E» ".2 n 91.2%? u: .r. .:...1| ‘ 1.1.3): t§al‘\.‘:§.vv :2"; ‘ t I 2 I Y .n .. nan c. o 2". :— 4 5:31.01! n} 1 a. Siziasi 1 . ‘ 33.11. . 5.3.0 1. (L. .21. 1.. .. .. . t 2“...) '1 It... f v-néq: F. Subtiiia!» . VO‘ 1 "1‘. i {A \1 13‘35‘... 7912513 ’1 00V LIBRAR—Y ' Michigan State University This is to certify that the dissertation entitled Grafting Seiberg—Witten Monopoles presented by Stanislav Jabuka has been accepted towards fulfillment of the requirements for Ph-D-‘degreein Mathematics flaw Major professor Date March 19. 2002 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:/CIRCIDateDue.p65-p.15 GRAFTING SEIBERG-WITTEN MONOPOLES By Stanislav J abuka A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPY Department of Mathematics 2002 ABSTRACT GRAF TIN G SEIBERG-WITTEN MONOPOLES Stanislav J abuka We demonstrate that the operation of taking disjoint unions of J-holomorphic curves (and thus obtaining new J-holomorphic curves) has a Seiberg-Witten counterpart. The main theorem (theorem 5.10) asserts that, given two solutions (Am/2i), 2' = 0,1 of the Seiberg-Witten equations for the Spine-structures. WE,- 2 E 69 (E (8) K ‘1) (with certain restrictions), there is a solution (A, 1/1) of the Seiberg—Witten equations for the Spine—structure WE with E = E0®E1, obtained by “grafting”the two solutions (At/(pi)- ACKNOWLEDGEMENTS I have had the help of many people while working on my thesis and its ultimate completion is in no small measure thanks to them. Firstly I would like to thank the person most closely involved with my day to day struggle with mathematics during the last seven years: my thesis advisor, professor Ron Fintushel. I am deeply indebted to Ron for his mathematical guidance, for waking my interest for the subject in the first place and for his never waning positive and optimistic approach to all problems, mathematical and otherwise. I would like to thank all the faculty members, staff members and graduate students at the Department of Mathematics at Michigan State University with whom I have had the pleasure to collaborate and share the workplace. Special thanks are due to professor Petar Novaéki, my 10-th and 11-th grade highschool mathematics teacher. It was he who set me on the path of my present career. His enchanting way of teaching mathematics turned me from a math-averted to a math-admiring student. He was the first mathematician I met. Lastly, I want to thank all my family and friends for their support. My parents, Stanislav and Vjenceslava Jabuka, who have always put my education before their own well being and comfort, as well as my brother, Kristijan Jabuka, for being there. My aunt and uncle, Frida and Otto Stein, for their continuing care and constant encouragement. Zeljko Calopek (Zac) for remaining the best friend one could hOpe iii for, even at a distance of 6000 miles. Most of all, special thanks to my wife, Michelle Wilson, for sticking with me though thick and thin, for her patience and understand- ing during times of my mathematical struggles as well as for sharing the joy when the math goddess was sympathetic with me. iv TABLE OF CONTENTS 1 Introduction 1 2 Seiberg—Witten Theory 6 2.1 Spiri,C—structure on 4-manifolds ....................................... 6 2.2 The Seiberg-Witten equations ...................................... 10 2.3 The moduli space ................................................... 18 2.4 The Seiberg-W’itten invariant ....................................... 25 3 Gromov-Witten theory 31 4 Gauge theory on symplectic 4-manifolds 36 4.1 Introduction ........................................................ 36 4.2 The anticanonical Spine-structure ................................... 37 4.3 The general case and S WX(WE) = GrX(E) ......................... 39 5 Grafting Seiberg-Witten monopoles 44 5.1 Producing the approximate solution ((1.21)) from a pair ((1,, 1b,) ....... 44 5.2 Inverting the linearized operators at ((1,, 112,-) ......................... 47 5.3 The linearized operator at (a, ’r) .................................... 52 5.4 Deforming ((1,212) to an honest solution .............................. 61 6 Comparison with product formulas 7 The image of the multiplication map 7.1 Defining (242,291) .................................................... 7.2 Pointwise bounds on S W” (Ag, 2,91) .................................... 7.3 Surjectivity of L, 4231’!) and deforming (A2, /,’.) to an exact solution References vi 64 67 68 71 73 77 1 Introduction In his series of groundbreaking works [12, 13, 14], Taubes showed that the Seiberg- VVitten invariants and the Gromov-Witten invariants (as defined in [15]) for a sym- plectic 4-manifold (X ,w) are the same. His results opened the door to a whole new world of interactions between the two theories that had previously only been spec- ulations. The most spectacular outcomes of this interplay were new results that in one theory were obvious but when translated into the other theory, became highly nontrivial. An example of such a phenomenon is the simple formula relating the Seiberg-Witten invariant of a Spine-structure W to the Seiberg-Witten invariant of its dual SpirzC-strtict1_1re l/V“, i.e. the one with 01(W“) = —c1(W). The formula reads SWXU’V“) = :t Sl/VX(W) When translated into the Gromov-VVitten language, this duality becomes GTX(E) = iGTX(K — E) (1) Here K is the canonical class of (X, w) and E E H2(X; Z) is related to W as cl(lV+) = 2 E — K . This is a highly nonobvious result about J-holomorphic curves, even in the simplest case when E = 0. In that case we obtain that GTX(K) : :t G'rX(O) = :i: 1, the latter equation simply being the definition of er(0). This gives an existence result of a J-holomorphic representative for the class K, a result unknown prior to Taubes" theorem. The formula (1) has recently been proved by S. Donaldson and I. Smith [4] without any reference to Seiberg-Witten theory (but under slightly stronger restrictions on (X,w) than in Taubes’ theorem). 1 In the author’s opinion, proving a result about Gromov—Witten theory which had only been known through its relation with Seiberg—Witten theory, without relying on the latter, has a number of benefits. One is to understand Gromov-Witten theory from within better. But also to possibly generalize the theorem to a broader class of manifolds. Recall that Taubes’ theorem equates the two invariants only on symplectic 4-manifolds. Both Seiberg-Witten and Gromov-Witten theory are defined over larger sets of manifolds, namely all smooth 4—manifolds and all symplectic manifolds (of any dimension) respectively. On the other hand, even within the category of symplectic 4—manifolds, one can hope for more nonvanishing theorems i.e. theorems of the type er(E) # 0 for classes E 7f 0, K. The techniques used by Donaldson and Smith are promising in that direction. The main result of this thesis is to prove an assertion in the same vein but going the opposite direction. Namely, on the Gromov—Witten side, given two classes E,- E H2(X; Z), i = 0,1 with Eo-El = 0 and J-holomorphic curves 2,- with [23,-] = P.D.(E,:), one can define a new J-holomorphic curve E = 201.121. By the assumption Eo-El = 0, the two curves 2,- are either disjoint or share toroidal components (see [5]) In the former case, 2 is simply the disjoint union of 20 and El and in the latter case one needs to replace the shared tori with their appropriate multiple covers. This induces a map on moduli spaces MEEWEO) X M§r(E1) “Li M§r(E0 + E1) (2) This thesis describes the Seiberg-Witten counterpart of (2). That is, given two com- plex line bundles E0 and E1 (with certain restrictions, see theorem 5.10 for a precise 2 statement) and two solutions (A,, 2,9,) of the Seiberg—Witten equations for the Spine- structures WE, = E, 69 (E, <8) K”), 2' = 0,1, with Taubes’ large 7‘ perturbation, we show how to produce a solution (429) = (A0, 290) - (A1, 291) for the SpinC-structure W E with E = E0 (8) E1. We say that (An/9) was obtained by grafting the two solu- tions (A,,2,9,), the choice of this term will be justified by the construction of (A,2,9) described in section 5. The operation of grafting induces the following commutative diagram: Mi""'(E0) X Mir”’(E1) —"’ Min/(E0 ‘8 E1) 9] [e (3) M§’(E0) xM§r(El) —”—» M§’(E0+E1) Here the map 6 : MirW(E) --+ MEYE) is the map described in [12] that associates to each solution of the Seiberg-Witten equations an embedded J-holomorphic curve. We call the map in the top row of (3) the grafting map. The monOpole (A,2,9) is constructed out of the the two monopoles (A,, 2,9,). The key observation here is that for the large T version of Taubes’ perturbation, a solution (B, Q9) of the Seiberg-Witten equations for the Spine-structure WE is “concentrated” near the zero set of fl 0, the E component of (,9. That is, the restriction of (B, <29) to the complement of a regular neighborhood of a‘1(0) converges pointwise (under certain bundle identifications) to the unique solution (A0, \/7: tie) for the anticanonical S pine-structure W0 2 Q EB K ‘1. This is used to define a first approximation of 2/2 by declaring it to be equal to 2,9, in a regular neighborhood V, of 0:1(0) and equal to fiuo on the complement of V0 U V1. Bump functions are used to produce a smooth spinor. The first approximation of A is simply the product connection A0 (8) A1. The contraction mapping principle is then evoked to deform this approximate solution to an honest solution of the Seiberg- Witten equations. The author has learned the techniques employed in this article from the inspiring work of Taubes on gauge theory of symplectic 4-manifolds, most notably from [13]. The thesis is organized as follows. In sections 2 and 3 we review the basics of Seiberg-Witten and Gromov-Witten theory. Since the emphasis of the thesis is mainly on Seiberg-Witten theory, most claims in section 2 come with proofs while in section 3 we refer the interested reader to the available literature. Section 4 points out the specifics of Seiberg-Witten theory on symplectic 4—manifolds. It explains important bounds that a Seiberg-Witten monopole satisfies and that will be used amply in the later chapters. It also explains Taubes’ theorem equating the two invariants on symplectic manifolds and proves some important corollaries. Section 5 contains the bulk of the work presented here. It explains how to define an “almost” monopole (A’, 29’) from a pair of monopoles (A,, 2,9,), 2' = 0, 1. It analyzes the asymptotic (as "r —» oo) regularity theory for the linearized operators L(A,,¢,) and deduces a corresponding result for L(A'.w’)- The latter is used in combination with the contraction mapping principle to obtain an “honest”monopole (A,2,9). Section 6 compares the present method of grafting monopoles to the one used in exploring Seiberg-Witten theory on manifolds X which are obtained as a fiber sum: X = X1#2X2. Finally, section 