donaldson 1989 connected sums of self-dual manifolds and deformations of singular spaces

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Connected sums of self-dual manifolds and deformations of singular spaces

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Nonlinearity2 (1989) 197-239. Printed in the UK

Connected sums of self-dual manifolds anddeformations of singular spacesS Donaldsont and R Friedman#t Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK$ Department of Mathematics, Columbia University, New York, NY 10027, USA

Received 8 July 1988Accepted by R Penrose

Abstract. We give general conditions under which the connected sum of two self-dualRiemannian 4-manifolds again admits a self-dual structure. Our techniques combinetwistor methods with the deformation heory of compact complex spaces. They arerelated on the one hand to the analytical approach which has been used recently byFloer, and on the other hand to the algebro-geometric esults of Hitchin and Poon. Wegive specific examples involving the projective plane and K3 surfaces.AMs classification scheme number: 32699

1. IntroductionThe decomposition of the 2-forms on an oriented Riemannian 4-manifold intoself-dual and anti-self-dual components has many striking geometrical consequences.One of these is the existence of a special class of Riemannian 4-manifolds, theself-dual or half conformally flat manifolds. To define these we think of thecurvature tensor of a 4-manifold as a symmetric tensor in A@ A and, given anorientation, decompose the 2-forms into the f elf-dual parts (eigenspaces of the* -operator)The self-dual and anti-self-dual parts W+,W - of the Weyl curvature are defined tobe the components of the Riemann curvature tensor in sg(A+),sg(A-) respectively,where s; denotes the symmetric trace-free 2-tensors. (The traces of the curvature onA+, A- are both equal to half the scalar curvature, and the components in A+0 A-can be identified with the trace-free Ricci tensor.) A self-dual metric is one forwhich W- is everywhere zero. The condition depends, like the Weyl curvature andthe * -operator on A, only on the conformal class of the Riemannian metric.

Self-dual manifolds can be studied using techniques of complex geometry via thePenrose twistor construction. This interprets a self-dual manifold as a set of lines inits associated twistor space, which is a three-dimensional complex manifold. In the

$ Research partially supported by NSF grant DM S 87 03569 and the Alfred P Sloan Foundation.0951 -771 5/89/01 O m ~ ~ $ 0 2 . 5 0 9 8 9 IOP Publishing Ltd and L M S Publishing Ltd

A= A+ CBA-. (1.1)

197


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198 S Donaldson and R Friedmancontext of Riemannian geometry this theory was first developed in detail by Atiyahet a1 [l] , nd we shall recall th e main points in 03 below. A s well as the Riema nnian(or more strictly conformal) structure, the solutions of many field equations on aself-dual manifold, notably th e self-dual Yang-Mills equ atio ns, can b e translatedinto holomorphic da ta on the twistor space.Until recently only a few examples of compact self-dual manifolds were known;these were:

(i)(ii) Th e Fubini-Study metric o n CP2 (and quotients of its non-compact dual(iii) The Yau metrics on a K3 surface I?, with the stan dard orientatio n reversedSo it was a great advance when Y S Poon constructed, using twistor methods,families of self-dual structu res on t he co nnected sums CP2# CP2 and CP2# CP2#@P2, 20,21]. Even more recently A Floer has shown, using quite differenttechniques, that there are self-dual metrics on the connected sums nCPz of anynumber of copies of CP2 [6]. The purpose of the present paper is to give areasonably general theory for constructing self-dual structures on connected sumsusing twistor techniques. If X1,X z are self-dual manifolds with twistor spaces Z1,Z2we look for metrics on the connected sum X I # X , which are close to the givenstructures outside a small neck, where the connected sum is made. More preciselywe find a twistor translation of this idea; constructing a certain singular complexspace 2 using Z1,Z2 and look ing for twistor spaces m ad e by small smoothings of 2.One of our main results is that if such smoothings exist then they always representthe twistor spaces of self-dual stru ctu res on th e connected sum (see 94 for precise

statements and results).There is a general theory of deformations of singular spaces, extending theKodaira-Spencer-Kuranishi theory for complex manifolds, which was developedby, among others, Douad y, Grau ert , Forster and Knorr and Palamadov. T he mostrelevant parts of this, together with other background in complex analytic geometryare summarised in 302 and 5 below. Applying the general theory to the situationthat arises with twistor spaces we get criteria under with X I # X z admits a self-dualstructure. We refer to 00 5 and 6 for precise statements of the general results.Roughly, we associate with X1,X 2 vector spaces Xx,,x2which serve as obstructionspaces for the deformation problem. The connected sum admits a self-dualstructure if there is a zero of a certain map Y nto X xl 63 X x z , and we can obtaininformation about this through an explicit approximation Yz (which should bethought of the leading term in Y as the neck of the connected sum shrinks to zeroradius). As particular applications we get an alternative proof of Floers result.Theorem 1.2. For any n > 0 the connected sum n@P2 dmits a self-dual metric.

conformally flat manifolds, e.g. S4;CW);(these metrics are both self-dual and Ricci flat).

This is a simple consequence of our theory since, as we shall see, XCpz is zero.Moreover we shall show in 07 that for n = 2, 3 the self-dual stru ctures we constructagree with some of those found by Poon. More complicated examples, when theobstructions do not vanish, are furnished by connected sums with K as a summand.In $6 we shall prove th e following new resu lt.Theorem 1.2. For N >0 and n 2 2N + 1 the connected sum N K # nCP2 admits aself-dual metric.


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Connected sums of self-dual manifolds 199We shall now make some brief remarks on the relation of this work to other

points of view. First, we should admit that despite its length this paper is at the keypoints heavily dependent on the work of other authors. This is an inevitableconsequence of the fact that we are applying the substantial machinery ofdeformation theory. On the other side of the coin, however, some of our resultsapply to deformations of general normal crossing singularities, and it is interesting tosee how the abstract theory works out in detail for these examples (compare [19]).

Second, let us point out that there are clear and detailed parallels between ourapproach and that of Floer, which latter can in principle be extended to generalconnected sums. Both approaches construct self-dual structures by deformationtheory methods, and both give information about open sets in the moduli spaces ofsuch self-dual structures in terms of explicit parameters-the points where theconnected sums are made and certain gluing data. It seems certain that theconstructions are essentially equivalent and that both can be refined to describeprecisely what parts of the moduli spaces are constructed in this way. In thisdirection we explain in 03 how the obstruction spaces which arise for us as sheafcohomology groups agree with the obstructions one would expect to encounter inFloers approach.

Third, we should mention that all of this discussion can be carried over to thecase of self-dual connections, where the theory is rather better understood. Floersstarting point was the work of Taubes on the construction of concentrated self-dualconnections [24,25]; in general we know that there is an obstruction theory for theexistence of such connections, that we obtain in this way coordinates on neighbour-hoods of certain points at infinity in a compactification of the moduli space, and thatthe same techniques apply to self-dual connections over connected sums [3]. Weshall explain at the relevant points how the ideas of this paper can be taken over togive a twistor approach to some of this theory (here one can compare the recentwork of Gieseker [ lo]) .

Fourth, we remark that deformation theory of singular spaces has previouslybeen applied by P N Topiwala in [26] to construct the twistor spaces of Yau metricson K3 surfaces. Our approach follows his in general outline but differs in detail; forexample Topiwala emphasises the Ricci-flat condition which leads to a special classof twistor spaces. However, one could hope that the two could be united into ageneral theory, since Topiwalas construction can be interpreted as a connectedsum across the RP3s rather than the S3 s . In this direction we mention the fact,pointed out by P J Kronheimer [MI , that Poon finds a moduli space of self-dualstructures on 2CP2 hich is diffeomorphic to an open interval ( 0 , l ) . One end of thismakes up the structures found by our method, or Floers, as a connected sum with asmall neck. The other end can be interpreted as a degeneration of 2CP2 f a broadlysimilar kind involving a generalised connected sum across a copy of RP3 (in fact2CP2= V U w p where V is a disc bundle over S 2 with boundary Kip3).Thisexample suggests that it might be possible to develop a general theory forcompactifying moduli spaces of self-dual metrics, analogous to that known, thanksto the results of Uhlenbeck [27] for connections.

2. Deformations and formal neighbourhoodsIn this section we will summarise briefly some of the techniques and formalismwhich will be used throughout the paper. These have to do with linearisations of


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200 S Donaldson and R Friedmanvarious geometric objects, the linear or infinitesimal versions being expressed interms of sheaf cohomology. Fo r all details we refer to [ 7 , 8 , 15 ,16 , 181.

2.1. ExtensionsOne of the simplest examples in which cohomology appears is the problem ofclassifying e xtensio ns of vector bundles. Let W be a compact complex manifold andE, E be holomorphic vector bundles over W. Th e exact sequences of holomorphicbundlesover W are classified ( up t o equivalence) by th e vector space H 1(Hom (E, E )). Tosee this we no te that locally in W any seque nce (2.1) splits holomorphically, and twosplittings differ by an automorphism

(2.1)+ E+E+ E+ 0

A :E @ E+E @ E A(e, ) = (e + a ( e ) , e rr )

(2.2)of the formwhere a is a bundle map from E to E . If we cover W by patches W =UU , andchoose splittings on each patch, the comparison of these splittings on the overlapsU , n U, yields a Cech cocycle: (a ampE H o m ( E , E) \uwnu f ihich represents theextension class in H(Hom(E, E)).2.2. Deformations of complex manifoldsA deformation of the compact complex manifold W parametrised by a (possiblysingular) complex space T, containing a base point to , is a complex space W andholomorphic ma p

p : W + T (2-3)such that the fibre p- ( t o ) is biholomorphically equivalent to W. Technically, werequire p to be a flat map-in most of ou r exam ples T will be smooth and then wecan suppose that p is a fibration of differentiable manifolds. Also we are onlyconcerned he re with arbitrarily small neighbourhoods of the base point to in T , so itwould be more accurate to talk about germs of spaces and maps.Given one deformation over T as above and a holomorphic map f : S , so)+( T , to)we get an induced deformation f * ( W ) ver S . A deformation is versal orsemi-universal if any other can be induced from it by a suitable map f. It isuniversal if the m ap is unique. T he m ain result of the Kodaira-Spencer-Kuranishideformation theory for compact complex manifolds is that a semi-universaldeformation always exists, and can be described in terms of cohomology.Let Ow e the sheaf of holomorphic vector fields on W . The global sections@(Ow) orm the Lie algebra of the group of holomorphic automorphisms of W.The higher cohomology enters as follows: there is a holomorphic map @ whosedomain is a neighbourhood of 0 in H1(Ow) nd which maps to H 2 ( O w ) ;with @ andd@ both vanishing at 0 , such that

