

ther rational nor elliptic, such as the hypersurfaces in CP3 of degree d ˜ 5. This

collection is poorly understood. Fortunately, it is fairly "small" in the following

sense: For any fixed integer b, there are only finitely many simply connected

Gompf: On the topology of algebraic surfaces 43

˜˜ b

â€¢â€¢ ,rfaces of general type (up to diffeomorphism) with second Betti number

1.

111

The "largest" class of simply connected algebraic surfaces is the simply con-

...˜(Â·t,(˜d elliptic surfaces, which will now be described. (See also [3], [15], [16],

I˜()I.) Without loss of generality, we will restrict attention to minimal ellip-

UC' Hurfaces. (Arbitrary elliptic surfaces are obtained from these by blow-ups,

f,r',. (:onnected sums with CP2.) In general, an elliptic surface is a compact,

.'ulIlplex surface with an elliptic fibration, a holomorphic map 1r : V ˜ C onto

A "olllplex curve (i. e., Riemann surface), such that the generic fibers of 1r are

1I˜lIipt,ic curves," i.e., complex tori. By elementary arguments, 1r will have only

ttult.(tly many critical values, whose preimages are called singular fibers, and away

tâ€¢â€¢Hil these 1r will be a fiber bundle projection with torus fibers (called regular

n'If˜I'H). Note that if V is simply connected then C must be the Riemann sphere

l 1'1 = 8 2 , since any homotopically nontrivial loop in C would lift to a nontrivial

'..up ill V. Note also that for any simply connected, closed 4-manifold, the Euler

t˜I'RI',\et,eristic X ˜ 2, since bl = b3 = o.

'I'he simplest example of a simply connected elliptic surface is the rational

AllllJtic surface, which we denote by VI. To construct this surface, we begin with a

'PIU˜l'ic pencil of cubic curves on CP2. Tha.t is, we let Po and PI denote a generic

I,all' of homogeneous cubic polynomials on C3 â€¢ For each t = [to, tl] E Cpl, we let

If; 41rllote the zero locus of toPo +t l PI in CP2. For all but finitely many values of

'. I'i will be a smoothly embedded torus in CP2. The base locus B of the pencil

(1', I' (- cpt} is the set of points in CP2 where Po and PI simultaneously vanish.

Nhlt'.' t.his is a generic pair of cubics, the base locus consists of exactly 9 points.

(tl'-'˜I'ly, B c F t for any t E CPl. For any z E CP2 -B, however, there is a unique

'f 4'cJlltnining Z, as can be seen by solving toPo(z) + tIPI(Z) = 0 uniquely (up to

Ilinl.,) for t. Now we blow up each point p E B. (See, for example [15].) That is,

W˜ clrl(Â·t.{˜ p from Cp2 and replace it by Cpl, thought of as the set of complex

˜",,˜,f'llt. lines through p in CP2. If the new cpt is suitably parametri˜ed, each

/t', will intersect it precisely in the point t. Thus, the 9 blow-ups will make the

I"t". JI', disjoint so that they fiber the new ambient manifold CP2 #9 CP2. This

ht â€¢till' .., ..tional elliptic surface Vi, with the elliptic fibration 1r : Vl ˜ Cpl given

t.\; t.

I'˜ I â€¢

Wp obtain other elliptic surfaces by a procedure called fiber sum. Suppose

r nod IV are elliptic surfaces. Let N C V be the preimage under of a closed

1r

Gompf: On the topology of algebraic surfaces

44

2-disk containing no critical values of 1r. (Thus, N is diffeomorphic to T2 X D2 .)

Let <.p : N c..--. W be a fiber-preserving, orientation reversing embedding onto

a similar neighborhood in W, and let M be obtained by gluing V - int N to '

W - int<.p(N) along their boundaries via the map <.pIaN. We call M the fiber

sum of V and W along ep. We construct manifolds Vn , n ˜ 2, by taking the fiber

sum of n copies of Vi. This turns out to be independent of all choices (except n).

In particular, the map ep may be changed by s˜lf-diffeomorphismsof N, using

the monodromy of the bundle part of VI. Although this fiber sum construction

is not holomorphic in nature, the manifold Vn actually admits an (algebraic)

elliptic surface structure whose fibration is the obvious one. Note that since Vi

has Euler characteristic X = 12 and signature u = -8, we have X(Vn ) = 12n and

u(Vn ) = -8n. As an example, V2 is diffeomorphic to the K3 surface.