7 proves a converse to theorem 5.10. It explains which monOpoles in the Spine-structure WE can be obtained as products of monopoles (A,, 29,) in the SpinC-structures WE“ 2': 0,1 with E0®E1= E. 2 Seiberg—Witten Theory This section is an introduction to Seiberg-Witten theory. If defines all basic concepts and provides the statements and proofs of the bare bone theorems needed to get the gauge theory machinery going. The author has learned most of the material in this section from professor Tom Parker during a one-semester course on Seiberg-Witten theory taught at Michigan State University, as well as from his excellent accompanying notes [10]. The exposition in this section relies heavily on these notes and I would like to take this opportunity to express my gratitude to him for having done a superb job. 2.1 Spine-structures on 4-manifolds The Seiberg-Witten equations are a pair of coupled, partial differential elliptic equa- tions for a pair consisting of a connection A and a positive spinor 2,9. This subsection is concerned with defining the notion of a positive (as well as a negative) spinor. Definition 2.1 Let V be a (real) vector space of dimension n and let 9 : Sym2(V) —> IR; be a metric on V. The Clifi‘ord algebra C(V,g) associated to the pair (V,g) is the algebra whose underlying vector space is C(V,g) = ®pzov®i and whose multiplication law is subject to the relation :1: - y + y ' .2: = —2g($, SL‘) 1 for all any 6 V (here 1 is to be viewed as 1 E R = V®° Q C(V,g)). In particular, if e1, ...,e,, is an orthonormal basis of V, then C (V, g) is generated by elements of the form e,l - - e,,c with i, < < ik and 0 S k S n (with e0 2 1). This implies that the dimension of C (V, g) is 2". Notice also that the e, satisfy the relations e,-e,=—-1 and e,-ej=—ej-e, fori#j (4) As vector spaces, C (V, g) and EBLOA‘V are isomorphic but not so as algebras, the reason being the first relation in (4) which in EBLOA'V doesn’t hold. Example 2.2 The Clifiord algebras of Euclidean spaces are well known: 3 £3 % “:5 3 MR(8) ED Maw) M306) n+8 C(R",g) ®1WR(16) OONODCJ‘QCADMr—I § vb v In the above table, 9 denotes the Euclidean metric on R" while NIFUC) are the k x k matrices with entries belonging to the field 1F. The definition of the Clifford algebra associated to a vector space extends without difficulty to vector bundles V ——> X over manifolds, giving rise to the Clifford algebra bundle (which we will still denote by C (V, g)). The most important example for us will be that of the tangent bundle TX of a Riemannian manifold X. We will denote the associated Clifford algebra bundle simply by C (X) (suppressing the vector bundle and metric from the notation). Recall that a Hermitian vector bundle is a complex vector bundle equipped with a Hermitian metric. Definition 2.3 A Spine-structure on a smooth 4-manifold X is a Hermitian, rank 4 vector bundle W —> X together with a bundle map of algebras :1: H :12. :C(X) —) End(W) (5) called Cliflord multiplication. We also demand that (512)“ = -:L‘. for :1: E TX Q C(X) (here * signifies taking the adjoint operator). Since C(X) is generated by TX, it suffices to define Clifford multiplication on elements of TX subject to the relation my. + ya. = —2(:I:,y) Id. This observation becomes particularly useful when trying to verify that a given bundle map TX ——> W defines a SpinC-structure on X. By abuse of terminology, we will often call the bundle W —-> X itself a Spine- structure, keeping the Clifford multiplication in the background. Every Spine-structure W —+ X admits a splitting W = W+ 69 W‘ into two complex, rank 2 bundles called the positive and the negative spinor bundles. The splitting comes about as follows: for :1: E X, let e,, i = 1, ...,4, be an orthonormal frame of TX in a neighborhood of m. It is easy to check that e = (e1.e2.e3.e4.) is independent of the choice of orthonormal frame and satisfies the relation e2 = Id. Thus, the eigenvalues of e. E End(Wx) are :L-l and WE are the associated eigenspaces, they fit together to give the bundles Wi. In the special case when X admits an almost-complex structure J, there are two canonically defined Spine-structures called the canonical and anti-canonical Spine- structure of X. We will define below only the anti-canonical Spine-structure being the one we shall use in subsequent chapters. The definiton of the canonical Spin?- structure is left as an (easy) exercise to the interested reader. Definition 2.4 The anti-canonical Spine-structure W0 2 W0+€BWOT —+ X associated to an almost-complex structure J on TX compatible with the Riemannian metric g (i.e. such that g(v, J(v)) = 0 for all v E TX), is defined to be W0+ = A0’0(T’X) ED A0’2(T*X) WO‘ = A0'1(T*X) (6) with Cliflord multiplication given by v.0 = \/§ ('05,, /\ a — as) v e I‘(TX), a 6 mm) (7) In the above, ”03,1 2 (21* + iJ(v*))/2 E A0'1(T*X) denotes the (0,1) projection of v“ E T‘X, the dual ofv E TX. That (7) indeed defines a Clifford multiplication on W0 follows from the remark after definition 2.3 and the following easy check: v.v.a = x/2v.(v5,1/\ a — iva) = an, A a) — 22,5. A (w) = —2[lv(t’6,1) /\ a — 1’5‘1A(tv0)+ v6,1/\(Lva) 9 : -]’U]20 VUEF V /\ V (as L = A2 W+). Given a connection A on L, we will label the corresponding S pine-connection by VA. Observe that if A1 and A2 are two connections on L with A2 — A1 = a E iQ‘b, then VA2 — VAI = a/2. Spine—connections preserve the parity of the spinor bundles Wi, that is, if a E I‘(Wi) and v E F(TX), then Vv(a) E I‘(Wi) as well. This can been seen as follows: recall the element e : e1.e2.e3.e4. E C(X) defined for a local orthonormal frame e,. Since the individual endomorphisrns e,. change the parity of Wi, e. will preserve the parity. Now, if a E I‘(W+), then ea = a. On the other hand we have e.Vv(a) — V,.(e.a) = e.Vv(a) — Vv(a) = Vv(e).a E I‘(W+) Write Vv(a) = 6+ +,B‘ with [3i E Wi. Then €.fii = :tfii and so the above equation reads s+ — s- — (8+ + 5*) = —25- e W+ 11 which immediately gives 5‘ = 0 and thus Vv(a) = fi+ E W+. The case a E W’ is treated in the same way. Definition 2.6 Let W ——> X be a Spine-structure and let x E X be an arbitrary point. Let e,, i = 1, ...,4 be an orthonormal frame in a neighborhood U of It. For a connection A on L =det(W+), we define the operator DA for a section a E Ill/V) with supp(a) Q U, as: 0,,(0) = e,.V:a a 6 Par) (10) As is easily checked, this definition is independent of the choice of the orthonormal basis e, and thus defines a global differential operator DA : I‘(W) —> HIV) called the Dirac operator associated to A. Since the SpinC-connection VA preserves the parity of the spinor bundles W i and the endomorphism e,. reverses it, it follows from the definition that the Dirac operator reverses parity, i.e. we get two operators Di : I‘(Wi) —+ NWT). In most cases, where the chance of confusion is little, we will omit the superscript :1: from these Dirac operators. The Dirac operator is defined on a much broader domain than just the set of smooth sections F (W). Since it is a first order differential operator, all that is required of a section, for it to lie in the domain of the Dirac operator, is that it should have one derivative. Thus, we get a whole panoply of Dirac Operators (which will all still be denoted by DA) acting on the various Sobolev spaces: DA : LP’q(W) —+ Lp‘l'q(W), p Z 1 (for a definition and basic properties of Sobolev spaces see for example [1, 12 2]). With this definition understood, the following proposition summarizes some important properties of the Dirac operator. Its proof can found in [3]. Proposition 2.7 Let W = W+ 69 W‘ be a Spine-structure on X. Pick a connection A on L=det(lV+) and let Di : Lp*q(Wi) ——> Lp‘l'q(W¥), p Z 1, be its associated Dirac operator. Then the following hold: 1. The Weitzenbock formula: -— + A A 1 1 + DADA'l/JZV V ¢+ZS¢+§FAlp (11) where 3 denotes the scalar curvature of the Riemannian metric g and F; is the self-dual part of the curvature FA of the connection A on L. 2. The operators Di are elliptic operators. In particular, elliptic regularity applies to them: llwllm s Gallant—1.. + Hill.) (12) The constant C only depends on the pair (p, q) and the Riemannian metric 9, but not on 2,9. 3. The index of D: can be calculated by the Atiyah-Singer index theorem: 1 Ind(D:{) = dim (KerDj) — dim Coker(Dj) = 4 (L - L — ox) (13) with ox = b+ — b" being the signature of X. 4. The Dirac operator obeys the unique continuation theorem: If DA29 = 0 and 29 vanishes on an Open set, then 29 E 0. 13 The second important ingredient of the Seiberg-Witten equations is the bilinear map q : W+ ® W+ —+ iA2'+(T*X) which we now define. Let v E A§’+(TX) for some :1:EX. Lemma 2.8 The endomorphism v. : W: —> W: is traceless and skew-hermitian. Furthermore, the assignment v +——> v. is injective. Proof. We proof the lemma here for the case when X admits an almost-complex structure J, the general case can be found in [3]. It is a somewhat tedious but straightforward calculation to see that v. is traceless in the case when W 2 W0. The general case now follows from this special case together with the definition (8). Namely, suppose v. = [2.3-J] in some basis (1,, i = 1, ...,4, of (WO+)_,,. Let E be any complex line bundle and (,0 E F(E) with 90(1) 74 0. Then we still have that v. = [v,,,-] in the basis (,9 <8) (2, of (E (8) W0+)1.. In particular, tr(v.) = 0. The fact that v. is skewhermitian follows directly from the definition 2.3. To prove injectivity, suppose that v. = 0 for some v E TxX. Apply v. to both sides to obtain v.v. = 0. But by definition 2.1 we know that v.v. = —|v|2 Id. Thus v. = 0 implies |v| = 0 which in turn shows that v = 0. I Denote the space of traceless, skewhermitian endomorphisms of W+ by End5(W+). The above claim showed that v +—> v. defines a monomorphism e : iA2’+(T*X) —» End3(w+) (14) It is easy to calculate that in fact both vector spaces have dimension 3. This leads to the conclusion that the above monomorphism O is actually an isomorphism 14 Let now 29 E I‘(W+) with W+ = E 69 E (X) K‘1 and define the quadratic map q : I‘(l/V+) (8) F(W+) #2123? as air/1119i ‘ 1P2 155) 792 “AI (109.19) = 9‘1 (15) 2.9129; tot/22 29; — 291 29:) In the above, 29 = (191,292) with (91 E NE), 192 E F(E ® K'l) and 29;“ is the dual of 102' (i.e.. WW) = (1+!)qu We are now ready to define the Seiberg-Witten equations. Let W = W+ 69 W “ —> X be a Spine-structure and let u E iQ2'+ be an arbitrary self-dual two form. The Seiberg-Witten equations are a pair of coupled equations for a pair (A, 2,9) where A is a connection on L =det( W +) and 29 E F(W+). The equations read: DA(’I,(J) = 0 Ft = (109.29) + u (16) A spinor 29 satisfying D A(l9) = 0 is called a harmonic spinor. In the second equation, EX is the self-dual part of the curvature form F A of the connection A. The form it serves as a perturbation parameter of the equations. A solution (A,29) of (16) is called reducible if 29 :— 0 and irreducible otherwise. For later reference, we calculate the linearized Operator 14A,.) : I‘(i A e W+) —» m A“ e W‘) (17) associated to a solution (A, 29) of (16). Recall that by definition, for a pair (b, (0) E 15 F(i A 69 W+) we have (1 L(A.19)(ba97)= a (DA+tb(¢ + UP): F/iub “ (1(19 +15%?!) + t9?) — A) The right-hand side expression is easily calculated using the formulas 1 DA+b€ =DA5 + 5 9-6 FL, =F; + (N) In conclusion, we find the following expression for LOW): 1 L(A,t9)(ba 4?) : (Dr-199 + 5599,61“? — 4W}, $0) — (1W, 19)) (18) We conclude this section with two useful lemmas. Lemma 2.9 If (A,29) is a solution to the Seiberg- Witten equations (16) of class at least C2, then 29 and F; satisfy the pointwise bounds , s lwli gm {0,—, + llullco} 8 an. 3mm: {Ilium—Z +2||9llco} Va: e X (19) Proof. If 29 E 0 then (19) holds trivially. Thus, assume that 29 ¢ 0. In that case, |219|2 attains a global maximum at some point :1: E X with |r9|x > 0 and Ali/9|?r Z 0. On the other hand we have A1912 = d* an? = d': 2Re «viz/2,9) = Me («WW/‘99)) — QIVAW = We +319 — $153.99) — 2IVA19|2 16 = ”3W — Re (mm) + 919,21») — 21W)? 