T = @-(O)


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Connected sums of self-dual manifolds 201is the base of a semi-universal deformation of W . If H o(@,) = 0 this is actually auniversal d eformation . (In futu re we will ignore th e fact that Q need only b e definednear 0 and just w rite Q, :H ' ( O w ) + H2 ( @ , ) . )Th e cohomology grou p HI(@,) enters into the deformation theory through theKodaira-Spencer m ap , defined as follows. Le t p : 7 j f + A be a deformation of Wover a disc ACC. Then, identifying p-'(O) with W , we get an exact sequence ofholomorphic vector bundles over W :

O+C+T*W(,+T*W-+O (2.4)since the normal bundle of W in 7 j f is trivial. This leads to an extension class inH'(Hom(T*W, C)) = H'(@,), as described in 02.1. Th e Kodaira-Spencer ma passigns this class in H' to the deformation or, more generally, for any deformationover a base S it is a natural linear map from the tangent space ( T S) , , to If1(@,).For th e versa1 deform ation it is just t he derivative of th e inclusion m ap taking T toH'.HI(@,) is the Zariski tangent space to T at 0, and so to the moduli space ofcomplex structures at the given point, insofar as the latter exists.To go in the other direction we introduce infinitesimal deformations. Let A be asubmanifold of a complex manifold X. he nth formal neighbourhood A(") f A in Xis the space A equipped with the sheaf of rings:

OA(n)= Ox/$:" ( 2 . 5 )where S A is the ideal sheaf of A in Ox. or ex am ple if A is the origin in X = C, (")carries the truncated polynomial ring: C[ t ] l t n + ' .By an 'nth-order thickening' of Awe mean a sheaf of rings on A which locally has this form. That is, the structuresheaf R of the space is locally isomorphic t o

for some vector space V. By a first-order deformation of W we mean a first-orderthickening W(' )of w and map p :W(')+ T( ') ,where T ( ' ) s a thickened point; forexample, W(' ) could be the first formal neighbourhood of W in a genuinedeformation.Now given a class e in H'(C3,) we construct the corresponding extensionO + @+ E T T*W+ 0 and let R be the sheaf of rings

R = {(f, )f E ow, Y E O ( J 9 I df =d4).Then R represents a first-order deformation of W (over the 'double origin' in C ). Inthis way we get a universal first-order deformation over the double origin inH'(@,). The map Q, represents t he obstruction to extending this thickened space toa genuine deformation. T he second-order term, Q 2 say, in the T aylor series of Q hasa simple interpretation. It is the quadratic map Q2(v) = 4 U , v] given by combiningthe Lie bracket on vector fields with the cup product H' 63H1+H 2 . In concreteterms, if we cover our manifold W by charts U , and a class in HI(@,) is representedby a cocycle ( vmp)f vector fields on th e intersections, we let f n p ( t )= exp(t va8) nddeform the manifold, to first order, by composing the overlap maps ,gap, which gluethe charts together, with the f & ( t ) . The deformed maps gas ( t ) satisfy theconsistency condition g L Y P( t )g S Y ( t )g n y ( t ) to first order in t on the triple overlapsand the t2 term gives a Cech representative for Q2.


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202 S Donaldson and R Friedman2.3. Deformations of submanifoldsNow suppose that W is a compact complex submanifold of an ambient manifold V(which need not be compact). Let v w + W be the normal bundle. We look atdeformations of W among complex submanifolds of V , .e. subspaces W of V x T ( Ta parameter space) for which the projection map p : W + T is flat and withp- '( t , ) = W x to . First-order deformations are parametrised by H o ( v W ) nd obstruc-tions are encountered in H ' ( v w ) . The complete picture is described by a map@ : H o ( v W ) +H ' ( v W ) , much as before. More generally we can consider deforma-tions of the pair (V, W). We let Ov,wbe the sheaf of holomorphic vector fields on Vwhich are tangent to W along W. This fits into a exact sequence:

o+ O,,+ Ov + O w ( v w ) + 0. (2.7)If V is also compact the groups H ' ( O v , w ) have the same significance for thedeformations of the pair as the Hi(@,) o in the absolute case. (If the versa1deformation Zr of V is smooth one can study this relative case by consideringdeformations of the copy of W in Y ; cf 34.)2.4. Vector bundlesIf E+ W is a holomorphic vector bundle the first-order deformations of E areparametrised by H'(End E ) and obstuctions are encountered in H2(EndE ) . Thesemi-universal deformation (a locally free sheaf over T X W ) has base T =


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Connected sums of self-dual manifolds 203where the h,, are a system of transition functions for E relative to localtrivialisations. (In Dolbeault cohomology EE is the class of the curvature of acompatible connection on E, the curvature being an End-E-valued (1,l) form.)

2.5. Real structuresIf W is a complex manifold, as above, and Ow s its structure sheaf, the ringed space(W , ow) s again a complex manifold which we will just denote by W. ocalholomorphic functions on W are the conjugates of holomorphic functions on W. Areal structure on W is a C" involution a :W-, W which induces an isomorphism ofringed spaces (W , Ow)+ (W , ow). f W has such a real structure a we can look atreal deformations of (W , a): that is, commutative diagrams

W y PI I (2.10)T * Twhere W+ T is a deformation as before and a, t are real structures, agreeing withthe given one on W.Lemma 2.1. Let a be a real structure on the compact complex manifold W andsuppose that the versa1 deformation W is actually universal. Then the deformationyd+redhas a real structure t.Moreover the action of t on the Zariski tangentspace is compatible under the Kodaira-Spencer map with the natural antilinear mapa* rom H' to itself, induced by a. (Here Zreddenotes the reduction of an analyticspace 2 .)Proof. This is a simple formal consequence of the universality assumption. Themap p :W-, T induces a map, also called p , from Ydo Tred,For any reducedcomplex analytic space Z we can define 2 analogously to A? above. Then p yields amap p :pd-+red.which is a deformation of W. sing a we may identify this witha deformation of W. By the versality of W we get a Cartesian diagram:

restricting to the given a on the central fibre. Iterating a we obtainWred 02 ' fI I

where now a2 s the identity on the central fibre. We now invoke the fact that ? isthe universal deformation. This implies that t2 s the identity o n the base space, andso the deformation has a real structure. By naturality the involution t acts by a* nthe Zariski tangent space H ' ( O w ) .


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204 S Donaldson and R FriedmanRemarks. (a) In our application of this idea we will be able to get around theuniversality assumption-see 94.

( b ) We can also show that the em bedding of Tred n H 1 can be chosen so that zis precisely given by the conjugation map U* on the ambient space. Moreover wecan suppose that Y is chosen to b e a 'real ' ma p, compatible with th e conjugationsU* on H 1 and H 2 . For th e quadrat ic par t Y2 his follows from the compatibility of c7with a bracket on vector fields; one should note, however, that Y itself, and theembedding of T in H ' , is not canonical so some care is needed.

(c) The proof of lemma 2.1 shows that if l f + S is an arbitrary deformation ofW with real structures U, z over a reduced base then the deformation and realstructure can be pulled back from T by a unique map. It seems likely that versa1deformations exist in the category of real spaces over (possibly non-reduced)complex spaces with real structure (suitably defined in case there are nilpotentelements in the structure sheaf). If this were true we could avoid any universalityassumptions.( d ) W e can, o r cours e, set up similar statem ents for the deformations of bundlesand submanifolds in the presence of real structures.

2.6. Description of formal neighbourhoodsLet W(") e an nth-o rder thickening of a com plex manifold W , for example a formalneighbourhood of W in an ambient manifold V. We will describe briefly a theory,due to Griffi ths [ l l ] , which comp ares W'") with a 'flat' model. First the tangentbundle T of W'"' is defined; it is a vector bundle over W. (If W'") is the formalneighbourhood of W in V , and n 2 1, T is the restriction of 7V o W.) W e have anormal bundle vw and an exact sequence

O- TW+ T +.vw+ (2.11)of bundles over W. The nth-order formal neighbourhood F(") of the z ero section invw is the flat model. T he stru cture sheaf S'") of Fc") is

S'"' = (33 O w ( s j ( v & ) ) . (2.12)j S nWe have inclusions W("-l) c W'"), F("- ' ) c '"' and if R'") is the s tructure sheaf ofW'") the kernel of R(")-. R("- ' ) is identified with O W ( f ( v & ) ) functions vanishing t oorder n - 1 along W ). Suppose we have found an isomorphism of W("-')withF("-'), we look for the obstruction to extending this to an an on W("-').Th e casen = 1 is special; if we assume is th e identity the n an isomorphism a, is equivalentto a splitting of the exact sequence (2.11). The obstruction is the extension class w 1in H ' ( T W 631 k ) . If n > 1 we have fixed a splitting T = TW Cl3 vW. canbe written $J = 1+ I where 3 :S("")-.s"(v&) is a derivation, q ( f g ) = q ( f ) g +f q ( g ) . Such a q!~ actors uniquely as

Th e automorphisms of S'") fixing ~ " ( v ; ) nd inducing the identity on

where d is the natura l derivative and a is an Ow map. Now we can certainly find a,,locally in W, and comparing the different choices on the overlaps U, n U, of a


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Connected sums of self-dual manifolds 205suitable cover we get a Cech cocycle

(a@1 a m / 3 E O(T @ . n ( G ) l L l , " U J .This represents the obstruction class w, in H ' ( ( T W (33 v W ) 3 s s " ( ~ & ) ) . If it vanishesthe possible choices for a, form an affine space modelled on H o ( ( W @ vW) 3

More explicitly, if W'") is the formal neighbourhood of W in V , we choose localcoordinates for V around W of the form (zi;A)where the zi are coordinates on Wand the EA are in the normal directions. Two choices of such coordinates compare by

(2.13)where we have omitted subscripts i, for clarity. A cover by such coordinatesystems in which the e,, e , vanish to order n - 1 along W ( 5 ' = 0) gives atrivialisation of and then the e , ,e , on the overlaps are a cocycle, for thetangential and normal components, respectively, of 0,. The matrix-valued functionh ( z ) represents a transition function in the normal bundle.