To construct further examples, we introduce an operation called logarithmic.

transform. Let V be an elliptic surface, with N C V as before, a closed tubular I:

neighborhood of a regular fiber. In the smooth category, a logarithmic transform

is performed by deleting int N and gluing N R:S T2 X D2 back in, by some dif- ˜

feomorphism t/J : T2 x SI -+ aN. The multiplicity is defined to be the absolute

value of the winding number of 1r 0 t/Jlpoint xS l as a map into 1r(8N) ˜ S1. A '

logarithmic transform of multiplicity zero destroys the fibration 1r and the com-

plex structure, but any positive multiplicity p can be realized by a holomorphic:,

logarithmic transform. This changes the fibration by the addition of a singular ,:'

fiber called a smooth multiple fiber, a smoothly embedded torus which is p-fold "

covered by nearby regular fibers. :;

!

Let Vn (PI, Â· Â· Â· ,Pl:) denote the manifold obtained from Vn by logarithmic:j

transforms of multiplicities PI, ... ,Pl:. The diffeomorphism type of this manifold.!

is completely determined by n and the unordered k-tuple {PI, ... ,Pk}. (This is .:i

due to the monodromy of the bundle part of Vn and the symmetries of T 2 x D2. ;;

See, for example, [13].) In particular, we may add or delete Pi'S equal to one'

without disturbing the diffeomorphism type, since the trivial logarithmic trans-"

form (regluing N by the identity map) has multiplicity one. If no Pi equals zero,

Vn (PI, Â· Â· Â· ,Pk) will admit algebraic surface structures which are elliptic. Further-

more, any minimal elliptic surface over 8 2 with nonzero Euler characteristic will

be diffeomorphic to some Vn(Pl, ... ,Pk), (Pl, ... ,Pk ˜ 2) [17], [23]. The mani-

folds Vn(Pl, ... ,Pk) which are simply connected are precisely those which can be

= 0, q = 1) by adding

put in the form Vn(p, q), p, q relatively prime (including p

45

Gompf: On the topology of algebraic sUlfaces

Vn and Vn(p).)

or deleting Pi'S equal to one. (Note that this includes

The homeomorphism classification of the manifolds Vn(p, q) (p, q relatively

Iu'ilne) is a corollary of Freedman's Classification Theorem for simply connected,

topological 4-manifolds [8]. For a fixed odd n, the manifolds Vn(p, q) all

Â«')oHcd,

fuJI into one homeomorphism type, that of #2'11-1 Cp2 #10'11-1 CP2. For fixed

.˜vc'n n, there will be two homeomorphism types, distinguished by the existence

C˜r It spin structure. If p and q are both odd (and n is even) then Vn(p, q) will

h,tlluit a spin structure, and it will be homeomorphic to #!n K3 # 1'11-1 52 X 8 2.

2 2

(H,hcrwise it will not admit a spin structure and be homeomorphic to

-II 'In-1 Cp2 #10'11-1 CP2.

The diffeomorphism classification of the manifolds Vn (p, q) is much more

l'cÂ»lllplex, and only partially understood. It is well-known that the algebraic

""!'fa.ces V1 (p) (p ˜ 1) are rational, and hence diffeomorphic to cp 2 #9 Cp2.

(A t.opological proof of this appears in [13].) Each Vn(O) is diffeomorphic to

II '/.,,-1 cp2 #10'11-1 CP2 [13]. FUrther results require Donaldson's invariants from

theory [5], [6]. In the n = 1 case, Friedman and Morgan [9] and Okonek

!hl1˜e

tUHI Van de Yen [24] showed that each diffeomorphism type is realized by only

n..it.(˜ly many V1(p, q) (2 ˜ p < q), and in particular, no two of the manifolds

l'd2,q) q = 1,3,5,7, ... are diffeomorphic. For n ˜ 2 (p,q ˜ 1), Friedman

"ltd Morgan [10] showed that the product pq is a smooth invariant, implying a

Milililar finiteness result, and showing that no two of the manifolds Vn(p) P =

(It 1,2, .â€¢. are diffeomorphic. (The P = 0 case follows from the decomposition

'I. (()) ˜ # Â±Cp2, together with Donaldson's theorem [6] that a simply connected

nIIJ.,ohraic surface cannot be decomposed into two pieces with bt > 0.)