3 £192 — 2194+ In - W = We; — 2192 + lul) (20) The calculation used the easy to check formula (q(2,9,29).29,19) = 2 [29]4 as well as the Weitzenbiick formula (11) in going from line 2 to line 3. At the maximum point :2: E X we must have —s/2 — 2le: + lulx Z 0 leading to the desired formula ['29]: g —s/4 + lull/2. Since a: is the maximum of |29|2, the first inequality in (19) follows. For the second inequality, use the second Seiberg-Witten equation together with MCI/2,29)? = 2 WI“: IFXII 5 910.94%: “fl/1|: = film: + luls s g fimax {0, _4 + llfillCO} + War 8 4 + 21191100} - (21) S max {llrtllcm— Lemma 2.10 If (A, 29) is a solution of the Seiberg- Witten equations (16) of class at least L12, then (A,i9) is gauge equivalent to a C°° solution. Proof. This follows from the usual bootstrapping process and the fact that the Seiberg-Witten equations are elliptic. Elliptic regularity of the Dirac operator implies ”19”” S CHI/2H2 (22) showing immediately that 29 E L1":2 for all p 2 1 and thus 29 E CO0. As for A, choose first a gauge such that d‘a = 0 where a = (A — A0) and A0 is some C°° base 17 connection. Then elliptic regularity of d“ + d+ gives llallp,2 S C(llalh + |ld+a||p—1,2) S C(Hallz +H#||p—1,2 +||q(29,'¢1)||p_1,2) (23) This last inequality together with (22) shows that A E U”2 for p Z 1 and thus also AEC°°.I 2.3 The moduli space In order to define invariants of the smooth structure of X from the Seiberg—Witten equations, one needs to ensure that the moduli space of the solutions of (16) has certain compactness properties. This and other properties of the moduli space is what we will study in the present section. The gauge group underlying Seiberg—Witten theory is Q =Map(X,.S'1) (since S1 = U(1) is the structure group of L =det(W+)). The gauge group acts on pairs (A,29) of connections on L and sections of W, in the following manner: g-AzA—2g-1dg g - 29 =99 g E Q (24) Notice that the first equation shows that the action of g on a Spine-connection VA is given by g - VA := VQ'A = VA — g“1dg, implying immediately that F9.A = FA. It follows directly from the definition that q(g - 2,9, g - 29) = q(29, 29) since |g| = 1. That the Seiberg-Witten equations are invariant under this action of Q can easily be checked. We give here the check for the first equation in (16), the invariance of the second equation is obvious: D9-A(9 ' 19’) : e,.(VZ;A(g 1(1)) 18 = e,.((v;‘, — g‘lde.g)(gw)) = e..(v:(g '19) - dew) = 6,,(de,29 + gVQw - den/2) = ei-(QVQ'l/J) = 9 19/109) (25) From this we see that if the Seiberg-Witten equations (16) have at least one solution, then they necessarily have infinitely many of them. This leads to a big redundancy of solutions: from a point of view of trying to formulate invariants of the smooth structure of X, the solutions (A, 29) and g- (A, 29) carry exactly the same information. Thus we are motivated to henceforth identify solutions that differ by a gauge transformation. The Seiberg-Witten moduli space is the set of these equivalence classes of solutions. Before continuing the discussion, we first introduce some more notation. Let A0 be a connection on L which we will referred to as a base connection (but which at this point is completely arbitrary). Assume throughout that p 2 1. We denote by E(L) the space .AP’2(L) EB LP'2(W+) with Ap'2(L) being the space of L“2 connections on L. Let E*(L) = Ap’2(L) EB (Lp'2(W+)\{0}). The configuration spaces of the theory are B(L) = E(L)/g and B*(L) = E*(L)/g. The space E(L) is an intermediate of sorts of the spaces E(L) and B(L) (same is true of its *-analogue) and is defined as E(L) = {(14119) E EMMY/1 - A0) = 0} (and 3"(13) = {(24.19) E 23—714) | (NA - A0) = 0})- 19 We also introduce the moduli spaces —S W' MX (L) ={(A,2,9) E [3*(L) | (A, /) solves (16) } $719212) ={ e E*I } -—S W Mid/(L) =Mx (Ll/9 (26) (we suppress the perturbations form it from our notation for the moduli spaces, but keep in mind that different choices of u give rise to different moduli spaces). The condition d*(A — A0) = 0 is called the Coulomb gauge fixing condition. While this condition doesn’t determine a unique gauge for A, it reduces the number of possible gauges considerably: if (A,t9') is a solution of (16) with d*(A — A0) = 0, then the subgroup 9,, Q Q of elements 9 for which d*(g - A — A0) = 0, is homotopy equivalent to H1(X; Z) x S1 (the S1 factor corresponds to the constant maps). In particular, the moduli space Mill/(L) can also be expressed as H§W(L)/go. Proposition 2.11 The moduli space Min/(L) is compact. Proof. Without loss of generality, assume that p 2 5 (every L1:2 solution (A,29) of (16) is C")0 according to lemma 2.10). Choose a sequence (An,29,,) E M§W(L) = AAV§W(L)/go. Recall that A0 denotes our base connection and set a7, = An — A0. Clearly, the connections An are bounded if and only if the 1-forms an are bounded. Note that d"an = 0 and FA“ -—- FAQ +d+an. The operator (1* +d+ : i0], —+ “22692193? is elliptic and Karen Uhlenbeck’s theorem applies to it: there exist gauge transforma- tions gn E 9,, (chosen from the H1(X; Z) part of go) such that a; = g, - An — A0 = 20 an — 2g;1 dg,, obey the inequality [[01,]po S C(1+||(d‘ + (Fla; lip—1.2) = C(1+ lld+ahllP—1.2) S C(1+ ||(1('¢'1.,195.)||p—1.2) (27) In the above, C is a constant depending only on the Riemannian metric, the S pinc- structure and A0 and its precise value may change from line to line. The spinor 29;, is given by "29:, = 9,, ~29". Agree to rename the (a',,,29:,) back to (an, 29,,). Observe that 2 D An = on. + 2 D A0. Using elliptic regularity of D A0 applied to 19,, gives llcé'nllpa SC(||DA0¢’n||p-1,2 + ll¢n||2) SC(]]an-¢'n]]p—l,2 + llw1lll2) (28) Recall that the Sobolev space L5’2 on a 4-manifold embeds (compactly) into the space of C"2 maps. Because of that, lemma 2.9 applies and we get the n-independent bounds: [[z,9,,|]q g C. In particular, the L4-bounds induce LI’Q-bounds (these two norms are equivalent on a 4-manifold). We now repeatedly use equations (27) and (28) to improve on these bounds (this process is called bootstrapping): (27) => Hanllm SC(1+ ||(1('l9n,¢n)||1,2) S C(1+ Ill/)nllg) S C (28) => lllljnllm SC(|Ian-29n||1.2 +||19n||2)S C(llanll2.2 ' III/Julie +1) S C (27) => llanllse SC(1+IIQ(’I//n,'99n)||2,2) S C(1+ III/alga) S C (28) => “19an,2 :C(llananllee +ll‘19’nll2ls C(llanllza ' ll¢nll2,2 +1) S C (29) 21 These inequalities serve as the base of the induction process that completes the proof of the preposition: from (27) and (28) we see that if (an, 29") is a bounded sequence in LP‘”, then it is also a bounded sequence in L192. However, the Sobolev embedding Lip-1'2 H L1":2 is compact and so (an,29,,) E LP‘I’2 has a convergent subsequence. By induction on p we conclude that the sequence (a,,, 29") has a convergent subsequence in each D”, p Z 5. It still remains to take the quotient of the moduli space under the S1 component of go. This however preserves compactness as S1 is a compact group. I Our next aim is to show that for “good ”choices of ,u in ( 16), the moduli space M§W(L) is a smooth manifold of finite dimension. To see this, consider the map SW : B‘(L) ——> LP_1’2(l’V_ EB A2’+(T*X)) given by emf/(All?) = (DAM): F2.) (30) Recall that a linear map between Banach spaces is called Pfedholm if its image is closed and both its kernel and cokernel are finite-dimensional. A map between Banach spaces is called Fredholm if its differential is Fredholm at each point. Lemma 2.12 The map Sl’V is a smooth Fredholm map, transverse to the linear subspace ((0,)1.) E Lp’2(li" EB A2‘+(T*X))} at all (A, 29) solving (16) with 29 ¢ 0 and With dI(A — A0) = 0. Proof. To prove the Fredholm property, we first consider the differential d(,,,,,,)S W of SW at a point (A, 29). We incorporate the gauge group action by choosing a gauge fixing condition: d*(A — A0) = 0 (recall that A0 is a base connection on L). For 22 (A, 29) E Adieu/(L) we find the said differential 1 d<.4.e)SW(b, 90) = (DMD + 512-19. d+b - (109. so) - tr(vp. '19)) (31) The gauge condition d*(A — A0) = 0 translates into db = 0. The first component of (30) is evidently smooth while the second component is smooth since both d+ and q(29, 29) are linear, bounded maps (this fact uses the Sobolev multiplication theorem). Thus SW is smooth. The Fredholm preperty is established by observing that (l(,.(,,,,)SLV is a compact perturbation (such perturbations preserve the preperty of being Fredholm) of the operator DA 69 d+. The Fredholmness of the later follows from two facts: one being part two of proposition (2.7) (elliptic operators on compact manifolds are Fredholm) and the second being an explicit calculation of the kernel and cokernel of d+| Ker (at): Ker (d+lKer (d.)) EH1(X) Coker (d+lKer (01.)) 