We will need a few simple facts about these obstructions 0,. First, suppose thatW moves in a one parameter family W, in V , as in 02.3, t E A and WO= W. Then thederivative of this family of submanifolds is a normal vector field:

2/ E H 0 ( V W ) . (2.14)On the other hand the derivative of the family, viewed as abstract manifolds lies inH'(0, ) . This is equal to the image of 2r 3 w1 under the map

I p ( Y w ) 3 H ' ( Z w 3 Y&)+ H ' ( T W ) = H'(0,) . (2.15)We also have a family of normal bundles Yr+ W,. Suppose w1 = 0 and we choose asplitting Tvlw= ZW @ vW.Then to first order in t we get an identification of W,with W, and we can regard the time derivative of vr as an element a,(v,) EH1(End vW). The exact sequence (2.9) gives a way to express this rigorously, atleast for the projective bundles which is all w e need in our application.)

f (y&) ) .

( 2 ' 7 5 ' ) = ( ~ ' ( z ) ,( z ) 5 )+ ( e t ( z ' , 5'1, en(z ' , 5 ' ) )

L e m m a 2.2. a,(v, ) is equal to the image of v 3 w2 under the composition of naturalmaps:

f m w ) 3 H 1 ( T v I w (8 S2(Y&)-+ H1(vlw 8 (Yw 3 s2 (&))-+. . .. . . = H I( Tvlw 3 Y&)+ H 1 ( v W 3 Y&) = H'(End vW ) .

The other point we need to take up is the variation of the 0, under change inthe trivialisation an-1 f W("-l). Specifically, suppose we change a,-' by an elementy of H 0 ( Z w 3 s s " - ' ( v & ) ) H ~ ( T ~ D " - ~ ( Y & ) ) .ecall from 02.4 that there is acanonical element 5, ofH'(End vW3 Qk).L e m m a 2.3. Under a change of a,,-l by y, the normal component of w, changes bythe image of y 3 5 under the natural map:@(Tw 3 s"-'(Y&)) 3 H'(End vW3 Qk)+ H1(End vW3 S"-'(Y&))-, . . ,

* . .- 1 ( Y , 3 S"(Y&)) c H1(T 8 "(Y&)) .The verification of lemmas 2.2 and 2.3, which is largely a matter of notation, is leftto the reader.


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206 S Donaldson and R Friedman3. Review of twistor theory3.1. Twistor spacesThe Penrose twistor transformation is a one-to-one correspondence betweenconformal classes of self-dual 4-manifolds X and twistor spaces Z, which arecomplex manifolds of dimension 3 with certain special properties. We will outlinethe relevant part of this correspondence, referring t o [1 ,3 ,1 3, 22 ] for most proofsand details.If X is an oriented Riemannian 4-manifold its twistor space 2, viewed as a C"6-manifold, is defined to be t he unit sph ere bundle of &+X. So the re is a fibrationn : - X with 2-sphere fibres n-'(x) = L, = S 2 . Th e f ibre Lx s naturally the spaceof compatible complex structures on T,X (with the opposite orientation to the usualone). The Riemannian connection gives a splitting of the tangent spaces of Z intohorizontal and vertical subspaces and there is then a tautological almost-complexstructure o n Z (in the vertical directions we take the standard complex structure onS2) . This almost-complex structure depends (like Z itself) only on the conformalclass of t he m etric on X and is integrable if and only if X is self-dual. From now onwe suppose that this is the case.By construction, the fibres L, are holomorphically embedded curves in 2,isomorphic to P', and with norm al bundle:

YLz = q ) 63 O(1). (3.1)More intrinsically, let V * + X be the spin bundles (which always exist locally onX ) . They a re complex 2-plane bundles with structure group SU (2). Then :

L, = P(v;) (3.2)YL,= n*(v:)8OL,(l). (3.3)

In addition 2 posesses a free antiholomorphic involution (or 'real structure') (T,which is th e antipod al ma p on each fibre of n. he lines L, can be described then asth e 'real' lines in Z , preserved by (T.Conversely suppose that Z is a complex threefold such that:(i) the re is a free anti-holom orphic involution (T o n 2 ;(ii) there exists a holomorphically embedded curve L = P' in 2, preserved by (Tand with normal bundle vL= O(1) CB 4 1 ) .Then H 1 ( v L ) 0 and H o ( v L ) s four dimensional so, by the deformation theory forem bedded sub-manifolds (02.3) th e set of deform ations of L inside Z (with the sam enormal bundle) may be parametrised by a four-dimensional complex manifold X c(not in general compact). The involution U acts on X c ; le t X c X c be the realanalytic 4-manifold of real lines in 2. If x is a point in X and L, the correspondingline, the tangent space to X c a t x is

TX," = @(YL,) = H o(O(1) 63 a(1)).Thus we can w rite

TX:= V: (8V ;for two-dimensional spaces V:, V ; and the identification is natural up to scalars.This means that the tangent space of Xc as a natural complex conformal structure


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Con nected sums of self-dual manifolds 207(i.e. the equivalence class of symmetric forms on V'3 V- given by the tensorproduct of basis elements in A'V', A2V-), and this restricts to a Riemannianconformal structure on X . It can be shown that metrics in this conformal class areindeed self-dual, and it is not hard to verify that this construction Z+X is inverseto the previous one X+ Z [3].Example . Let X be a Euclidean 4-space E with the flat metric. The twistor spaceZ(E) is the total space of the bundle:v+ 3 O(1)- P'= P ( v - ) .Here Vc , V- are the spin bundles of E. The four-parameter family of lines is justthe space of holomorphic cross-sections of the holomorphic fibration of Z(E)over P'(which should not be confused with the twistor fibration n). or every point x in Ethe inclusion of L, in Z is biholomorphic to the inclusion of the zero-section in thetotal space of the bundle O(1) G3 41).

For a point x in a general self-dual space we may compare the jets of theconformal structure around x with this flat model, and the remaining part of theWeyl curvature Wr,' gives, or course, the leading, second-order, term in theexpansion of the structure about x . In the twistor space we can look at theobstructions to trivialising the formal neighbourhoods of the line L, c Z , asdescribed in 03.4. Not surprisingly these ideas correspond, and we get a twistordescription of W + .

First, the obstruction to trivialising the first-order neighbourhood L(') lies inH'(TZ 3 vL1) = 0, so a trivialisation exists. The choice in making this trivialisationis Z$( TZ 3 v i ' ) = V + 3 V-. Given such a choice the obstruction to trivialising L(')lies in H'(TZ 3 s ' (vz ) ) , which again vanishes and the choice in making the secondtrivialisation lies in Ho(TZ3 s ' (v 2 ) ) = s:(V'). Then we meet an obstruction w3 tomaking a third trivialisation in H '( TZ 3 s3(vZ) )= H ' ( v L3 s 3 ( v i ) )= s3(V') 3 V c .Here we have identified the spin bundles with their duals, which amounts to a choiceof metric at x in X , within the conformal class. Now we have an irreducibledecomposition:

s3(V+) 3 v+= S"V+) G3 S2(V+).So the obstruction o3has components w, say, in s4(V'), s2(V+) respectively.

Now the choice in trivialising L(')corresponds to the choice of the first jet of ametric, in the conformal class, on X (see [3], p 387). Then the Weyl tensor W + sdefined as an element of s;(A+) = s 4 ( V + ) .Proposition 3.1. For any choice of trivialisation of L(') there is a unique trivialisa-tion of L(') for which the component n of o3 anishes. The remaining part w of w 3then corresponds under these natural isomorphisms to the Weyl tensor W + .

For the first part of this we use the description in 03.4 of the action of sectionss E Ho(TL 3 s2(v i ) ) on the obstructions o3 H 1 (vL 3 s3 (v i ) ) :

0 3 + 0 3 + 5,s.The class 5; in H'(End v 3 S2L) is id 3 [L]where [L] s the fundamental class of L inH'(S2;) = C. It follows that s-., &,s gives the standard map s ' ( V + ) ~ s 3 ( V + )3 V + =


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208 S Donaldson and R Friedmans3(V') 63 V')* corresponding to multiplication V + 3sz(V')+ s ' ( V + ) , so thereis a unique way to remove n . The identification of w with W + , one directly, is arather lengthy calculation which we leave for the interested reader.

3.2. Self-dual connectionsA connection V on a (vector) bundle E over a Riemannian 4-manifold X is self-dualif its curvature F E A$(End E ) satisfies * F = F . Self-dual connections over self-dualmanifolds can be described in terms of holomorphic data by the well known Wardcorrespondence. If E + Z is a holomorphic bundle over the twistor space of such amanifold which is holomorphically trivial over each real line L, we can construct aself-dual connec tion as follows. W e define vector spa ces E , for x in X by

E, = Ho(L,; ElL) , . ( 3 . 6 )These fit together smoothly, because the restrictions of E are trivial, to form avector bundle over X (a 'direct image' bundle). Moreover E has a naturalconnection which turns out to be self-dual. The exact sequence

0 + E l L 6 3 v ~ + E p - - E I L + o (3.7)together with th e vanishing of H i ( E I LC3 v i ) , shows that a trivialisation of 81, has aunique extension to a trivialisation of l ? l L ~ ~ ~ .e define a connection V o n E by therule tha t Vs vanishes at x E X if t he correspo nding section s of E is com patible w iththe natural extension from L, to L:'). Conversely a self-dual connection o ver X liftsto a holomorphic bundle over 2. For unitary connections we need 'real' bundles E ,with a holomorphic isomorphism t:E* --$ a*E inducing a positive Hermitian formon the P ( L x ; lL , ) . Th ere is a one -to-on e correspondence between isomorphismclasses of such bundles with real structure and unitary self-dual connections, m odulogauge equivalence.Just as for the self-dual spaces themselves, the curvature F + of a self-dualconnection on X has a twistor description. The obstruction to extending atrivialisation of E I p to lies in

H' (End E 63s2(vE,)).This space is isomorphic to End E , 63A: and the obstruction goes over to thecurvature under th e natural isomorphism.

3.3. Linear field equationsTh e cohomology groups of various sheaves on the twistor space Z of a 4-manifold Xhave a translation (the 'Penrose transform') into solutions of linear differentialequations on X. he case relevant to th e obstructions in the deformation theory isx; = H y Z , !&63 K Z ) . (3.8)

xx = H * ( Z ,0,) (3 .9)If X , and hence 2, is compact this is dua l, by the Serre duality theore m, to

which is the home of the obstructions described in 02. O n the other ha nd, s tayingwith the Riemannian 4-manifold X , we can regard the map which assigns the


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Connected sums of self-dual manifolds 209anti-self-dual Weyl curvature to a conformal structure as a nonlinear differentialoperator:(This involves various natural identifications made using the fixed given self-duallnetric on X , cf [6] 03.) We let D be the linearisation of this operator about 0 (i.e.the given metric), and D * he formal adjoint operator. So D * s a second-orderlinear differential operator:

(3.10)- :r(A+3 A-)+ I'(s:(A-)).