'rhese results about elliptic surfaces are quite surprising from a topologist's

\'if'wpoint. Observe that we have many families (one for each odd n and two

ru.. ('nch even n), each of which contains only one homeomorphism type, but

lullllit.ely many diffeomorphism types. We may interpret each family a sin-

88

.,.1" I.()]>ological manifold which admits infinitely many nondiffeomorphic smooth

rucÂ·t,ttres. This contrasts strikingly with topology in dimensions ˜ 4, where

HI

c'.alupact manifold admits only finitely many diffeomorphism types of smooth

11

(In fact, high dimensional smoothing theory would predict that a

f.f 1 "cÂ·ttires.

""lIply connected, closed 4-manifold should admit no more than one diffeomor-

[26]

,.1&itHll type of smooth structure.) Furthermore, a classical result of Wall

III,pliPH that all members of a given family will be smoothly h-cobordant, so we

46 Gompf: On the topology of algebraic surfaces

have infinite families of counterexamples to the smooth h-Cobordism Conjecture

for 4-manifolds. In Section 4, we will further analyze the topology of elliptic

surfaces, and use these to construct other surprising examples.

3 CONNECTED SUMS OF ALGEBRAIC SURFACES

We begin by considering connected sums of algebraic surfaces with ratio-

nal surfaces. Observe that connected sum with CP2 is the same as blowing-up,

which keeps us within the category of algebraic surfaces. Thus, we should not

expect too much information to be lost during this procedure. In fact, Donald-

son's invariants are stable under blow-ups [9], [10], so that our infinite families of

distinct elliptic surfaces remain distinct after sum with any number of Cp2's. In

contrast, sum with +Cp2 is much more damaging. Mandelbaum and Moishezon 1

[20], [23] showed that if M is a simply connected elliptic surface, then M # Cp2 .j

always decomposes 88 a connected sum of Â±CP2 's. They obtained similar results J

for many other algebraic surfaces, including hypersurfaces in CP3 and complete 'j

intersections, which are those algebraic surfaces obtained as transverse intersec-

tions of N hypersurfaces in CpN+2. Mandelbaum has conjectured that for any

simply connected algebraic surface M, M # CP2 should decompose as # Â±CP2.

Sum with 52 x 8 2 is similarly damaging. In fact, Wall [25] showed that if

M is a simply connected non-spin 4-manifold, then M # 52 X 52 is diffeomorphic

to M#S'lx$l, where S2'X5 2 denotes t.he twisted S2-bundle over 8 2 , which is

diffeomorphic to Cp2 # Cp2. (An analogous phenomenon occurs in dimension 2:

If M2 is nonorientable, then M # 8 1 X 8 1 is diffeomorphic to M # 51 XSI where

8 1 X8 1 , the Klein bottle, is diffeomorphic to Rp2 # Rp2 .) It follows immedi-

ately that if M is a simply connected elliptic surface or complete intersection,

and if M is nonspin, then M # 52 X 52 ˜ # Â±CP2. If M is spin, such a decom-

position cannot occur (since M # S'l X 8 2 will be spin and # Â±Cp2 will not),

but we might expect a similar decomposition into simple spin manifolds. In

fact, Mandelbaum showed that for M a simply connected, spin elliptic surface,

M # 52 x 52 decomposes 88 a connected sum of K3 surfaces (with their usual

orientations) and 8 2 x 8 2 's. This suggests the following:

A 4-manifold M dissolves if it is diffeomorphic to either

Definition.

o.

8 2 ) for some k,l ˜

#k Cp2 #t Cp2 or Â±(#k K3#l 8 2 X

Note that for any given M, at most one of the two possibilities can occur, and

this, as well as k,Â£ and the sign (Â±) are determined by the (oriented) homotopy

47

Gompf: On the topology of algebraic surfaces

type of M. (In fact, the intersection form suffices.) We can now state the results

Â«.[ Mandelbaum and Moishezon for elliptic surfaces concisely: If M is a simply

.'C Hlnected elliptic surface, then M # Cp2 and M # 8

2 X 52 dissolve.

We now turn to more general connected sums. It can be shown that for M, N

Niluply connected elliptic surfaces, M # N dissolves [14]. This result still holds if

1\4 and N are also allowed to be complete intersections other than Cp2, provided

thn.t at least one of M, N is not spin [12]. One is free to conjecture that M # N

.liHHolves for M,N any simply connected algebraic surfaces except CP2. These

"'˜Hll1ts, as well as those of Mandelbaum and Moishezon, are proven essentially by

,˜I('lllentary cut-and-paste techniques. Ultimately, they rely on various versions

t˜r 't. lemma of Mandelbaum [21] which shows how -to decompose fiber sums and

rrlnt,ed objects into ordinary connected sums in the presence of an 8 2 x 52 or

,&",/. XS2. A unified discussion of the results for elliptic surfaces appears in [14].