2H2’+(X) (32) Here H1(X) are the harmonic l-forms and H2’+(X) are the harmonic, self-dual 2- forms on X. The transversality of SW to the subspace {(0,}1) E LP'2(W' EB A2’+(T"X))} is equivalent to the surjectivity of the first component of (31). Suppose that it is not onto, then there exists (9 E Lp‘1*2(W‘) that is L2 perpendicular to the image of the map (b, (p) +——> DAcp + %b29. Said differently, the equation 1 (DAV? + 519-19, ¢>L2 = 0 23 holds for all b and (,9. Choosing b = 0 and integrating by parts shows that D Ad) 2 0. A contradiction is now produced by constructing a special 1-form b: locally b is defined as b = Re ((29, e,.29) ei (where {e,} and {e’} are a dual pair of local orthonormal frames of TX and T*X). Using DA29 = 0 and DA29 = 0 one can verify that d*b = 0 and an easy calculation shows that lb]2 2 [(9]2 [29]? Using this b, with (p = 0 leads to o = (wens = llalle = [X lei" W Since 29 is assumed to be nonzero, (9 vanishes on at least an open set and thus everywhere by the unique continuation theorem for the Dirac operator. I Invoke now an infinite-dimensional analogue of Sard’s theorem (due to Smale [11]) guaranteeing that the set of regular values of a Fredholm map between Banach manifolds are of second category (i.e. the countable intersection of open, dense sets). Agree from now on to choose the perturbation parameter ,a in the Seiberg-Witten equations to be such a regular value of the map SW. Under such a choice, the moduli space Mill/(L) is a smooth compact manifold. To conclude this section, we calculate the dimension of the moduli space. Since MV§W(L) 2 SW “1(0,u), the dimension of Hill/(L) is given by the index of the differential of SW and we calculate this dimension first. From (31) we see that Index (d(A,,,,)SW) = Index DA + Index d+ The first summand on the right hand side is provided by (13) while the second follows from (32) to be: Index(d+) 2 b1 — b+. Putting these together we obtain 1 1 Index (d,,.,,,,,SW) = 1(L- L —— a.) + b, — V = Z(L . L — W + b‘ + 4b,) 24 Since Mj§rW(L) = M§W(L)/go and dim 9,, = 1, we find that , 1 1 dim Mi,” (L) = Z“ - L — W + b" + 4b,) — 1 = Z(L . L — 30X — 2a,) (33) Here e X and 0,», are the Euler characteristic and the signature of X. 2.4 The Seiberg—Witten invariant This section defines the Seiberg—W’itten invariant coming from the moduli space of gauge equivalent solutions of the Seiberg—Witten equations (16). To begin with, let W —> X be a Spine-structure of X and denote the determinant line bundle of W +, as usual, by L. Recall that the moduli space Mill/(L) is a com- pact, smooth, finite—dimensional manifold and as such, it carries a fundamental class [M§W(L)] in its top dimensional (non-zero) homology group. The Seiberg—Witten invariant will be obtained by pairing this homology class against certain cohomol- ogy classes of the configuration space 3*(L). Our next task is to understand the cohomology group of B‘(L). Recall the definitions of B~*(L) and go: B'*(L) = {(A,29) 6 Ba) |d*(A — A0) = 0} and go 2 H1(X;z) x 51 Choose to view B*(L) as the quotient E*(L)/go. The space E(L) is an affine space and hence contractible. On the other hand, the set E(L)\E*(L) = {(A, 0) E E(L)} is an affine set of infinite codimension in E(L) implying that the open set EXL) is also contractible. But the action of go on E*(L) is free and thus the quotient E*(L)/go is the classifying space for go. This space is homotopy equivalent to the product of the 25 Eilenberg-MacLane space K (H 1(X ; Z), 1) with Cliff”. To summarize, we have proved the following Lemma 2.13 The space B*(L) is homotopy equivalent to T”1 x (CIP’OO where b, is the first Betti number of X. Denote the generator of H2( X be a Spine-structure on X with determinant line bun- dle L. Let d = (L2 — 30X — 2eX)/4 be the dimension of the moduli space Mill/(L). Then the Seiberg—Witten invariant SWX(L) is defined to be: 1. SWX(L)=0 ; ifdis odd ord<0. 2. SWX(L) = (hd/Q, [Mill(L)]) ; ifd is even. Remark 2.15 The above definition pairs [M§W(L)] only the with the correct power of h E H2(B*(L); Z). In the case when X is not simply connected, this second coho- mology group has a richer structure and other pairings with [M§W(L)] are possible, leading to more invariants. However, we will restrict ourselves only to the case of definition 2.14. Our next objective is to justify the use of the word “invariant ”in the above definition. The Seiberg—Witten moduli space M§W(L) depends, a priori, on the choice of the Riemannian metric g as well as the choice of a (generic) perturbation form 9.. The dependence on the metric g manifests itself on several levels: it defines a Levi-Civita connection and thus determines what a S pine-connection is. This in turn gives rise to the Dirac operator. On the other hand, the metric determines the Hodge 26 star operator * : Q} —> Off and so determines the projection 7r : $23, —+ $233+ which is used via F I =2 7r(F,.,) in the Seiberg-Witten equations. While the moduli space itself may change when altering those two choices, it turns out that these different moduli spaces carry the same homology class [MiW(L)] and thus determine the same Seiberg-Witten invariant. Lemma 2.16 Assume that X is a 4-manifold with b+ Z 2. Let W —+ X be a Spine- structure on X with determinant line bundle L. Then the Seiberg- Witten invariant SWX(L) doesn’t depend on the choice of (a generic, see proof below) Riemannian metric g and perturbation form ,u. Proof. We first describe the set of “bad”pairs (g, u), i.e. the ones for which the Seiberg-Witten equations admit reducible solutions. If (A, 0) is such a solution, then the second equation in (16) gives (F A —— u)+ = 0. Denote by rri(g) the orthogonal projection from {23, onto the (anti-) self-dual, harmonic 2-forms ”Hi(g). Likewise, denote the orthogonal projection from {23, onto the harmonic 2—forms H(g) by 7r(g). Then we have (FA — (0+ = 0 st 7er(9)0511 — 9) = 0 E 7r(g)(FA — 9) E 7LHg) (34) On the other hand, L), :2 rr(g)(i F A / 2a) is independent of the choice of the particular connection A since it is the unique harmonic representative of L. We conclude that (g,,u) is a bad pair if and only if rr(g)(iu/27r) E L), + H‘(g). The space ’H‘(g) has codimension b+ in H(g) = Codomain (7r(g)). This shows that the set was) = {e e 032 | ed) (5",,- s) e 1,. + H‘(g)} <35) 27 has codimension b+ inside of 2523:. Consider now two pairs (g,, u,), i = 1, 2. Take a path (g,, u,), t E [1,2], connecting (91.91) to (92,;12) and define the set W(L; (g,,,u,)) as W(L§ (gm/11)) = {(stll) I ll E W(Ligt)7t E [112]} Since by assumption b+ Z 2, the set r2 (W(L; (g,,,u,))) has codimension at least 1 inside of i Sig? (er is projection onto the second coordinate). It is shown in [11] that a generic path (g,, u,) will intersect W(L; (g,, u,)) in at most finitely many points and each of these is a transverse intersection point. It follows now, by invoking again the results from [11], that the set MiWIL; (9211111)) = {(21.9) I (As) solve (16) for any of the pairs (emit E II. 2]} is a compact, smooth, oriented manifold of dimension d + 1 (d = dim Min/(D), providing a. cobordism between the Seiberg-Witten moduli spaces obtained from using (91,;11) and (92,}.t'2). In particular, the homology class [M§W(L)] is independent of the chosen “good” pair (9, ,u). I In the case b+ = 1, the Seiberg-Witten invariant does depend on the choice of the pair (9, ,a). That dependence is well understood and completely described by the next lemma (cf. [9]), which we give without proof. Lemma 2.17 Let X be a 4-manifold with b+ = 1 and let (.129 E 03? be a generator of the positive forward cone in H2(X;Z). Let W —> X be a Spine-structure on X with determinant line bundle L. Then the Seiberg- Witten invariants SWX(L; (91,,211)) 28 and SWX(L; (g2, (12)) calculated from two pairs (9,, (2,) are the same, provided the two expressions i . (L'ng,[X])—2—/flj/\ng let2 7? x have the same sign (notice that this sign doesn’t depend on the specific choice of (99). Definition 2.18 An element L E H2(X;Z) is called a Seiberg- Witten basic class if S WX(L) 71$ 0. Proposition 2.19 The Seiberg- Witten invariants of X have the following properties: I. If X admits a metric of positive scalar curvature, then SWX(L) = 0 for all L E H2(X; Z). 2. The number of basic classes ofX is finite. 3. There is an inherent duality in Seiberg- Witten theory, namely that of replacing L by —L. The Seiberg- Witten invariants of a dual pair of Spine-structures are related by a simple relation: swan = (APEX awn—L) (36) Proof. The first two claims follow readily from (19). We give a proof of claim 3 only in the case when X admits an almost—complex structure J and refer the reader to the general case to [7]. Let L 2 2E — K, then the associated Spine—structure has positive spinor bundle W+ = E 69 (E <8) K”) and we write spinors 29 E I‘(W+) as 29 = (a, B). The duality L I—> —L corresponds to replacing E by E <8) K”. A direct 29 check shows that if (A, (0.3)) is a solution of (16) for the Spine-structure L (with perturbation form u), then (A, ((3,0)) is a solutions of (16) for the SpinC-structure —L (with perturbation form -—;1). Here A is the dual connection of A. This gives a diffeomorphism between MiI'V(L) and MiIV(—L) and thus S WX(L) = :I: S WX(—L). The correct sign is calculated by a, somewhat tedious but straightforward, comparison of the orientations of Mill/(L) and Mill/(-L), and is omitted here. I we finish this section with a few examples: Example 2.20 1. The Seiberg- Witten invariants of (CI?2 and 29 x S2 calculated using a pair (g,,u) with |n| small and with 39 > 0, are all zero. 2. The Seiberg- Witten invariants of the simply-connected, elliptic surfaces E(n) without multiple fibers are: n — 2 ( ) ;L = in, F 2 class of regular fiber SWX(L) — q 0 ; otherwise 3. The only basic classes of complex surfaces of general type, are :EK (cf. [7]). The invarinat for each class is i1. 30 3 Gromov-Witten Theory In this section we give a very brief introduction to the theory of counting embedded J-holomorphic curves in symplectic 4-manifolds. We omit proofs and instead refer the interested reader to the vast and comprehensive literature available on the sub- ject. At the onset we would like to point out, as there are many Gromov-Witten theories in existence today, differing from each other in the types of objects they are counting (e.g. embedded J-holomorphic curves, immersed J-holomorphic curves, J-holomorphic maps, etc.), that we will follow the approach of Taubes [15]. Definition 3.1 Let (X,w) be a symplectic 4-manifold. A triple (112,9, J) consisting of the symplectic form to, a Riemannian metric g and an almost-complex structure J, is called a compatible triple if g(u,v) = w(u, J(v)) u, v E TX (37) It is easy to see that any two members of a compatible triple, uniquely determine the third member. An important consequence of (37) is that J becomes an orthogonal map, i.e. awn), J(v)) = 90% v) Also, observe that (u,v) I——> g(u,v) + iw(u,v) defines a Hermitian metric on the complexified tangent space TCX. Given a symplectic form a), an abundant supply of compatible triples always exists (cf. [6]). The canonical classes K1 and K 2, associated to almost-complex structures J1, J2, each of which belongs to a compatible triple, are the same. The canonical class only depends on w. 31 Definition 3.2 Let C <—> X be an embedded symplectic submanifold of X. We say that C is a J-holomorphic curve if the tangent space TIC is a complex subspace of TIX at every point x E C. The genus g Of a connected J-holomorphic curve C is determined by its square and its pairing with the canonical class, as given by the adjunction formula 2g -— 2 = [C]2 + K . [C] (38) Another important property that J—holomorphic curves share with holomorphic curves is that they intersect each other locally positively. Namely, if C, and C2 are two dis- tinct J-holomorphic curves, then [C1] [C2] 2 0 and each point x E ClflCg contributes positively to that intersection number. This is a result of Dusa McDuff and can be found in [5]. For a given E E H2(X;Z), set (1: %(E2—E-K) (39) where K is the canonical class associated to to. Introduce Ad as the set of pairs (J, 52) with J an almost-complex structure compatible with w and Q a set Of d distinct points Of X. It has the structure Of a smooth manifold inherited from the Frechet manifold C°°(End(TX)) x Symd(X). Each J -holomorphic curve C comes equipped with a linear Operator DC : C°°(NC) —2 C°°(NC®T0’1C) Obtained as the linearisation of the generalized Cauchy-Riemann op- erator 8—6:. Here No is the normal bundle of C in X (which is also a complex subspace 32 of TX). In the case when C contains all points of I), let er : C°°(NC) -—+ EDPEQNP be the evaluation map associated to Q. If (1 = 0, we say that DC is non-degenerate if Coker(DC) = {0}. In the case (1 > 0, DC is called non-degenerate if DC 69 6’00 3 COO(NC) —’ COOUVC ® TO'ICI @PEQ Np has trivial cokernel. A pair (J, 9) E Am, m 2 0, is said to be generic if the following five conditions are met (see [15] for more details, especially on the definition of n-non—degenerate which is immaterial for the present discussion and we omit it): 1. For a fixed class E E H2(X;Z), there are only finitely many embedded J- holomorphic curves representing E and containing d points Of Q. 2. For each J-holomorphic curve C, the Operator DC is non-degenerate. 