D :T(s;(A-)) + (A' 3 A). (3.11)Now suppose that h is an element of X: = H 1 ( Z ,Q: 3 K z ) and L is a twistor

line in Z. Then we have two restriction maps:and H ' ( L , 5 2 : 3 K Z I J - - j H 1 ( L , Q ; @ K z ( L ) = H ' ( L , 0(-6)) (3.13)and the latter space can be identified with the fibre of &A-) at the correspondingpoint of X . In this way h yields a section of sg(A-) over X . The following fact is nowtrue for any twistor space (not necessarily compact).Proposition 3.2. The construction above sets up a one-to-one correspondencebetween X$ nd the kernel of D * n r(si(A-)).

H ' ( Z , Q : 3 K z ) + H 1 ( L , Q : @ K Z ( L ) (3.12)

The relevance to the deformation problem is that in the compact case the kernel ofD * an be regarded, using the L 2 inner product, as the dual of the cokernel of Dand this latter is the space where the obstructions to deforming the solution of thepartial differential equation W-(g)= 0 lie. Proposition 3.2 is an instance of thegeneral Penrose transform, but unfortunately it has not been considered explicitly inthe published literature. For the case when X is conformally flat, or a domain in@P2,t can be deduced from the general transform for homogeneous bundles givenin [4], xtending the results of [5]. Or one can use arguments like those in [13]. Thecorresponding theory for self-dual connections has been treated more extensively.In place of D* e now have the operator

d$ :A+(End E ) + A1(End E ) (3.14)(see, for example, [ l] and [3]) and there is a Penrose transform taking the kernel ofthis operator to the space

X:,v = H ' ( Z , End E 3 K,)dual to H 2 ( Z ;End E ) . This isomorphism is explained in [22, 0111. In fact, in bothsituations the whole Dolbeault complex on the twistor space can be related to acomplex on X . For metrics this is the complex formed, in Floer's notation, by thedifferential operators L and D [6,03]. (We shall not in fact use proposition 3.2 inour proofs.)

4. Twistor interpretationof connected sums4.1. The flat modelThe connected sum of two oriented 4-manifolds XI, X , is defined as follows. Wepick open neighbourhoods U,! of points x i in Xi nd coordinate charts which identify


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210 S Donaldson and R Friedmanthe U ; with ope n neighbourhoods U, of 0 in R4. It is convenient t o think of the U, assubsets of distinct Euclidean spaces E i , which we can in turn identify with ( T X , ) , .W echoose an orientation reversing linear isometry

p :El+ E2 (4 .1)and identify E,\{O} with E2\{0} by a map ip,A:

where A is a positive real parameter. When A is sufficiently small iP,* gives anorientation-preserving diffeomorphism between a pair of annular regions in theand the connected sum is formed by removing small balls about the points xi andidentifying these annuli.It is an elementary fact that the connected sum of conformally flat manifoldsagain has a conformally flat structure. This follows from the conformal property ofthe inversion maps ip,h. Since conformally flat 4-manifolds a re self-dual, an d so havetwistor spaces, we can translate this special case into twistor language. We beginwith the punctured Euclidean space IE\{O}. Its twistor space W has a holomorphicfibration:

h- :W -+ P ( V- ) = CPl(cf the example in 03). Each fibre is a copy of C2\{O} so itself fibres over CP'. t iseasy to se e that th e two can b e combined in a holomorphic fibration, say

C * 4 W- h + x h - P(V') X P(V- ) = QE. (4.3)Geometrically, QE = S2 X S2 is the Grassmannian of oriented 2-planes in IE and Wcan be naturally identified with the space of pairs

(2-plane n n E, point in n).Then h+X h- maps such a pair to its plane n.However one looks at i t , W is thetotal space of the holomorphic C* undle associated with the line bundle 4-1, 1)over QE. To describe the connected sum in terms of twistor spaces we m ust identifythe map j p , h :Wl+ W,, between copies W.of W attached to E i , induced by ip,h. FirstA2((p) induces an identification:

~2 :&E, QE,.p also gives an isomorphism between the line bundles O(--l, 1) over Q,, and41, -1) over Q,, covering p2 (no te tha t p2 switches the factors in the e,,).p , A sgiven by identifying the W. fibrewise, using on each pair of C* ibres a map of theform z I+ A/z . When A is real jp ,A espects the real structures on the W and thetwistor space of the connected sum is made by identifying the appropriate parts ofWl. under jp , * .Fo r our purposes, the best way to think of this identification goes via the blow-upof the twistor spaces. Let 2, be the blow up of the twistor spaces ZE, long the linesn-'(O). Z , contains a holomorphically embedded copy of the surface Q,, theprojectivised normal bundle of n-'(O). The fibration h+ x h- extends to Zi,exhibiting this space as the line bundle O(1, -1) over Qi.We use p2 to identity the


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Connected sums of self-dual manifolds 21 1Q i and form the singular space with a 'normal crossing'

z =z, 2,.Qi

Explicitly, Z is the union of the two 'axes' in the total space of the rank-2 bundleE = O(1, -1) G3 4-1, 1) over Q = Q , = Q 2 . The map p fixes a du al pairing b etweenthe two factors and thus a qu adratic form q o n E , i .e.q ( & =&I E E O(1, -11, 11 E O(-1, 1).

Z is the variety { q ( & q ) = 0} inside the total space of E . We now deform this to asmooth sub-variety Z(A)= { q (5 , 7) = A}, which can be identified with the twistorspace of the connected sum formed using fo r A > 0. (Actually we can take A < 0,but changing the sign of A just corresponds to replacing p by - p . ) Of course sincethe construction is local we can think of the twistor space of the connected sum ofany conformally flat 4-manifolds in the same way as a canonical deformation of asingular space formed by identifying the blown-up twistor spaces along a surface Q.4.2. Standard deformationsNow let X I ,X , be any compact self-dual 4-manifolds, Z , , 2, their twistor spacesand x i points in X i . L et Z j be the blow-up of Z , along n;'(xi). Suppose we fix anorientation reversing isometry p between the tangent spaces (TX,),s above . pidentifies the exceptional divisors Q i in Zi. e form the singular space Z byidentifying the Q i in Z i ;we will write Q for each of the Q ,. M ore precisely, Z has astructure sheaf consisting of local holomorphic functions on th e Z i which agree alongQ. p also gives us a dual pairing between the normal bundles of Q in Z1, Z , . W ewill denote them by O(1, -l) , O(-1, 1) respectively. The zi have antiholomorphicinvolutions induced by those on the Z i and, since p is 'real', we get anantiholomorphic involution 17 of Z . The discussion of 04.1 motivates the followingdefinition.Definition 4.1 . A standard deformation of Z consists of:( a ) a smooth complex (n + 3)-dimensional manifold $ and a proper holomor-phic map p :$+ S where S is a neighbourhood of 0 in 62";

( b ) an isomorphism of complex spaces p - l (O ) = Z ;(c ) antiholomorphic involutions 17 on 3 and S, compatible under p and such( d ) 171~-1(,,) s the given involution on 2.that:In addition we assume that near any point of Q c Z c 3 there are local coordinatesz l , z,, 23, 2 4 , tZ , . . , , o n $ such that p(zl, . . . , ,) = (zlzz, tZ ,. . . , Hence thesingular fibres of p are precisely those lying over the smooth hypersurface t l = 0 inC", here f l is th e first coordinate fun ction.

The main result we shall prove in this section is the following theorem.Theorem 4.1 . If $+ S is a standard deformation of a singular space 2 formed byidentifying the exceptional divisors in blown-up twistor spaces, as above, then forsufficiently small vectors s in the fixed locus S" of the involution 17, not lying in the


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21 2 S Donaldson and R Friedmanhypersurface {tl = 0}, the fibre p - ' ( s ) is the twistor space of a self-dual conformalstructure on X I # X 2 .

So if Z admits a standard deformation the connected sum X1# X 2 admitsself-dual m etrics.Remark( a ) In a standard deformation we can always choose coordinates on the base Sso that a is given by complex conjugation of coordinates t, . This is a standardconsequence of th e implicit function theo rem . In dee d, let { t , } be a set of coo rdinatefunctions on C"= TS and set tl' = $ ( t ,+ t p ) . T h en t,' is holomorphic and has thesame differential at 0 as t,. Hence {t,'} is also a coordinate system, and in thesecoordinates a is just conjugation. In particular th e set of rea l points S" is locally justR", and the real non-singular fibres are a copy of R"\Rn-' .( b ) There is no loss of generality in supposing that the base of a standarddeformation is one dimensional, for we can always restrict to a a-invariant line in Stransverse to the hypersurface { t l = O}. For simplicity we will give the proof oftheorem 4.1 for this one-dimensional case.

(c) It is easy enough to see from the proof of theorem 4.1 below that theself-dual structures found on the connected sum are small deformations of the'wedge' of the original structu res on th e X , in the following sense: for any compactsets K , c X,\{x,} there a re maps f , ( s ) rom K , to the 4-manifolds X ( s ) of real lines inp ( s ) such that the pullback by f , ( s ) of the self-dual conformal structure on X ( s )differs from the original o n K , together with all derivatives, by O(ls1).( d ) The main assumption we make is that the total space of the deformation issmooth. It is possible to dispense with this; if the given total space is singular onecan make a finite nu mb er of blow-ups to get a sm ooth space to which the argum entsof 04.3 below apply. However, this gives only a small increase in the generality ofthe resulting existence theorem for self-dual metrics.In the case when X , are conformally flat we can make a standard deformation inan elemen tary way by using th e constructions of 04.1 abo ve. It is worth m entioninganother example where we get an explicit standard deformation. Let Z1 be thetwistor space of X1 and blow u p the product Z1X C along the line n; ' (x , ) x {0} toobtain a fourfold 3, with a natural map p :3 4 C nduced by the projection m ap onZ , x C. The fibres p - ' ( z ) for z # O are naturally copies of 2 , while ~ ~ ' ( 0 )szlU Y , where Y is the exceptional divisor of the blow-up. This can be identifiedwith the blow-up of CP3-the twistor space of S e a l o n g a line. In short is acanonical standard deformation for the pair of 4-manifolds X1, S4 and the 'new'conformal structures we obtain on the X ( s ) are just copies of the original on e on X1.The construction is the twistor analogue of a family of conformally equivalentmetrics g, on X1 which expand small balls about x1 into large 'balloons' anddegenerate. in the limit as &--to, in to the wedge X1v S4.4.3.Proof of theorem 4.1Begin by noting two families of real lines in the blow up of 2,-the pre-im ages oftwistor lines nT1(x) or x # x l and the real lines, in the linear system lO(1, 1)1, o nthe ex ceptional divisor Q. Th e lat ter correspond to the graphs of maps f W 1 +P1such tha t f ( - l /Z ] = - 1 / f ( Z ) , that is to th e isometries of the rou nd 2-sphere. So these