The case of connected sums with compatible orientations seems harder. For

'˜)(Iunple, there is no known example of irrational algebraic surfaces M 1 , â€¢â€¢â€¢ ,Mk

Ruch that #˜=1 Mi dissolves. It is conceivable that such sums never dissolve,

Itlld perhaps such connected sum decompositions (for simply connected, minimal

"TIl,t.ional surfaces) are even unique. However, the usual Donaldson invariants

will vanish for these sums, making analysis of this situation difficult.

The crucial dependence of the topology on orientations is even more graph-

h'M,lIy illustrated by the following result. Suppose M is made as a fiber sum of

two (-Uiptic surfaces with nonzero Euler characteristic, but assume that the sum

1.'v.'rHes orientation (i.e., the gluing map <p preserves orientation). If M is simply

.Â·utlll(˜cted, then it dissolves [14]. Of course, fiber sums with the usual choice of

..ri(,llt.ation are elliptic, so they never dissolve (except for the rational case and

\', . K3). In practice, this difference arises from the "negativity" of most ir-

I"tiC)11al algebraic surfaces. Most (and perhaps all) simply connected, irrational

"IKftl .('aic surfaces contain embedded spheres with negative normal Euler number,

hut rpw (and perhaps no) such manifolds contain embedded spheres with positive

u.,rtllul Euler number. (In fact, such spheres cannot exist if the algebraic surface

Itu" f,lw > 1. Otherwise, by blowing up we could obtain a sphere with normal

:11).'.. number one. A tubular neighborhood of this would be diffeomorphic to

1

'

,.1 -{point},

and we would have a connected sum decomposition of an algebraic

c

IHIl f",("(˜ into two pieces (one of which is CP2), both of which have bt > o. This

t nlltrndicts a theorem of Donaldson [6].) When we connected sum or fiber sum

Gompf: On the topology of algebraic surfaces

48

an algebraic surface with one of reversed orientation, this typically introduces

spheres of positive normal Euler number. It is the interaction of positive spheres

with negative spheres which provides the 8 2 x 8 2 summand required for the ap-

plication of Mandelbaum's lemma. This also explains why CP2 behaves more

like a typical algebraic surface than Cp2 does: CP2 contains embedded spheres

with normal Euler numbers +1 and +4, but no negative spheres. One further

example of this phenomenon is the following: If V is a simply connected elliptic

surface and M is a nonorientable 4-manifold, then V # M is diffeomorphic to

W # M for some W which dissolves [14]. (Roughly, this is because positive and

negative are indistinguishable in a nonorientable manifold.)

4 NUCLEI OF ELLIPTIC SURFACES

The new theory of Floer and Donaldson (for example, [2]) motivates the

study of homology 3-spheres in 4-manifolds. Specifically, if a 4-manifold is split

into two pieces along a homology 3-sphere, we can understand Donaldson's in-

variants of the 4-manifold in terms of certain invariants of the pieces which take

values in the "instanton homology" of the boundary homology sphere. In this

section, we will show how to split an elliptic surface along a homology 3-sphere

with known instanton homology, in such a way that one piece is very small but

still contains all of the topological information of the elliptic surface. This "nu-

cleus" is itself an interesting 4-manifold with boundary. (For more detail, see

[13].)

We begin with any Vn(Pl' ... ,pic). We will find embedded in this the

Brieskorn homology sphere E(2, 3, 6n - 1), whose instanton homology was com-

puted by Fintushel and Stern [7]. This splits the manifold into two pieces. The

small piece, or nucleus Nn(Pl' ... ,Pk) has Euler characteristic 3 (compared with

12n for the ambient space). Its diffeomorphism type depends only on n and

Pl, ... ,Pk, and doesn't change if we add or delete Pi'S equal to 1 (just as with

ñòð. 2 |