3. There are no connected J -holomorphic curves representing the class E E H2(X; Z) containing more than (1 points Of Q. 4. There is an open neighborhood of (J, I?) in Ad such that each pair (J’, (2’) from that neighborhood satisfies conditions 1-3 above. Furthermore, the number of J’-holomorphic curves containing (1 points of Q’, is constant as (J’, (2’) varies trough the said neighborhood. 5. If E2 = K - E = 0 then each Of the finitely many J-hOlomorphic curves in E containing d points Of Q, is n-non-degenerate for each positive integer n. 33 _ 1'." The set Of generic pairs (J, 9), which we denote by .789, is a Baire subset of Ad. For the choice of a generic pair (J, (2), each J-holomorphic curve containing all the points of Q, is assigned a weight 5(0). The weights for genus g 2 2 curves are always i1, however, weights of J-holomorphic tori may be other integers as well. The definition Of E(C) is not an all together simple matter and the reader is referred to the excellent account [15]. We only give here the definition Of 5(C) for the case d = 0 (and thus (I = Ql) and g 2 2. In that case, the operator DC : C°°(NC) —+ C°°(NC ®T0'1C) has the form Dcs=dcs+us+n§ VEF(T0’1C),uEF(TO‘1C®N®2) Here E E F (N) is the dual section of s and the sections V and u are determined by the almost-complex structure J. Find a path of Fredholm, index zero, Operators D, : C°°(NC) —> C°°(NC ® TO'IC) connecting DC to the complex linear Operator Dos + us. Such a path, if chosen from a suitable Baire set of generic paths, will have only finitely many singularities, i.e. there are points t1 < < tn with dim Ker(D,,) = 1 and dim Ker(D,) = 0 for t 76 t,. The weight E(C) is defined as (—1)". While 72 may depend on the chosen path D,, its parity does not, and thus E(C) is well defined. The Gromov—Witten invariant is now defined in the following manner: Definition 3.3 Let E E H2(X; Z) be a cohomology class and pick a pair (J, Q)from,7”9. Let MS?(E) be the moduli space of all J -holomorphic curves passing through each point of Q and homologous to the Poincaré dual of E. Then the Cromov-Witten 34 ‘u-vv I‘l' “I 11"-. l"- invariant Crx(E) of E is defined as can?) = 2 5(0) (40) CeM§’(E) An argument similar tO the one used in section 2.4 to Show the independence of the Seiberg-Witten invariant Of the choice Of the pair (g,u), shows that the Gromov- Witten invariant is independent Of the choice of the pair (J, 9) E ,7’89. 35 4 Gauge theory on symplectic 4—manifolds 4. 1 Introduction While Seiberg-Witten theory is defined for all smooth, compact 4—manifolds with b+ Z 1, it has some additional features on 4-manifolds which possess a symplectic structure. The two most outstanding Of these are the fact that there are always Seiberg-Witten basic classes on a symplectic 4-manifold and their spectacular relation with the Gromov-Witten invariants. Both of these results are due to Taubes. Let (X ,w) be a symplectic, smooth, compact 4-manifold with symplectic form to and pick a compatible triple (19,9, J) The symplectic form (9 induces a splitting of A2:+ 2: A2:+(T*X) as A“ S R-w6BAO'2 (41) which will be used below to write the curvature component Of the Seiberg-Witten equations (16) as two equations, one for each Of the summands on the right-hand side of (41). It proves more convenient and natural for the purposes of this section, to de- note the Seiberg-VVitten invariant Of the Spine-structure WE = E 69 (E 8) K ‘1) by S W X(VVE) rather than SWX(L) (with L = 2E — K). It also proves convenient to write the spinor 29 E Nil/E) in the form ¢=\/7:(asfi) GENELAENE‘EK—l) where r 2 1 is a parameter whose significance will become clear later. With this 36 notation, the map q from (15) can be calculated to be ir ir _ — (109.29) = —8—(Iozl2 - Ifil2)w+ E(afiwfi) (42) 4.2 The anticanonical Spine—structure Among the first spectacular results in Seiberg-Witten theory was Taubes’ theorem [16] saying that the Seiberg-Witten invariant Of the anticanonical Spine-structure on a symplectic manifold is equal to $1. More is true: the equations have exactly one solution (A0, fl - uo), uo E I‘(Q), for the choice of ir #=FX0——8‘w (43) in (16) and for r >> 1. The purpose of this section is to describe the solution (A0, \/2—‘ uo) and its linearized Operator. The pair (A0, 9? - uo) is characterized (up to gauge) by the condition (V0110, 11.0) = 0 (44) (where V0 is the Spine-connection induced by A0) and can be Obtained as follows: let uo be any section of Q 69 K ‘1 with |u0| = 1 and whose projection onto the second summand is zero. Likewise, let A be any connection on K “1 and let VA be its induced S pine-connection on WOJr = Q 63 K ‘1. Set a = (uo, VAuo). This defines an imaginary valued 1-form as can easily be seen: a + ('1 = (VAuO, uo) + (no, VAuO) = dluol2 = 0 Define the connection A) on K ""1 by A0 = A — a which induces the S pine-connection V0 = VA — a on W0+. This connection clearly satisfies (44). With the choice of u as 37 in (43), the Seiberg-Witten equations (16) take the form DAL/J =0 + + ir 2 2 7:7. — ‘ F. — F... = ,0ch —1-IeI >ev + ZIee + as) <45) Since the B—component of uo is zero and since [a] 2 lug] = 1, the pair (A0, uo) clearly solves the second equation of (45). The fact that is also solves the first equation relies on the closedness of w as well as (44). Taubes [16] showed that there are, up to gauge, no other solutions to (45) and, as we shall presently see, that the solution (Amuo) is a smooth solution in the sense that the linearisation of (45) at (A0, uo) has trivial cokernel. These two facts together prove the Theorem 4.1 Let (X,w) be a symplectic manifold. Then SWX(W0) = :I:1. Define S : L1’2(iA1 Q VVJ) ——> L2(iAO Q i115“ Q W0“) to be the linearized Seiberg- Witten Operator for the solution (A0, uo). Thus, for (b, (50, 52)) E L1'2(iA1Q(QQK“1)) we have (DAO(€0,€2) + g bfuo, S(b, (€0,52II = d+b " \/7—"Q(€,U0) - \/7_"Q(uo,€)a (46) f. \/§ Let S* : L2(iA0 Q iA2'+ Q WO‘) —> L122(i1\1 Q W0+) be the formal adjoint of S. The (Ff) +2 Im(u“0£2)) following proposition and corollary are proved in [13], section 4. Proposition 4.2 Let S and S“ be as above. Then the operator SS“ on L2(il\0 Q 38 iA2’+ Q WOT) is given by 1 55* = Zvow“ + no + fin, +% (47) where V0” is the adjoint of V0 and where ’R,,i = 0,1 are certain r-independent endomorphism on L2(i(./\0 Q ASH) Q W70"). The proof is a straightforward calculation, terms Of the form D A0 D310 are simplified using the WeitzenbOck formula for the Dirac operator. An important consequence Of (47) is the following: Corollary 4.3 With S and S" as above, the smallest eigenvalue A1 of SS* is bounded from below by r/16. In particular, S is invertible and S“1 satisfies the bounds _ 4 _ llS 1yl|2 S —,r||y|I2 and HS ll/IIm S Cllyll2 (48) f where C is r-independent. 4.3 The general case and SWX(WE) = CrX(E) Consider now a SpinF-structure WE = E (8) DVD on X. The connection A0 on K '1 and a choice of a connection B0 on E together induce a connection 3892 (8) A0 on E®2 <8) K‘1 = c1(W§) by the product rule B§2®Ao(991®992®¢) = Bo(¢1)®992®¢+901®Bo(992)®¢+¢1®W2®A0WI The space of connections on EL32 <8) K ‘1 is an affine space with associated vector space iflk. With the choice of a base connection B832 @940 in place, we will from now 39 on regard solutions to the Seiberg-Witten equations as pairs (a, 29) E i9], x I‘(W§ ) rather than (A, 9) E Conn(E®2 (8) K”) x I‘(W§), the relation between the two being A = 3892 (8 A0 + a We will agree to use henceforth the choice of ,u in (16) to be ir f1 2 —§ + F410 (49) For 29 E F(E (8) (Q Q K"1)) we will write 29 = fi(a ® umfl) with oz E I‘(E) and ,B E F(E (8 K ‘1) and an as in the previous section. With these conventions understood and with the use of (41), the Seiberg—Witten equations become D029 =0 F“ = i—Tdalz — W — 1)w (50) a 8 F022 = i—Tafi a 4 Here F351 is the orthogonal projection of 2 F $0 + d+a onto A”. We also use this section to remind the reader of several useful bounds that a so- lution (a,29) of the Seiberg-Witten equations satisfies. These bounds are provided courtesy Of [12] and their proofs rely solely on properties Of the Seiberg-Witten equa- tions. A solution (a, 29) of (50) satisfies the following bounds: lal S1 + Q ,. 40 C C’ W 3 7(1— lal2)+ ,—3 (51) IVAOIE. S Cfi exp (—%;dist(x,a—l(0))) , x E X |1—[a(x)[2[ S Cexp (—-\é—;dist(x,a“1(0))), x E X The constants C and C’ appearing above only depend on E and the Riemannian metric g but not on the particular choice of r. The inequalities (51) (together with a monotonicity formula which we don’t need for our discussion) are the basis of [12] where Taubes shows that every solution of the Seiberg-W’itten equations gives rise to an embedded, possibly disconnected, J- holomorphic curve. The converse of this fact is also true. Namely, in [13] Taubes shows that every J-hOlomorphic curve with genus g 2 2 can be used to construct a Seiberg-Witten monopole. Tori are special cases, not every torus gives rise to a Seiberg—Witten monopole, but certain collections of tori tOgether do. In any case, the following theorem holds and is a magnificent culmination of the interplay between Seiberg-Witten and Gromov—Witten theory: Theorem 4.4 (Taubes, 1996) Let (X,w) be a symplectic manifold with b+ Z 2 and E E H2(X;Z). Then SWx(WE) = er(E) (52) Some of the most immediate consequences Of theorem 4.4 are summarized in the following corollary. Recall that a manifold X is said to have Seiberg-Witten simple type if for all basic classes L of X, the dimensions of the corresponding moduli spaces Mill/(L), are all zero. 41 Corollary 4.5 1. The Poincare dual of the canonical class K of a symplectic manifold can be represented by an embedded J—holomorphic curve. 