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Connected sums of self-dual manifolds 21 3lines are parametrised by a copy of SO(3) which we can identify more invariantlywith the real projective space P ( (T X l ) , , ) . (Recall that SO(3) is diffeomorphic toRIP3.) The former set of lines are parametrised by Xl\{x l} . Any collection of linesin the smooth manifold Z 1 has a natural topology induced on it and it is easy tocheck that the induced topology on

GI= Xl\{Xl} U P (T X 1 )is that of the real blow-up of X I at xl. We leave this exercise to the reader.Similarly, in the singular space Z = Z l U,& we have three families of lines:pre-images of the ordinary twistor lines in Z1, Z2 and the lines in Q. The naturaltopology on the union of these three families is that of G = GI U G,, where Gi rethe real blow-ups of the X i and P = P ( T X l )= P ( T X 2 ) , where the identification ismade using p. This is a real singular variety, with a normal crossing along P , andtransverse to P , G is locally modelled on { xy = 0} c R2. We observe that theconnected sum X I # X , is the natural smoothing of G obtained by changing thestructure transverse to P according to the model { x y = E } for E # 0. (The choices ofsmoothings E > 0, E < 0 correspond to the choices f p which have the same actionon the projective spaces.) To prove theorem 4.1 w e have to show that thesmoothings of the twistor space Z contain spaces of lines parametrised by thesesmoothings of G.

We now turn to the smooth total space 3, and consider G as a set of lines in 3.To understand the deformations of a line L of G in the ambient space 3 we have tolook at the normal bundle vL of L in 3. Let us suppose for simplicity from now onthat S is one dimensional, so vL has rank 3. If L does not lie in Q its normal bundlein Z is O(1) @ 6'(1), the same as the normal bundle of twistor lines in Z1,Z2.So wehave an exact sequence:

0- O(1) @ O(l )+ YL O+ 0.This sequence splits since H'(O(1)) = 0. So in this case vL = O(1) @ O(1) @ 0.f L isa line in Q we have an exact sequence:

O + YL,Q YL vQlL+ 0 .The normal bundle Y ~ , ~f L in the surface Q is O(2), since the self-intersectionnumber of L is 2. On the other hand the normal bundle Y, of Q in 3 is spanned bythe normal bundles of Q in Z 1 ,Z, which are O(1, -1) O(-1, 1) respectively. SovQ = 4 1 , 1) @ 6'(-1, l ) , and vQIL= O@ 6'. SO this sequence also splits, sinceH' ( O( 2) ) = 0, and in this case vL = O(2) @ O@ 0.So for all lines L we have H 1 ( v L ) = O and according to the theory sketched in02.3 there is a universal local deformation modelled on a neighbourhood of 0 in@(v L ) = C 5. Moreover these are universal deformations of all nearby lines, and sothese local deformations fit together to give coordinate charts on a complexmanifolde f lines in 3. The function p :3" c C must be constant on each ofthese lines (since CP" is compact), so induces a holomorphic map r :e+ . Alsothe involution a of 3 acts on the set of lines, and shrinking if necessary, itinduces an antiholomorphic involution, which we also call a, of e.he fixed pointset ,$? of a (the real lines in the family) forms a smooth five-dimensional real analyticmanifold, containing G.

The key to the rest of the proof of theorem 4 .1 is the following fact.


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214 S Donaldson and R FriedmanLemma 4.1. Th ere is an neighbourhood N of G inwith normal bundle in 3 not isomorphic to O(1) 9 O(1) 9 6 are th e l ines of G in Q.such that th e only lines L' of N

(As we have seen above, the lines in Q have normal bundle O(2) 9 09 .)Pruof. Consider the versal deformation of a line L of G, modelled on H o ( v , ) . T hecondition that a bundle over CP' be isomorphic to O(1) 9 O(1) 9 6 is open (forbundles E of the given topological type, it is equivalent to the conditionfF'(E(-2)) = 0). So if L is not one of the lines in Q we can suppose, by restrictingthe local deformation, that all the other lines in the deformation have the normalbundle O(1) 9 O(1) 9 6'.f L is one of t he lines in Q with vL= O(2) 9 O90 e askwhich points in the deformation space represent lines with this 'jumping' normalbundle. For this we apply the theory of 002.3 and 2.4. The versal deformation ofthe bundle E = O(2) 9 O90 ver P' is modelled on H'(End E ) = H1(6( -2) 90(-2)) = C2. Th e deformations just deform the direct sum in to extensions:

o+ 0630- F+ O(2)+ 0and it is easy t o see th at t he only bundle in the versal deformation no t isomorphic toO(1) 9 4 1 ) 9 6 is that corresponding to the point zero in C2. Now the exactsequence O + O(2)+ T 3 I L + vL+O splits so the first formal neighbourhood of L in$ is trivial. Choose a trivialisation; then according to 02.6 we have a linear map:

y, :H o (v,) + H' (End vL)representing the derivative of the family of normal bundles. If yL is surjective itfollows that there is a smooth submanifold HL of codimension 2 in U, consistingexactly of the nearby points with jumping normal bundle. But we know that thefamily of lines in Q is three dimensional and has jumping normal bundle so it lies inH L;hence this family m ust account fo r all of H,. an d, taking a finite cover of G in 3by these local patches, lemma 4.1 follows.It remains the n to see that y, is surjective. As stated in lemm a 2.2 y, is inducedby multiplication with the component of the second-order obstruction o2 n:

H ' ( L ; vL3 s2(vE))c H ' ( L ; T 3 3 s2(vE)) .Now the second-order neighbourhood of Q in % differs from t he flat mode l Y$" by anelement of:

H ' ( Q ; s 2 ( v Z )8 3 ) = H 1 ( O ( - 2 , 4))(cf corollary 5.1 below). But the restriction map from H ' ( Q ; s2(v;) 3 T 3 ) toH ' (L ; s2(v;) @ T 3 ) (induced by vZIL+ v i ) is zero and this means that the secondformal neighbourhood of L in 3 is isomorphic to the corresponding neighbourhoodof L in Y$" (the total space of O(1, -1) 9 0(-1, 1) over (2). So it suffices to provethe surjectivity of 'yr. for the flat model o r, equivalently, that th e induced family ofnormal bundles is versal for the defo rmations of O9O@ O(2). T he flat model is th etotal space of O(1, -1) 9 0(-1, 1 ) and on e sees easily that it is enough t o show thatthe deformations of L in the total space of O(1, -1) contain a versal deformation ofthe normal bundle 09 (2) (and symmetrically in the other factor).

Write W, for the total space of O(1, -1 ) over Q and consider th e one-param eterfamily of 'vertical' deformations L, (s E C) of L given by the trivialisation of41, - l) IL. We choose inhomogeneous coordinates (q,2) o n Q = P'x P', and let


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Connected sums of self-dual manifolds 21 5wi l /z i . Using these coordinates, trivialise W, over {zl # m, 2, # m} via coordinateszl, z2, ; call this open set U , = C2X @. Glue in another copy of C2 C, ithcoordinates w,, ,, ' which trivialises Wl over {zl = 0, 2, = 0}, and giving coordin-ates on an open subset U, in Wl with transition functions

wi = l / Z i t' = (z,/z,)t.Now let z be an inhomogeneous coordinate on L and consider the m ap L x @+ W,defined by z1 = 2, = z, t = t' = s; aking L X {s} to L,. Over U , , d / a z 2 and d l d t spanthe normal bundles of the lines L, and over U, we can use d / d w 2 and d l d t ' . T h eresulting transiton function in the normal bundle of L, is

~s = ( 1 /z2 s ir)0 1 '

Thus the derivative of @s a t s = 0 is (: I t ) and this represents the generator ofH1(End(O@ O(2)), since l / z represents the generator of H 1 ( O ( - 2 ) ) . This concludesthe proof of lemma 4.1.End of proof of theorem 4 .1 . Let % c 3 X 2F be the universal family of lines. Thuswe have comm utative diagram:

in which all maps are compatible with real structures. By construction %+ 2F is aP' bundle over Zc, nd thus is smoo th overF.ext we claim that the map %+ %is smooth, at least in a neighbourhood of the real locus, and possibly after shrinkingS. To see this, let L be a real line in Z and choose a point x in L. The differentialT % , x - fT,,x fits into two exact sequ ences:0- TLx T % , x + p ( v L ) + oo+ TL,x+ Ts,x+ L,X -+0.

After shrinking S if necessary, we have that vL s generated by its global sections, soTQiXmaps onto T,,,, and hence % is smooth over Z. Th us f or sufficiently small s nS"\O, q - l ( s ) and r - l ( s ) are smooth, while at 0 q- l (O) and r-'(O) have normalcrossings. Moreover r at 0 has the local form xy = t. Everything is compatible withthe real structures so the same is true of r :2?+ S". In particular, r - ' ( t ) nZ = 2, s asmooth 4-manifold for t E S"\O.A standard argument shows that r :2 - S" is proper . Thus X , is a compact4-manifold, for t E S"\O. For such t , a point in X , is real line L in p - ' ( t ) = Z , , andwe know by lemma 4.1 that the normal bundle of L in Z is O@ O(1) @ O(1), if t issmall. But the normal bundle of Z, in 2 is trivial so we have an exact sequence:o + v L , z , + o @ O(l)CBO(l)+O+O

and this shows that the normal bundle of L in Z, i s O(1)@8(1) . Moreover theinvolution on 2, induced by (J is fixed-point free for small t (since it is so o n 2,J.


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21 6 S Donaidson and R FriedmanThus 2, is the twistor space of a self-dual manifold containing X , and , since X, scompact, this must actually be X,. Finally, since r :%+ S" as the local form xy = t ,we see that X , is given by the standard smoothing of X,, as discussed at thebeginning of the proof, and is thus diffeomorphic to X1 X,. This completes theproof of theorem 4.1.5. Deformations of singular spaces5.1. General theoryThere is a theory of deformations of a co mp act, reduced complex analytic space 2which is quite parallel to the theory for manifolds summarised in 91. The roles ofH'(O,) there are now taken up by groups:

TI, = Ext'(Qk, Oz) (5.1)where Q i is the sheaf of Kahler differentials o n 2. The se are the 'global ext ' groups:T i classifies extension s of sheaves of OZmodules:

0- 6"+ 9- Qi-+ 0 ( 5 4over Z . It is clear then how to modify the d escription of 92.2 to define th e derivativeof a deformation in Ti-we take the extension class of the sheaf of Kahlerdifferentials on the total space g1 estricted to the central fibre. Again T i gives th euniversal first-order deformation and there are obstructions expressed by a mapQ,: T i + T : ; @- ' (O) is the base of a semi-universal deform ation. Also TO, is the Liealgebra of the group of holomorphic automorphisms of 2.

We can also define sheaves TI, of Oz modules:TI,=X(QI,,Z).

These are related to th e T I , by a spectral seque nce E;sq+ TpZ'" withE$l9= H p ( Z , 2%).

In the smooth case zI,= 0 for i >0, z is the sheaf of vector fields and we recoverthe previous set-up. (In general z$ is the sheaf of derivations of Oz.) he singularsituation that we need to consider; as described in 94, is perhaps the simplest afterthe sm oo th cas e, and for gre ater clarity we will consider a gene ral class of problems.Let V , be compact complex manifolds of the same dimension n and supposether e ar e hypersurfaces in V , which are biholomorphically equivalent. Fixing anidentification between them, we will denote each of the by W . L et vr+ W be thenormal bundle of W in V,. Then let V be the singular complex space obtained byidentifying VI, V, along W in a 'normal crossing'. So V is singular on W and locallyequivalent to {zlz2= 0}c Cn+' ther e. In this case the sheaf z$ is zero (this is true forany locally complete intersection: roughly the 2 measure the local obstructions toextending first-order deformations and these obstructions are easily seen to vanishfor comp lete intersections). M oreove r the sheaf z is supp orted on W c V and is thesheaf of sections of the line bundle Y,@ Y,+ W . (Roughly, this is because the mostgeneral deformation of {zlz2= 0}c en+'s t o zlz2= E ( z3 , . . , z , + , ) . )Ou r spectralsequence then am ounts to the exact sequence:O + H ' ( V ; z $ ) - + T T : + H o ( W ; Y , @ Y , ) ~ H ~ ( V ;$ ) + T $ - + H ' ( W ; Y , @ Y ~ ) .

(5.3)


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Connected sums of self-dual manifolds 21 7The subspace H 1 ( V ; b) corresponds to the deformations of V which are locally

trivial, i.e. for which the singularities remain locally a product. The image in theprojection T : +H o ( V ; :) measures to first order the change in the singularities.Suppose in particular that v1C3 v2 is the trivial holomorphic line bundle over W (andfix a trivialisation). Then V is called d-semistable [9]. In this case H o ( v l8 2) = Cand a deformation of V over the disc A has a smooth total space ~ d :'"+ A providedthe image of the derivative of the deformation in H 0 ( v 1C3 v2) = C is non-zero. Inthis case there are local coordinates around the singular points of V c V,( t , zl, . . . , z ,) say, such that such that JC is given locally by t - zlz2 [9]. Inparticular the fibres V ( t )= ~ d - l ( t )are smooth for t # 0.

We have, then, a good theoretical understanding of the conditions under which'standard deformations' of our singular twistor space exist. In 335.2, 5.3 and 6 belowwe will spell out in more detail the implications of these. There are naturalextensions of this theory to pairs of spaces, to vector bundles and to real structures,all very much as in 32, and we will mention these at the appropriate places.

5.2. The unobstructed caseWe continue to discuss our d-semistable space V , as above, although we are ofcourse particularly interested in the case when the components V , are the blown-uptwistor spaces Zj and W is the quadric Q. We leave the discussion of real structuresto 36 .The first task is to analyse r& Let V ' be the normalisation of V (the disjointunion of VI, V,) and q :V'+ V the obvious map. Sections of r; on an open setQ c V can be identified with holomorphic vector fields ( u l , U,) on the twocomponents of q - l ( Q ) such that the v ihave zero normal component along W andthe tangential components agree there. (This must be modified in an obvious waywhen q-'(S2) is contained in one of the V,..) That is, we have an exact sequence ofsheaves on V :o + ~~+q*@,~~,,,,*-,i*O,-,O (5.4)(see [9, P 881).Now the cohomology of Ov.,wluw,ver V ' agrees with that of its direct imageover V. On the other hand the former sheaf is the direct sum of O,,,, and OV2,+.which fit into exact sequences

0 -+ OK,, +0, , i+ (5 .5 )The long exact cohomology sequence of (5.4) then gives the following.Proposition 5.1. Suppose Hp(v,)0 for all p and i = 1, 2. Then there is an exactsequence:

In particular if also l ip (@, ) 0 for p = 1, 2 then: H 2 ( r V ) H 2 ( O v , ) i3 H2(Ov,) .O - , H ' ~ ~ , ) + H ~ ( O , ) i3Ho(O,)+HO(Ow)--t~'(r,)-, . . . etc.


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21 8 S Donaldson and R Friedmanall vanish then (5.3) and (5.4) imply that T $ = O , the deformation problem isunobstructed and we can lift the third term C to T : to get smoothings of V. All ofthe conditions ( a ) , ( b ) and ( c ) are satisfied in the twistor case; we now go a stepfurther to trace back to the twistor spaces Zi.Proposition 5.2. F or 2 = 2 ,U Q2, where zi re the blown-up twistor spaceszi- Zi e have:4i

H2 ( t $ ) = H 2 ( Z * ;O,,) @ H2(Z2; eZ2)and an exact sequence:O+T$+HO(Z1; 0z , ) @ H o ( Z 2 ; Oz2)+Ho(Q; O Q ) + H ' ( Z ; t$)

+H'(Z1;@Z1,L1)@ H1(Z*;@Z2,L2) -+ 0.This is just a combination of the exact sequences above and the isomorphisms:H y Z ; 0.2)= H ' G ; ,,I,,) for all i.The latter follow from the Leray spectral sequence applied to the q, (alsoH 2 ( Z , ; O , , , ) H 2 ( Z , ; 0,) since the normal bundle of the line L, has vanishingH ' , H 2 ) . Th e terms in this exact sequenc e have a natu ral interpretatio n. As we haveseen H1(Z;$) represents deformations of Z preserving the form of the singularityand the sequence says that these a re given by deformations of the pairs (Z,,L,) ( thelast term in the sequence) and deformations of the map identifying the quadricsurfaces in the Z, ( the term H0(0,)),odulo the automorphisms of 2, ixing L, ( theterms H o ( Z , ; e,,,>).

As far as existence of smoothings goes we have, in the unobstructed case, thefollowing.Corollary 5.1. f H 2 ( Z , ;0,) vanish for i = 1, 2 there is a complex standarddeformation of the singular twistor space 2.

5.3. Obstructions and formal neighbourhoodsLet q, , W be varieties as in 005.2 and 5.3 and denote the obstruction spacesH 2 ( q ;0,) by 2;.W e want to understand the composites of th e differential:

d 2 : H " ( v 1 2 ) + H 2 ( V ;t;)in the exact sequence (5.3),with the restriction maps (cf proposition 5.1)

r i : H 2 ( V ; O,)+ 2;.We suppose we are in the d-semistable case when the product of the normalbundles is trivial, so we have to find the elements

pi = (&)l 2;.Recall from 02 that the first-order neighbourhoods of W in the v. re described byclasses o(ti) n H'(vT @ T w ) . So CO!') can be regarded as lying in H'(vl @ Tw), ythe hypothesis on the normal bundles. The inclusion of T w in rV2induces a map:

Hl(V2 @ T w ) + H ' ( v 2 @ W2lW) . (5 .6 )


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Connected sums of self-dual manifolds 21 9Finally the exact sequence O + TVz -+ TV2[W]-+TV2[W] +0 furnishes a boundarymap:

d 2 : H 1 ( v 2@ TV2lw)-+ H2(TV,) . (5.7)(Here W2[ ] denotes the tensor product TV2@ 2, where 2 is the line bundle overV associated with the divisor W.) Putting (5.6) and (5.7) together, we get anelement, which we will just denote by d2(m(11)), n 2:= H2 ( T V2 ) .Interchanging thetwo manifolds we obtain a corresponding element of X:.Proposition 5.3. p1= d, (o$ ' ) ) and p 2 = d 2 ( mi2 ) ) , n %':, %; respectively.

To prove this we go back and recall the definition of the differential d,, whichoriginated in the spectral sequence relating the global and sheaf ext. In general wehave

d, :HO(&( Ce , 9)) H2( 5%"3,9)) (5.8)for any sheaves, Ce , 9. his map can be defined as follows (cf [12, pp 722-61): thestalks ofU(%,) orrespond to equivalence classes of extensions of the stalks ofCe by those of 9. global section in H o ( @ ( Ce , 9))s given by a cover V = U U, andextensions

0 4 1 ,+Y,+%I u,+o (5.9)which are equivalent on the overlaps UmB U , n U,, so there are diagrams

(5.10)

Now 0 r$,@ differs from on the triple intersection UnSyy an automorphismwhich respects the extension: so by an element q j W p y of Hom(Ce I U,py, 9 U,@,,),Then ($J,~,,) is a Cech cocycle representing d 2 { S , } in H 2 ( % ' m ( % ,9)).f thiscohomology class vanishes the can be modified to give compatible gluing maps;the S, fit together to give a global extension, representing an element of Exr'(3, S),and the choice of suitable r$ap corresponds to an ambiguity El1(&(%, 9)).hisexplains the early terms in the spectral sequence.Let us now implement this procedure in our example. For the generating element'1' of Ho(@(S2:, O v ) ) = H O ( v l @ ,) =C w e can take a cover of the formU , = V,\W, U,= V2\W, {U,},+,,, a cover of a neighbourhood of W in V. Wedecompose each U, into components U:, U i - o p e n sets in Vl, V, respectively-meeting along W. Choose coordinate systems which identify the U , with neighbour-hoods of Y = {6 = 0, 9 = 0) in the hypersurface X = {g q = 0) in en+ ' .We fixcoordinates ( E , q ; l, . . . , z, , -~ ) on en+'which we will abbreviate for clarity to( 5 , q ; z ) We suppose our coordinate systems take the form 59 to the fixedtrivialisation of v1 @ v,. Then over U,, U , we form the trivial extensions Yl, 9,Q 1 $ O of Qb by Qv. Over X we have the following extension (the co-normalsequence):

(5.11)-. I~/I$+ sz&.+,l, +a: --+ 0.