2. There is a duality in Cromov- Witten theory relating the Gromov— Witten invari- ant ofE to K — E via Crx(E) = iCrx(K — E) (53) 3. Symplectic manifolds with b+ Z 2 have simple type. Proof. The first claim is a direct consequence of theorems 4.1 and 4.4, while the second follows from theorem 4.4 and proposition 2.19. The third point requires a bit more thought. If W3 is a Seiberg-Witten basic class, then by theorem 4.4, E is a Gromov-Witten basic class. Thus, its Poincaé dual can be represented by an embedded J-holomorphic curve 2 of genus 9. Also, the dimension Of the Gromov—Witten moduli space has to be non-negative: 1 dim MOVE) = 5(E2 — K . E) 2 0 (54) Combining (54) and the adjunction formula (38), one Obtains the following two in- equalities: E22g—1 and K-ESg—l (55) Let n 2 0 be the integer such that E2 = g — 1 + n and thus (via (38)) also K - E = g — 1 -- 71. Now we use, the already proved, second point of the present corollary, namely Cr(E) = :izCr(K — E). The conclusion is that the class K — E is also a basic class and so its Poincaré dual is represented by another J-hOlomorphic curve 2’. 42 Now one invokes the aforementioned result of Dusa McDuff [5] on the positivity of the local intersection of two J-holomorphic curves: [2] fl [2’] 2 0. Translated into cohomology classes this gives OSE-(K—E)=E-K-E2=g—1-—n—(g—1+n)=~2n§0 Thus n = 0 and E2 = K - E = g — 1. The Seiberg-Witten simple type is now an easy consequence: dim MSW(L) = ((2E — K)2 — (30 + 2e)) = (K2 — (30 + 2e)) = 0 shit—t Ali—t whereL=2E—K. I 43 5 Grafting Seiberg—Witten Monopoles This section is the heart of the thesis as it is here where we construct the grafted monopole (a, 29) for the S pine-structure WE from two monopoles (a,, 29,) for the S pinc- structures WE“ i = 0, 1 (where as usual E = E0 (8) E1). Recall from the introduction that the term “grafing” refers to the map described by the top row in the commutative diagram (3). 5.1 Producing the approximate solution (a,29) from a pair (610,190), (01, 791) Let E0 and E, be two complex line bundles over X. The aim of this section is to produce an approximate solution (a, 29) of the Seiberg-Witten equations for the S pinc- structure Whom, from two solutions (a0,290) and (a1,291) for the Spine-structures WE0 and W'E, respectively. Implicit tO our discussion are the choices of two ”base” connections Bo and B, on E0 and E1 and the product connection BO (8) B, they determine on E0 (8) E1. As before, we will write 29, = fi(a, ® uo,fi,), i = 0, 1, and 29 = (5(0 <8 umfi). We define (a,29) as a=ao+a1 a=a0®al (56) 5=ao®5i+ai®fio The first task at hand is to check how close (a, 29) comes to solving the Seiberg- Witten equations. We begin by calculating D029 locally at a point x E X. Choose an 44 orthonormal frame {e,}, in a neighborhood Of x and let {6‘}, be its dual frame. 00(19): fiDamo ® (218) uo + 0’0 (83,314“ 0118) 50) = fiDao(Oo 69 21.0) 23) (11+ x/fozo <8 Dal (ai <89 Hol+ + weivgfao (831+ 0129.30) —_— (519mm a ug) (a a1+ fiao ® Da,(a1® uo)+ + V2010 ® e‘.(V:},81) + “1‘59 ei-(VSWOH + (V3300) a as. +(vzga1)® e'lfio) = finaomo (81 21.0) (8) (11+ flat) (8) Do, (01 69 ”OH + x/f(ao ® Dalfii + 0183 Dao,30)+ + WMVQ‘f’ao) ® 6‘91 +(V3.101)® 62,30) 2 (Daolv120)® 01+ do 49 (13.11191)+ + fi((vgyao) e (2'91 +(V23a1) <8 6‘90) = VHVZMOI <8) (ii-51+ VRVSSO’II ‘8’ ei-fio (57) It is easy to see, using the bounds in (51), that the first term in (57) satisfies the following pointwise estimate : 7” [(V2'3010) ® (ii-(31]: S _<_Cr exp (——\éjdist(x, a51(0))) -exp (—%dist(x,af1(0))) (58) The second term in (57) satisfies the same bound. In order for the right hand side of (58) to pointwise converge to zero, it is sufficient and necessary that there exist some r0 2 1 such that for all r 2 r0, the distance from 061(0) to (kl—1(0) be bounded 45 from below by some r-independent M > 0. This condition, under the map O from (3), is the Seiberg-Witten equivalent of the condition that 2, = O(A,, 29,) be disjoint curves. Thus, from now onward we will make the Assumption: There exists an r0 2 1 and AI > 0 such that for all r 2 r0 the inequality dist(agl(0),af1(0)) 2 AI (59) holds. We now proceed by looking at the second equation in (50): i 8 i : F0151 '1' Falf1 — g7“ ([010]2 ' [01]2 — 1 _ [0'0]2 ' I131]2 — [011]2 ' [50]2 — 2(0051101ngollw Fal’l- 7“(lalg -1-I.3|2)w = i i = F3511" Fol;1 — g?" IC¥1I2(I00I2 —1—I30I2)w — §T|010|2(|01|2 ‘1—I51I2)w+ 2' i + g7~([(10|2 — 1)(|c11]2 -1)w + Zr (definaifiolw ,- 2' = §T"(1-|0z1|2)(lao|2 —1-|,30I2)w _ {3'7“(1-IC10I2XI011I2 — 1 - I131I2lw+ i i + -v (Ieor -1)(lall2— Us + ,2 wowed“: 00 From this last equation, and again using (51), one easily deduces that i IFJ’I-yUaIQ—1—I3I2lwl s (60) S Cr exp (—\gdist(x,agl(0))) -exp <—%dist(x,af1(0))) + % Finally, we consider the third equation in (50): i _ i F22 - Zrafi 2 F262 + Fff — 4T aoa1(aol31 + 0190) =—rc'i +—roF,B——rc12d ——ra2ci3 4 030 4 1 1 4 I o] 151 4 I 1] 0/0 46 i _ i _ = Z7” (1 — |01|2laofio + 17(1- I00I2la1/81 Once again using the bounds (51), we find from this last equation: IFS"? — grew s (61) 3 Cr exp (—%dist(z,aal(0))) -exp (“—gdist(xva?1(0))) + '5? TO summarize, we have proved the following Proposition 5.1 Let (a,29) be defined as in (56) and assume that there exists an r0 2 1 and WI > 0 such that for all r 2 r0, the distance dist(a{,’1(0),af1(0)) is bounded from below by 114. Then for large enough r and any x E X the pointwise bound below holds: i i I(Da(v), Fl:1 — g7“ (lal2 — 1— Inflows” — Event 3 (62) SIC? 5.2 Inverting the linearized Operators Of (a,, 29,) This section serves as a digression of sorts. The main result here is theorem 5.5, an asymptotic (as r —> oo) regularity statement for the linear operators L(a,,,,,,) (as defined by (18)). We start with two easy auxiliary lemmas: Lemma 5.2 Let L : V -—+ W be a surjective Fredholm operator between Hilbert spaces. Then there exists a (5 > 0 such that for every linear operator 6 : V -—+ 14/ with ||l(x)||w S (5 ||x||v, the operator L + t is still surjective. 47 Proof. Since L is Fredholm, we can orthogonally decompose V as V = Ker(L) Q Im(L*). Let L, be the restriction of L to Im(L"). Then L1 : Im(L*) —-> W is an isomorphism with bounded inverse LII. If the lemma were not true then we could find for all integers n 2 1 an operator 8,, : V —> W with [Ifnxllw S l/n- [lelv and with Coker(L + in) # {0}. Let 0 set y.n E Coker(L + in) with [lanIW = 1 and xn = Ll‘1(y,,). Notice that the sequence {xn},, is bounded by ||Lf1||. Since yn E Coker(L+€,,), yn is orthogonal to Im(L+€,,). In particular, ((L + l’,,)x,,, yn) = 0 This immediately leads to a contradiction for large enough n since (L xn, yn) = 1 and |<€n$mynllSllLI1||/n- I Lemma 5.3 Let V and W be two finite rank vector bundles overX and L, : L1'2(V) —~> L2(W) a smooth one-parameter family (indexed by r 2 1) of elliptic, first order, diflerential operators of index zero. Assume further that there exists a (5 > 0 and r0 2 1 such that for any zeroth order linear operator 3 : L1’2(V) —> L2(W) with ||€(x)||2 < 5II$II1,2; the operator L, + i is onto. Then there exists a r1 2 r0 and a (W > 0 such that for all r 2 r, the inverses of the operators L, are uniformly bounded by M; 236- IIL:1yII1,2 S MHZ/Ili- Proof. Notice that a universal upper bound on L,‘1 is equivalent to a universal lower bound on LT. Suppose the lemma is not true: then there exists a sequence r,, —> 00 and 33,, E L1'2(V) with ||x,,||1,2 = 1 and “L," $n[[2 < 1/n. Choose 12 48 large enough so that 1/n < (5 and define the Operator 8 : L1’2(V) -—> L2(W) by €(x) = —(x,,, x)1,2 - L,,,(x,,). For this i the assumption of the lemma is met, namely 1 II5($III2 S Ell-TIM < 5IIIII1.2 Thus the Operator L,” + i should be onto and into (since the index of L, + f is zero). But x1, is clearly a nonzero kernel element. This is a contradiction. I Recall that the set J of almost-complex structures compatible with the symplectic form id, contains a Baire subset Jo of generic almost-complex structures in the sense of Gromov-Witten theory (see [15]). Also, as in the introduction, let 9 : Mill/(WE) —-» M§:(E) (63) be the map introduced in [12] which associates an embedded J-holomorphic curve to a Seiberg—Witten monopole. Proposition 5.4 Let J be chosen from .70 and let (a, 29) be a solution of the Seiberg- Witten equations (50) such that O(a,29) doesn’t contain any multiply covered com- ponents. Then there exists a (5 > 0 and on re 2 1 such that for all linear operators 6: L1»2(2;A1e9 E a W3“) —> L2(tA0 (9911+ Q E 8) W5) with norm |l€(:v)|l2 < 6|I$ll1,2, the operator L(a.29) + 8 is surjective. Before proceeding, the reader interested in the proof of proposition 5.4 is advised to familiarize her/ himself with the definitions and notation in [13] since our proof will heavily rely on results proved therein. Proof. The proof is a bit technical and relies on the even more technical account from [13] on the connection between the deformation theory of the Seiberg-Witten 49 equations on one hand and the Gromov-Witten equation on the other. The idea is however very simple: for large r >> 1, a certain perturbation of the operator L (with the size of the perturbation getting smaller with larger r) has no cokernel if a certain perturbation of the linearisation of the generalized del-bar operator has no cokernel. The latter is ensured by the choice of a generic almost complex structure J from the Baire set J0 of almost complex structures compatible with 2.9. For the convenience Of the reader we restate here the parts of lemma 4.11 and a slightly modified version of lemma 6.7 from [13] relevant to our situation. Lemma 4.11 The equation Lq + nq = g is solvable if and only if, for each k, the equation Artur" + 261w) + me) = 22(9") + why) is solvable. In the above, I: indexes the set of components of O(a,29) and Ac), represents, roughly speaking, the Cauchy-Riemann Operator associated to the component Of O(a,29) with index k. The terms 771- and x(gk) are constructed from 27 and 9 re- spectively while 70 and ’71 are some auxiliary operators which depend on r (and whose norm gets smaller as r increases, they should be thought of as small correction terms). The assignment of 771 to n is linear i.e. for two operators 2) and 27’, we have (27 + We = 721 + 221.- Lemma 6.7’ The equation (Limy) + €)p = g has an L1:2 solution p if and only if there exists it = (u1,...,uk) E QkL1'2(N(k)) for which Ayuk + 25301) + (2(a) = Tilfvwkl + 251(9) 50 holds for each k. Here y is a J-holomorphic curve without multiply covered components and \Ilr(y) is its associated Seiberg-Witten monopole. As with the notation in lemma 4.11, the terms f), and x(g"') are constructed from f and g respectively. Similar tO the terms 7, from lemma 4.11, the terms a, serve as correction terms whose size diminishes as r grows. The proof of lemma 6.7’ is almost identical to that of the original lemma 6.7 in [13]. The only difference is in Step 2 where Taubes shows that one can write the equation Lw,.