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220 S Donaldson and R FriedmanThe first term is a rank-1 free sheaf over X , generated by

B = d(Er]) = r ] dE + l j dy.The middle term, which we will just denote by O;,,, is also free of rank n + 1, andthis is the basic non-trivial extension of SZ; by 6 Our coordinate system gives astandard splitting of the sequence (5.11), when restricted to X n { ] # O}. We justlift the forms dlj, dz-which generate !& away from r ] = 0-to the forms with thesame names on Cn+'. Similarly we get a splitting away from E = 0. Thus we can usethe coordinate systems to carry the sequences (5.11) over to the U,, and getextensions Y,. We also have identifications @la, G2 , of the Ye with Yl, Y2over theintersections u',\W, induced by the splittings described above. It remains to choosethe @ on the other double intersections. For this we apply the followingobservation, which will be used also in the proof of proposition 5.5 below.

Denote the hyperplanes { ] = 0}, { l j = 0} in Cn+l by H,, H2 respectively.Suppose we have biholomorphic maps 6 Hi i hich agree on the intersection( 5 = r ] = 0) . We seek a map F :Cfl+'+ Cn+' which restricts to the 6 n the Hi.(More precisely we should suppose the f;. are defined on neighbourhoods of acompact set in H , n H z , and seek an F defined on a similar neighbourhood-corresponding to the fact that we can always shrink our sets U,-but we will notintroduce extra notation for this.) We write

Lemma 5.1. Suppose

Then there is a F = (FE,F,, F , ) agreeing with the fi and such that FEF, = Er] .Proof. We can write

Then put FE= E'(1+ r ] P ( E , z ) ) ,4 = r] '( l+ ga(r],)). For the other component wewrite

z ( I )= z(*l,(z) + E d l ) ( E , 2) A2) z * ( z )+ T p v ' 2 ' ( r ] , 2)and put F , = z ( ' ) + z ( ~ ) - z * ( z ) . In the situation above the map F acts, via itsderivative DF, on Q;+, and the action preserves the 1-form B = d ( E r ] ) sinceFEF, = Er]. So F induces an automorphism of the extension (5.11), covering theaction of the fi: on a'. Applying this to the coordinate change maps on the U, n U,we obtain the desired identifications @,p:.Y,--,Yp.The vector fields qmB1nU k n Ui\W = U , n U n U, are found by examining the action of F on thecoordinate induced splittings of Q;+, over Hl\(H, r l H2) . Dually, we project theimage under D F of the normal vector field:

(5.12)


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Connected sums of self-dual manifolds 22 1to the 5 , z components. This is the vector field

dF d d F dE-(- g- - -)d q ag d q dz (5.13)Now the image r1d2(1)of d,(l) in WT is represented by the cocycle of vector fields(qaP1,vas,,)n the cover U , = V,\W, U: of V,. The qap1,iven in local coordinates

by (5.13), are meromorphic on U : I I U ; with at most simple poles along W. If allthese poles are absent the I),ol are holomorphic on U L n U ; and give a Cech1-cochain on the cover whose coboundary is just (qmp,,aPY),o r1d2(1) dependsonly on the poles of the qWpl. n fact the poles of qnpl re naturally local sections ofWlIw31 defining a class 6 in H 1 ( W ; 7'V 3 vl) and the 2-cocycle is the standardrepresentative for &(e) (cf (5.10)). So we have to compare 6 and U,. The conditionFgF, = &, implies that the normal component ( d F l d q ) E vanishes when 8 = q = 0.So the only pole comes from the tangential component:

But this is just the representative for the obstruction class ~ ( 1 2 )of the first-orderneighbourhood of W in V2given by the covering U:, so the proof of proposition 5.3is complete.(Note. One should compare proposition 5.3 with the very similar work of Perssonand Pinkham [19].)

We can think of this calculation slightly differently using spaces in place ofsheaves. Cfl+l is decomposed into the family of hypersurfaces:x, (E q = t } c ,+I.The singular space V is constructed by identifying open sets U , in X o on overlapregions Una by maps:

fap : U,, + U,,and identifying U,\W with domains in V,\W and V2\W by maps i These satisfy aconsistency condition

(5.14)If we have maps FaS between suitable domains in P',xtending the faa andpreserving the X,, e can attempt to use the identifications F,P[x,,, o p , (wherep i , p 2 are the obvious projection maps on the X,,# 0) to make a family of spacesV ( t )deforming V = V(0) .To do this we must have

imopi =iPopioFmyIX,* (5.15)When t = 0 this is just (5.14). The class in H 2 which we have calculated is the next,O(t), erm in (5.15). More precisely, the vector fields I ) f n p iare

. .1, = $3 ofas.

d- ( ipp i )F or8 1x, crOpi)-'d t

at t = 0. (5.16)When the class in H 2 vanishes we can choose the identifications to make a first-orderdeformation of V.


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222 S Donaldson and R FriedmanWe will now take the ideas of proposition 5.3 a little further, and relate the

quadratic obstruction Yz to the higher formal neighbourhoods of W. Let us firstobserve that, for n 3 2, the normal and tangential components of the classes o, nH ' ( ( T W @ vw) 63 "(v&)) have independent meanings; the normal componentvanishes if and only if we can find a cover by polydiscs with coordinate change maps:( z ' , 5')= ( z ' ( z )+ O ( P ) , ) E + O(P")). (5.17)Then we can talk about a trivialisation of the normal component of the nth-orderneighbourhood, and these form a set modelled on @(vw 63s"(v&)), provided atleast one trivialisation exists.Proposition 5.4. Suppose the first-order neighbourhoods of W in the v. re trivialand fix trivialisations. Suppose then that the normal components of the second-orderneighbourhoods are trivial. A choice of trivialisations for these normal componentsgives a natural splitting of the exact sequence (5.3) at the third term, i.e. a choice ofisomorphism:

T L = H 1 ( V ;z,) CB C.Proof , We continue with the notation set up above. We choose coordinate chartscovering neighbourhoods of W in the V , compatible with the hypotheses of theproposition, so that on overlaps U: they compare by maps 5 = f y a of the form

5' = W ) ( 5 0 ( E 3 ) ) 11' = W - l ( r + 0 ( v 3 ) ) (5 .18)= z * ( z )+ el(z)E2 + O(E3)= z * ( z )+ e2(2 )$ + O(q3). (5 .19)

Then choose Fwp, s above, and obtain vector fields qaal,qwa2 hich are, byproposition 5.3, holomorphic on u t p , U",. If the normal components of the qwpivanish on W and the tangential components agree there, qwolnd qwa2efine anelement xwo of ~Ov(U,fiU,) and ( x nP ) is a Cech cocycle with coboundary(qwai,wau).hen we can modify our maps by xwa o get an explicit globalextension representing a lift of 1E C o T :. One can check then that this lift doesnot depend on the choices made, subject to the given trivialisations of theneighbourhoods of W.

To see that the qwaiatisfy the given conditions along W we work in the standardcoordinates so that qwB1s

For the first component, the condition FEF, = Er] implies that F , has no (611)term (since Fq has no q 2 term) so this normal term vanishes along W. For the secondcomponent we get d 2 F , / d E d q which agrees with qaa2.his completes the proof ofproposition 5.4.

Finally we evaluate the quadratic map Y2 on the lift of 1E C o T : given by(5.20)

proposition 5.4.Composing with the maps to 2: we get elements:ai r i Y 2 ( 1 )E 2 ;.


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Connected sums of self-dual manifolds 223On the other hand we have a boundary map:

a, :H I ( rviJ (I )8 V I ) -+H 2 (W ,) Hf (5.21)constructed from the exact sequence:

O + r V 1 + r V 1 [ 2 ~ ] +w18 : I ~ ~ + O . (5.22)We can take the obstruction class in H'(W; Tw 8 - ' ) , replace v-' by Y:, lift tothe formal neighbourhood W(') using the trivialisation and apply dl to get anelement for which we just write a,(w%), in X:.Similarly with V,, V, interchanged.We have then a second-order version of proposition 5.3.Proposition 5.5. Suppose the hypotheses of proposition 5.4 hold and that we canalso choose a further trivialisation of the normal components of the third-orderneighbourhoods of W in V,, V,. Then:

0 1 = a l ( d 0 2 = a,(w%)in X : , X $ respectively.Proof. In this situation we have a first-order deformation of V and Y2(1) is theobstruction to extending to second order. 0, is represented by the cocycle of vectorfields formed from the t 2 term in the family of local holomorphic maps (5.15) on thecover. In our coordinates these are given by the t2 terms in

& (E , t / f ; ) F.(f , t / E ;2).Again only the poles are relevant. For the tangential component F, , notice from theconstruction in lemma 5.1 that we can suppose there are no mixed 5 , r] terms:F, = ~ ( " ( f ,) + ~ ( ~ ' ( r ] ,)- * ( z ) .In particular there is no Er]' term and we get a double pole from the r]' terms inz(')(r], ) . This is the standard representative for al(wz) . For the normal com-ponent, the condition f i F , = Er] means that FEcontains no terms in q 2 . Similarly thiscondition, together with the hypothesis that F, contains no q 3 term (triviality of thenormal component of the third neighbourhood) means that F. contains no Er]2 term.This implies that there are no poles in the t2 term in FE(& / E ;z ) and the proof iscomplete.

Turning now to the singular twistor space 2 = 2,Ue2, we have the following.Proposition 5.6. The formal neighbourhoods of Q in the zi admit trivialisations ofthe kind considered in propositions 5.4 and 5.5.Proof. The obstructions lie in

H 1 ( ( W , 0 ) @ 0(0 ,2 ) ) @ ob, - n ) )H'( 6 ( n - 1, 1- ) )

(tangential)

(normal)and

The first vanish when n = 1 and the second when n = 1,2 so the only task is to seethat we can remove the normal obstruction in H1 (0(2, -2)) = C3 to satisfy the


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224 S Donaldson and R Friedmanhypotheses of proposition 5.6. B ut H o ( T Q 3 Y;') = H 0 ( 0 ( 2 ,0 ) )= C3 acts on thesethird-order normal obstructions (even though the tangential part need not be trivialto second order) by multiplication with the fundamental class in H1(End(vQ)3Qb), as in lemma 2.3. We have to show that this map is an isomorphism. ButH'(End(v,) 3 Qb) = H'(G&) = C X @, and the curvature class is (-1,1) in thestanda rd basis (th e first Ch ern class of the normal bundle). I t is then easy to see thatthe multiplication map is indeed an isomorphism.