(y)P = g in the form Lp + 27p = g with L as in lemma 4.11 and with 77 = (fr: 2 w(q’, p). The difference here is that in our case one can write (L¢,(y)+t)p = g as Lp + r/p = g (with L again as in lemma 4.11) but with 77’(p) = fi2w(q’,p) + [(p). Since t is assumed bounded, lemma 4.11 applies to 27' in the exact same way as it applied tO the original 77 and the proof Of lemma 6.7 in [13] transfers verbatim to our case. Note also that the Operators (9f occurring in lemmas 6.7 and 6.7' are identical so in particular they continue to satisfy the bounds asserted in lemma 6.7 of [13]. According to lemma 5.2 there exists a (5’ > 0 such that A,, + l’ is still surjective if [It’ll < 6’. Choose r large enough so that ”(96” < (5’/2k. On the other hand, since 8,,(21) = rr(X255,k€(Zk, X1006,,,rykl)) (see (66) for a definition of X6.k in the present context) we find that “8,,” S C Hf”. Thus choosing 6 = (5’/2C ensures that Lanai) + t is surjective provided that Hill < 6. This finishes to proof of proposition 5.4. I Together, the last lemma and proposition imply the following: Theorem 5.5 Choose J E .70 and let (a,29) be a solution of the Seiberg- Witten 51 equations for the S pine-structure WE with parameter r. Assume that O(a, 29) contains no multiply covered components. Then there exists a r—independent AI > 0 and r0 2 1 such that for all r 2 r0 ||L_lv)~"€||1,2 S M||$I|2 (64) (0.19 5.3 The linearized Operator at (a, 29) In order to use the contraction mapping principle to deform the approximate solution (a,29) to an honest solution of the Seiberg-Witten equations, we need to know that L = Lump) admits an inverse whose norm is bounded independently Of r. We start by exploring when the equation L5 = g (65) has a solution 6 for a given g. Here {6 L1»2(1A1e(E0 (a E1®WO+)) and g e L2(2'A° eat/122+ e (E0 a E128 W;)) The idea is to restrict equation (65) first to a neighborhood of 0251(0). Over such a neighborhood the bundle E1 is trivial and, under an isomorphism trivializing E1, the equation (65) becomes a zero-th order perturbation of the equation L060 = go (with £0 and g0 being appropriately defined in terms of 6 and g). This allows one to take advantage of the results of theorem 5.5 about the inverse of L0 = L(,,O,,,,0). Then one restricts (65) to a neighborhood Of til—1(0) where the bundle E0 trivializes and once again uses theorem 5.5, this time for the inverse of L1 = L(a1,291)- Finally, one restricts tO the complement of a neighborhood Of (151(0) U 014(0) where both E0 and 52 E1 become trivial and L becomes close to S - the linearized operator of the unique solution (A0, \/r—‘ uo) for the anticanonical Spine-structure lVo. TO begin this process, choose regular neighborhoods V, Of 0:1(0), i = 0, 1 subject to the condition dist(V0, V1) 2 IV for some AI > 0 The existence Of such neighborhoods V, follows from our main assumption (59). A priori, as one chooses larger values of r, it seems that the sets V, may need to be chosen anew as well. However, it was shown in [12], section 5c, that in fact this is not necessary. An initial ”smart” choice Of V, for large enough r ensures that for r’ > r, the zero sets (1,—1(0) continue to lie inside of V,. Choose an open set U such that X 2 V0 U V, U U and such that Un(a,1(0)uQ,—1(0))= 0 Arrange the choices Of V, and U further so that 8V, is an embedded 3-manifold Of X and so that U (1 V, contains a collar 8V, x I . Here I is some segment [0,d] and (9V, corresponds to 8V, x {d}. For the sake of simplicity of notation, we shall make the assumption that for large values Of r, the sets 0:1(0), i = 0,1, are connected. The case Of disconnected zero sets of the a,’s is treated much in the same way except for that in the following, one would have to choose a bump function x,”- (see below) for each connected component. This complicates notation to a certain degree but doesn’t lead to new phenomena. Fix once and for all a bump function X : [0, 00) —> [0, 1] which is 1 on [0,1] and 0 53 on [2,00). For 0 < (5 < d/1000 define X5, : X ——> [0, 1] by 1 x E V,\(8V, X I) X21103) = X095) .2: = (at) E 6V.- X I (66) 0 :v 9! V.- Set V’ 2 V0 U U and VI’ = V, U U. Define the isomorphisms To : Q x Vd —> Elli/0' and T1 : (C X Vl’ —> E0Iv,' as T0()\,x) = 01(x) - A and T1(/\,IL‘) = ao(x) - )I. The isomorphism T1 defined here shouldn’t be confused with the isomorphism of the same name appearing in lemma 6.7’. While both serve tO identify a pair Of bundles, the bundles in question in these two situations are not the same. For i = 0, 1 define the operators M, : L1:2(21A1 e (E, e WJ); v.’) —, L2(iA0 69 2A“ 213 (E.- e Wo‘); V!) 1 and T: L1’2(il\l Q WOT; U) —> L2(iA0 Qi1'12’+ Q W0—; U) by demanding the diagrams L1'2(2'A1Q(E0 e E1 e1 WJ); V,’) L122(2'A1e(E.® W5”); W) L] (M. L2(1‘AO @ 2A”r 219(E0 (31 E1 (a W0"); V,.’) ,_r.-_ L2(iA0 eat/12d <9 (E, (a W0“); V,’) Ti and L1'2(1A1Q(E0®E1®WO+);U) 39—“ L1v2(1:A1eWO+;U) L] [T L2(2JA0 ea 2A“ e (E0 a E, 19 W5); U) J—T— L2(2:A° e 1111+ air/01(1) to be commutative diagrams. We now start our search for a solution 6 Of (65) in the form 5 = T0(X1006,0€0) + T1(X1006.1€1)+ T0T1 ((1 - X46,0)(1 — X42107?) (67) 54 Here .5, E L1*2(iA1Q(E,®W0+)) and r) E L1’2(iA1QW0+). Given a g E L2(iA0QiA22+Q (E0®E1®W0')), define g, E L2(iA°QiA2’+Q(E,-®W0’)) and '7 E L2(iA°QiA2’+QWO_) as 91' = TEI(X256,29) and ’7 = (T0T1)"1 ((1 — X256,0)(1 — X255,1)9) (68) It is easy to check that g, g, and 7 satisfy a relation similar to (67), namely 9 = T0(X1006,0go)'+' T1(X10025,191)'l‘ T0T1((1 — X46,O)(1 — X46,1)’7) (69) Putting the form (67) Of 25 and the form (69) of 9 into equation (65), after a few simple manipulations, yields the equation T0(X1006,o(Mo(€0) — T17301901210, 7)) — go))+ (70) +T1(X1006,1(Mi (£1) — T0719(dX425,1,77) ‘ 91))+ +T0T1((1— X46,0)(1 — X46,1) (T77 + ’P(dX1006,07€0) + P(dX1006,1,€1) — 7)) = 0 In the above, ”P denotes the principal symbol of L. This last equation suggests a splitting into three equations (each corresponding to one line in (70)): Mo(€o) — T1P(dX4a,o, 7)) = 90 MI(€1)— ToP(dX4a.1, 77) = 91 (71) Tr) + P(dX1005,07€0) + P(dX1006,1v€1) = 7 Equation (70) (and hence also equation (65)) can be recovered from (71) by multi- plying the three equations by T0 ' X10050, T1 ' X1005’1 and TOT] ' ((1 — X45,0)(1 — X45,1) respectively and then adding them. Thus, given a g and with g, and 7 defined by 55 (68), solutions 5, and n of (71) lead to a solution 5 of (65) via (67). However, the problem with (71) is that the Operators M, and T are not defined over all of X. We remedy this in the next step. Define new Operators : M,’ : Uta/11 e (E, 23) W6“)) —> L2(2‘A0 e A“ e (E.- e W0“)) and T’ ; Uta/11 (9 WJ) —. L2(2'A° e A“ e W5) (11,, = X2006,2'A’Ii + (1 — X2006,1)Li T, = (1 — X6,O)(1 — X6,1)T + (Xao + X6,1)S (72) Here L, : L0H“). Now replace the coupled equations (71) by the following system: 1146050) - ToP(dx4.s,o, 7)) = 90 Mi(§1) — T1P(dx45,1, 77) = 91 (73) T") + E(deoao, {0) + P(dX1006,17 51) = 7 The advantage of (73) over (71) is that the former is defined over all of X. On the other hand, solutions of ( 73) give rise to solutions of (65) in the same way as solutions of (71) did because X1006,i ' ll/I,’ = X1006,i ' (WI 1 = 0, 1 (1 — X46,0)(1"‘ X45,1)T’ = (1 — X46,0)(1“ X46,1)T 56 Lemma 5.6 For every 6 > 0 there exists an rE > 1 such that for r 2 r6 the following hold: “(Mi - L1)£E1II2 S 6Il3«‘z'll2 |l(T' - S)yll2 S 6l|9||2 Here x, E L1'2(iA1 Q E, (8) DVD“) and y E L1’2(il\1 Q W0+). Proof. The above Sobolev inequalities are proved by first calculating pointwise bounds for l(AzI,~’—L.,)x, [p and |(T’—S)y |,,, p E X. Notice firstly that [(llif—L,)x, [p = 0 ifp E V, and |(T’ — S)y [p = 0 ifp E U. For p E V, and for q E U, a straightforward but somewhat tedious calculation shows that |(M! — 91st S C (x/Fll — [0,-[2| + x/Flfivl lael + lVa‘asl) lxilp |(T’ — S)y I. s C(fill — Iaelil + fill - Iallzl + filfiel+ + 1/2 I61! + lV“°ael + IV“‘a1I) Iqu Squaring and then integrating both sides over X together with a reference to (51) gives the desired Sobolev inequalities. I The lemma suggests that the system (73) can be replaced by the system Lo(€b) — TOP(dX46,Ov 77') = 90 L1(51) " T1’P(d>(46,1,77’) = 91 (74) 577' ‘1’ P(dX1006,03 6(1) '1‘ P(dX1006,1,€l) = ’7 Lemmas 5.6 and 5.2 say that for r >> 0, (73) has a solution (£0,§1,n) if (74) has a solution (£6,£[, n’). It is this latter set of equations that we now proceed to solve. 57 Since S is onto, we can solve the third equation in (74), regarding £6 and {i as parameters. Thus 7), = 77156151) = S_1(7 " E(dX1006,0,€()) — E(deoahéi» (75) Recall that the inverse Of S satisfies the bound (48) IIS'lylle s 3—7llylle for y 6 L294" ea 121+ es Wo‘) We will solve the first two equations in (74) simultaneously by first rewriting them in the form: {6 =L61(go + T07701001210, 7((61’), 5,1)» 61 =L1’1(91 + T1P(dX4.s,1, 27756.61)» (77) To solve (77) is the same as to find a fixed point of the map Y : L2(iA1 Q WED) >< L2(iA1 93 WE.) —> L2(2'A1 ea WED) x L2(iA1 99 W31) given by 1466.61) = (78) =(chl(90 + Tf1T0T1P(dX46‘0, 77')), Lil(91 + T1P(dX46,11 7(D) with r)’ given by 75. The existence and uniqueness of such a fixed point will be guaranteed by the fixed point theorem for Banach spaces if we can show that Y is a contraction mapping. TO see this, let x, y E L2(i/\1 Q W50) x L2(iA1 Q WEI) be two arbitrary sections. Using the first bound of (48) and the result of theorem 5.5 to bound the norms of L," 1, one finds ||Y(:r) — Y(9)|l§ = 58 = llL61(9o + ToP(dX4vs,o. 77(0)) - L61(9o + ToP(dX4a,o. 27(9)))||§ + IILf1(91 + Tlp(dX45,la 77($))) - L1—1(91+ T173((1X4Is,1177(9)))II§ S00II7'I(’~F) — 77(9)”; + CIIITKI) — 77(9)”; SCIIS—l(p(dX1006.Oa y) - p(dX1006,07$) + E(deoanJ/l - E(dX1006,1.$))IIg C £7||$_ yllg (79) Choosing r > 2C, where C is the constant in the last line of (79), makes Y a contraction mapping. Thus we finally arrive at an L2 solution (56, 51). It is in fact an L1:2 solution because of (77). This, together with equation (75) provides a solution ({6,€[, 17’) of (74). As explained above, this gives rise to a solution (50,61, 77) of (73) and thus provides a solution 6 E Ll'2 E (M1 Q W5“) of (65). In particular, we have proved half of the following Theorem 5.7 Let (a,29) be constructed from (a,,29,) as in (56). Suppose that the (a,,29,) meet assumption (59) and that J has been chosen from the Baire set J0 of compatible almost complex structures. Then L(a,,,,) : L1’2(il\1 Q E0 <8) E1 (8) W0+) —+ L2(iA0QiA2'+QE0®E1®W0—) is invertible with bounded inverse IIL(;]¢)?JII1.2 S C ||y||2 for all sufiiciently large r. Here C is independent of r. Proof. It remains to prove the inequality “LEW/“1.2 S C ||y||2. Each of the two lines of (77), together with the bound (64) on L,” 1, yields: ||€§||1,2 S C(ll91ll2 +I|71’(€6.