Another approach to this is to use the discussion of the neighbourhoods of thetwistor lines in the Zi n 03 and show that the trivialisations there induce therequired trivialisations in 2;. In the same vein we can express the formula ofproposition 5.5 in term s of Z;. First, the obstruction w: lies in H '(O(4, - 2 ) ) which isnaturally isomorphic to s4(V:,), the space in which the Weyl curvature W:, lies.Second we know that the Serre dual of X f = H 2 ( 0 , , ) is H 1 ( Q k , 3 K 2 , ) and , asexplained in 03, we have an evaluation map:e , , :H ' (Q& 3 Kz, )+s4 (VJ

O n the oth er hand the o rientation reversing isometry p of the tangent spaces to the4-manifolds identifies s4 (VJ with s4(V:J so we get a natural pairing ( , p betweenthese two spaces (using the sympletic form o n either) .Proposition 5.7. (i) Under these natural isomorphisms 0; orresponds to theobstruction to trivialising the third neighbourhood of the line L x, and so, byproposition 3.1, to W:.(ii) The pairing between a class h in H ' ( Q k , 3 Kzl) and o1= al(w:) goes over,under thes e isomorphisms, to th e pairing

( i X , ( h ) > w:>,.W e leave the verification of this proposition as a (rath er lengthy) exercise for th ereader. Combined with the discussion of 03 it gives a description of the quadraticpart the obstruction entirely in terms of the 4-manifolds Xi,he Weyl curvaturetensors and the evaluation of the solutions of the equation D * s = 0. In our mainapplication in 06 below one could avoid this discussion by working throughout withthe twistor spaces. With regard to part (i) of proposition 5.7 , note tha t in general if

B c is a complex submanifold with trivial nth-order formal neighbourhood thanthe exceptional divisor in the blow-up of A along B has a trivial ( n - 1)th-orderneighbourhood. With regard to part (ii), note that in general if W is a surface in athreefold V with first-order neighbourhood trivialised then the adjoint of theboundary map

d : H ' ( T W 3 Y2)+H2(7V)on the Serre duals:H 2 ( T V ) * = H ' ( T * V 3 K,) H ' ( T W 3 Y 2)* = H'(T*W 3 K , 3 v-2)is given as follows. For any bundle S over V we have a restriction map:

H ' ( V ; S ) + H ' ( W " ' , S) = H'(W, S) (3H ' ( W , S 3 Y*)where the last isomorphism uses the trivialisation of W( ' ) .Now our adjoint map isform ed by taking the second comp onen t of this m ap, with S = T*V 3 Kv, nd using


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Connected sums of self-dual manifolds 225the restriction T*V+ T*W and the adjunction isomorphism Kw = Kv 63Y. T heproof that this is adjoint to the boundary map d is an application of the residuetheorem.

6. The main results and applicationsIn 44 we showed that self-dual metrics on the connected sum of self-dual4-manifolds could be obtained from suitable deformations of a singular twistorspace. In 65 we assembled the techniques required to study these deform ations andwe can now put th e two toge ther and spell out in detail some of the consequences.

6.1. The unobstructed caseLet XI, X , be compact self-dual 4-manifolds and suppose that X,,, x2 are bothzero. As we explained in 43.3 this is equivalent to the vanishing of the sheafcohomology groups H 2 ( Z i ;0,) or the twistor spaces Zi.f we choose points x i in Xiand an identification p of the ta ngent spaces at those points we can fo rm , as in 44,the singular space

z Z(x,, x, , p ) = 2 ,U 2,. (6.1)Z has a real structure 0. By corollary 5.1 we know that 2 can be s mo othed, sincethe obstructions vanish. These smoothings will give standard deformations, in thesense of 44, provided they can be chosen to be compatible with suitable realstructures. If Z has no infinitesimal automorp hism s we can use t he argum ent of 2.10,adapted in the obvious way to singular spaces, to get a real structure on theuniversal deformation space. However, in the cases of most interest to us theseautomorphisms are present so we ex tend th e theory by the following device.It is easy to se e that w e can choo se a finite set of disjoint real twistor lines {L,}in Z , none meeting the singular set Q, so that no non-trivial automorphisms of Zpreserve U,L,. (For each Zi he identity component in the group of suchautomorphisms is the complexification of the group of isometries of Xi reservingthe corresponding finite set of points.) So ther e is a universal deform ation S for thepair ( Z ,UL,). Th e involution 0 preserves the lines so this pair has a re al structure.O n the ot her hand t he exact sequences relating the de form ations of Z and thedeformations of the pair show that

T L L , = T2, ( 6 . 2 )and this lat ter vanishes by hypothesis. So the base space S for the relative problem ismodelled on T k , U L , nd we have an exact sequence:

0- U -, T & , U L , - , & + O . (6.3)Here U is $ H 0 ( v L , ) . This exact sequen ce is compatible with n atural rea l structureson all three spaces, induced by 0.Proposition 6.1. The universal deformation for (2 ,UL,) has a natural realstructure and th e local isomorphism of S with T i , U L ,an be chosen to be com patiblewith the two real structure s.


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226 S Donaldson and R FriedmanTh e proof follows tha t of lemma 2.1 precisely.We can now forget about the lines L, and just consider the real deformation ofZ over S.Of course this is induced from the versal deformation T of 2 by a mape :S+ T , the tangent space of T at the base point is T i and (6.3) just describes thederivative of e . Now by ou r discussion in 05 the total space of the versal deformationover T is smooth a nd th e set of points in T for which th e fibres are not smooth is asmooth hypersurface whose tangent space at the base point is

Ker{Tk+@(&) = C.} (6.4)Once again this map is compatible with real structures induced by a. This followsreadily enoug h from th e naturality of the various constructions. In turn, since S+ Tis a subm ersion, the points in S over which the fibres are not s mo oth form a smoothhypersurface, tangent at the base point to a a*-invariant hyperplane in Tk,,,..Then the family over S is a standard deformation of Z in the sense of 04 and bytheorem 4.1 any of the real points in S, sufficiently close to th e base p oint and not inthe hypersurface, yield a tw istor space fo r a self-dual metric on X I # X,. So we havethe following.Theorem 6.1. For any compact self-dual manifolds XI, X , with Xx,,ex2 both zerothe connected sum ad mits a self-dual metric.

6.2. ExtensionsBefore taking up t he question of the obstructions let us observe a num ber of simpleextensions of theorem 6.1. First, if Z( t ) ar e the fibres in a smoothing of th e singulartwistor space Z(0)we have the semi-continuity of hypercohomology [NI:

dim T 2 ( Z ( 0 ) ;Oz(oJa im-0 sup dim T 2 ( Z ( t ) ;O,,,,). (6.5)So, by proposition 5.2, we see that under the conditions of theorem 6.1 theconnected sum admits self-dual metrics with 0. Thus we can iteratetheorem 6. 1 to get the following.Proposition 6.2. For any set X I ,X 2 , . . . ,X,, of compact self-dual 4-manifolds withYexi= 0 for all i , the connected sum X I # X , # . . . # X,, adm its a self-dual metric.Corollary 6.1. (cf Floer [6]). For any n 2 0 he connected sum n@P2of n copies ofCP2 admits a self-dual metric.

Indeed the twistor space of CP2 is the flag manifold F and, using the embedding1Fc CP2x CP2, or the g eneral formulae of B ott, on e sees that

H2( E; 0,) 0 (6.6)so there are no obstructions in this case. In 07 we will go into the geometry of theseparticular tw istor spaces in more detail.

Instead of this inductive construction we can equally well make a simultaneousconnected sum by taking n twistor spaces Z1,Z 2 , . . . , Z, , , blowing up along n - 1twistor lines l i and identifying by maps pi in normal crossings to make a singular


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Connected sums of self-dual manifolds 227space with a singular set consisting of n - 1 copies of the q uadratic surface. In factthere are various different ways in which we may do this (corresponding to differentdegenerations of the self-dual metrics o n the connected sum ). Fo r exam ple, we cantake n - 1 lines in Z 1 and one line in each of the other spaces. If we allow the'connected sum of a manifold with itself' we can construct non-simply connectedmanifolds, and the general singular configuration we can consider is labelled by agraph, with multiple edges an d loops allowed, whose vertices correspond to self-dualmanifolds X i and whose edges correspond to one-point identifications in themanifolds. So here we are considering th e graph:

In any case the same ideas go through to show that suitable smoothings yieldself-dual metrics on the connected sum and to analyse the deformation theory of th esingular space. The reader will have no difficulty in modifying the proofs above. Inthe unobstructed case the local picture is described by T' which naturally fits in anexact sequence:n - 1i = l- , H 1 + T 1 + @ C i 4 O (6.7)

where the terms Ci are copies of C ssociated with the different singularities. Theversa1 deform ation T can be embedded in T' in such a way that the fibre over apoint t in T is sm oo th if an d only if t maps to a vector ( A l , A, , . . . , A n ) in CBCi withall components A i non-zero. The other factor H measures the deformations of thesingular space, i.e. of the data ( Z i , i, pi).6.3. Self-dual connectionsRecall that there is a one-to-one correspondence between self-dual connections overa self-dual manifold X and holomorphic bundles on the twistor space Z which aretrivial on all the real twistor lines (with real structure, if we want unitaryconnections). Suppose X 1 ,X , ar e self-dual manifolds as in 06.1 above, with XX, 0.Let El, E , be bundles over X I ,X 2 carrying self-dual connections V I , V2 and E , , 8,the corresponding holomorphic bundles ov er the twistor spaces Z 1 ,Z , . We at temptto construct a self-dual connection o n a bundle El # E 2 over X 1# X , as follows. Th elifts n:(Bi) to the blown-up twistor spaces are trivial on the exceptional divisors Q .If we identify th e bund les over Q by an isomorphism Y we get a holomorphic bundleE over singular space Z . (On any space with normal crossings a holomorphic bundleis given by a bundle on the normalisation and suitable identifications along thecrossing divisors.) We then try to deform this to a holomorphic bundle over thesmall smoothings; that is to find a bundle g+ Z which restricts to E on t he centralfibre. If we can do this 8 ives self-dual connections fo r the self-dual metrics on t heconnected sum constructed in theorem 6.1. Since E is trivial o n all the rea

donaldson 1989 connected sums of self-dual manifolds and deformations of singular spaces

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