€I)||2) (80) A bound for the second term on the right—hand side of (80) comes from (75) and the 59 L2 bound in (48): H9 (£6 €1)S||2 %(III I||2+ ||€6||2+||€il|2) (81) Adding the two inequalities (80) for i = 0, 1 and using (81) gives (1—\/—)(II€()II12.+II€III12)SC(IIQOII2'1'II91IIQ'I'7IIIII) (82) For large enough r, this last inequality gives a bound on the L1:2 norm of (£6,231) in terms of an r-independent multiple of the L2 norm of (go, 91,7). With this established, the missing piece, namely the L1:2 bound of 27’, comes from (75) and the L1:2 bound in (48): lla’llm S C (ll‘I/lle + l|€6||2 + lléillz) S C(IIVII2 + ||90||2 + ll91ll2) (83) It remains to relate the now established bound on (56,61, 27’) to a bound for (£0, {1, 77). To begin doing that, write the systems (74) and (73) schematically as f(€6»§i,n')=(9o.91.2) and Q(€o,£1.77)=(9o,91.7) Lemma 5.6 implies that for any 5 > 0 there exists a r, 2 1 such that for all r 2 r, the inequality ||(f — Q) xllg S e ||xI|2 holds. The established surjectivity of .F guarantees (by means of lemma 5.2) that Q is also surjective. The proof of theorem 5.7 thus far, also shows that [If-71]] S C where C is r—independent. Now the standard inequality ||9"1|| S llf‘lll + Hg"1 - f‘lll S llf‘1||+||f‘1|l ~ llg‘lll Ilg - TH implies the r—independent bound for [IQ—1]] llf’lll C - ||7"1||-I|9 -fll_1- CE _1 < II9 II _, 60 This last inequality provides L1:2 bounds on (50,51) and r) in terms of the L2 norms of (go, g1) and 7 which in turn imply an r—independent L1:2 bound on 6 = L‘lg in terms of the L2 norm of g through (67) and (68). This finishes the proof of theorem 5.7. I 5.4 Deferming (a, 29) to an honest solution The goal of this section is to show that the approximate solution (a, 29) can be made into an honest solution of the Seiberg-Witten equations by a deformation whose size goes to zero as r goes to infinity. To set the stage, let SW : lei’h'A1 ea wg) —> L2(iA0 ea ,A2,+ ea Wg) denote the Seiberg-Witten operator Swa, e) = (are, F: - Fr, — q(e, e) + ’gw) We will search for a zero of SW of the form (a, 29) + (a’, 29’) with (a’, 29’) E B(6). Here B (6) is the closed ball in L1*2(iA1 Q WE ) centered at zero and with radius 6 > 0 which we will choose later but which should be thought of as being small. The equation SW((a, 29) + (a’, 29’)) = 0 can be written as o = SWIa, 2)) + Lemma 9) + 9(a', 29’) (84) Here Q : L1'2(iA1 Q WE) —> L2(i/\0 Q iAQ’+ Q W5) is the quadratic map given by 99. e0, e) = (M96 + e2), €00.50]? — l92|2)w + 61450 e + a e.» (85) Lemma 5.8 For x, y E L1’2(il\1 Q WE), the map Q satisfies the inequality: IIQ(5E) - Q(y)ll2 S C (II$II1,2 + Ill/[[1,2III33 - llllia (86) 61 Proof. This is a standard inequality for quadratic maps and it can be explicitly checked using the definition of Q and the multiplication theorem for Sobolev spaces. We give the calculation for the first component of the right hand side of (85). Let x = (b,(9) and y = (C,(,9), then we have [lb-41 - C-vll2 =llb-(9 - 0.95 + 6.95 - C-WII2 S ”(b - Cle¢l|2 + IIC-(¢ - 4C2/62 ensures that Y is well defined. Lemma 5.9 The map Y : B(6) ——> B(6) as defined by (87) is a contraction mapping for r large enough and 6 small enough. 62 Proof. Let x, y E B(6), then using (86) we find IIYW) — Y(y)II1,2 S CIIQW) ‘ Q(l/)II2 S CII'I + ylll,2 II18 — Illlia (88) Choosing 6 < 1/2C makes C ||x + ylllg S 2C6 less than 1. I We summarize in the following: Theorem 5.10 Let (a,29) be constructed from (a,,29,) as in (56). Suppose that the (a,,29,) meet assumption (59) and that J has been chosen from the Baire set ,70 of compatible almost complex structures. Then there exists a 60 > 0 such that for any 0 < 6 S 60 there exists an r5 2 1 such that for every r 2 r5 there exists a unique solution (a, 2,9)+(a’, 2,9’) of the Seiberg- Witten equations (with perturbation parameter r) with (a’,2,9’) E L1’2(iA1 Q WE) satisfying the bound |[(a’, /’)||1,2 S 6. 63 6 Comparison with product formulas Before proceeding further, we would like to take a moment to point out the similarities and differences between our construction of (A,29) from (A,,29,) on one hand and product formulas for the Seiberg-Witten invariants on manifolds that are fiber sums of simpler manifolds. We begin by briefly (and with few details) recalling the scenario of the latter. Let X,, i = 0,1 be two compact smooth 4-manifolds and 2, H X, embedded surfaces of the same genus and with 20 - 20 = —):1 - 21. In this setup one can construct the fiber sum X = Xo#2,X1 by cutting out tubular neighborhoods N (2,) in X, and gluing the manifolds X,’ = X—,\TV_(Z_,) along their diffeornorphic boundaries. Under certain conditions one can calculate some of the Seiberg-Witten invariants of X in terms of the Seiberg-Witten invariants of the building blocks X, (see e.g. [8]). One accomplishes this by showing that from solutions (8,, (1),), i = 0, 1 on X, one can construct a solution (B, (1)) on X (this isn’t possible for any pair of solutions (8,, (1),) but the details are not relevant to the present discussion). This is done by inserting a ”neck” of length r 2 1 between the X,’ so as to identify X with X = X51) ([0,7‘] x Y) UX[ with Y = 6N(Eo) ’-‘_-’ 8N(21). A partition of unity {990,191} is chosen for each value 64 of r 2 1 subject to the conditions (,9, = 1 on X,’ (p, = 0 outside of X,’ U [0, r] x Y Iezl sgen [0.2:] W An approximation Q’ of Q is then defined to be Q’ = «pg Q0 + (p, Q, (similarly for B’, a first approximation for B). The measure of the failure of (B’, Q’) to solve the Seiberg-Witten equations can be made as small as desired by making r large. The honest solution (B, Q) is then sought in the form (B’,Q') + (b, 9)) with (b, (9) small. The correction term (b, (9) is found as a fixed point of the map (b2 ((1)) H Z(ba¢) : —L(—Bl’,q>') (Q(b)¢) + BIT) Here ”err” is the size of SW (8’ ,Q’ ) and L and Q are as in the previous section. Choosing r large enough and [|(b, (9)“ small enough makes Z a contraction mapping and so the familiar fixed point theerern for Banach spaces guarantees the existence of a unique fixed point. In the case of fiber sums there are product formulas that allow one to calculate the Seiberg-Witten invariants of X in terms of the invariants of the manifolds X,. The formulas typically have the form: swerve) = Z Son(WEo) - SWXIIWE.) (89> E0+E1=E Due to the similarity of our construction of grafting monopoles to the one used to construct (B, Q) from (B,, Q,), it is natural to ask if such or similar formulas exist 65 for the present case, that is, can one calculate S WX(WE,,® E,) in terms of S W X(WE0) and S l/VX(WE,)? The author doesn’t know the answer. However, if they do exist, they can’t be expected to be as simple as (89). The reason for this can be understood by trying to take the analogy between our setup and that for fiber sums further. In the case of fiber sums, once one has established that the two solutions (B,, Q,) on X, can be used to construct a solution (B,Q) on X, one needs to establish a converse of sorts. That is, one needs to show that every solution (B, Q) on X is of that form. It is at this point where the analogy between the two situations breaks down. It is conceivable in our setup, that there will be solutions for the Spine- structure (E0 Q E,) Q WO+ that can not be obtained as products of solutions for the Spine-structures E, Q W0+. Worse even, there might be monopoles that can not be obtained as products of solutions for any Spine-structures E, Q W0+ with the choice of F], j = 0,1 such that E = F0 Q F, and F,- 7é 0. Those are the monopoles where a‘1(0) is connected. Thus if a product formula for our situation exists, it must in addition to a term similar to the right hand side of (89) also contain terms which count these ”undecomposable” solutions. But then again, they might not exist. The next section describes which solutions of the Seiberg-Witten equations for the Spine-structure (E0 Q E,) Q W0+ are obtained as products of solutions for the Spine-structures E, Q W01”, E = E0 Q E,. 66 7 The image of the multiplication map This section describes a partial converse to theorem 5.10. Recall that e : M§W(We) -+ M929) is the map assigning a J-holomorphic curve to a Seiberg-Witten monopole. Theorem 7.1 Let E 2 E0 Q E, and let (A,29) be a solution of the Seiberg- Witten equations in the Spine-structure WE with perturbation term )1 2 FIG — irw/8 and with 29 = \/r_:(o( Q umfi). Assume further that J has been chosen from the Baire set J0 and that O(A,29) contains no multiply covered components. If there exists an r0 such that for all r 2 r0, a‘1(0) splits into a disjoint union a‘1(0) = 20 LI )3, with [2,] =P.D.(E,) then (A, 29) lies in the image of the multiplication map Min/(E10) X Milt/(E1) —* Min/(E0 ® E1) The proof of theorem 7.1 is divided into 3 sections. In section 7.1 we give the def- inition of (Ag, 29;) - first approximations of Seiberg-Witten monopoles (A,, 29,) for the Spine-structure WE, which when multiplied give the monopole (A, 29) from theorem 7.1. Section 7.2 shows that for large values of r, (A;,29§) come close to solving the Seiberg-Witten equations. In the final section 7.3 we show that L(A;,,,,;) is surjective with inverse bounded independently of r. The contraction mapping principle is then used to deform the approximate solutions (A2, 29f) to honest solutions (A,, 29,). Section 7.3 also explains why (A0,290) - (A,, 29,) = (A,29). We tacitly carry the assumptions of the theorem until the end of the section. 67 7.1 Defining (A2, 291-) The basic idea behind the definition of (A[, H) is again that of grafting existing solutions. For example, one would like 296 to be defined as the restriction of 29 to a neighborhood of 20 (under an appropriate bundle isomorphism trivializing E, ever that neighborhood) and to be the restriction of flue outside that neighborhood. This is essentially how the construction goes even though a bit more care is required, especially in splitting the connection A into A6 and A’l. To begin with, cheese regular neighborhoods V0 and V, of 20 and 2,. Once r is large enough, these choices don’t need to be readjusted for larger values Of r. Choose, as in section 5.3, on Open set U such that X=V0UUUV1 Also, just as in section 5.3, arrange the choices so that U (I V, contains a collar (9V, >< [0,d] (with (9V, corresponding to 8V, x {d}) and choose 6 > 0 smaller than d/ 1000. Assume that the curves 2, are connected, the general case goes through with little difficulty but with a bit more complexity of notation. Over U UV,, choose a section 70 E I‘(E0; U UV,) with I‘lol = 1. Choose a connection 8,, on E0 with respect to which '70 is covariantly constant over U U V,, i.e. Bo('yo(x)) = 0 Vx E U U V, (90) N otice that such a connection is automatically fiat over U U V,. Choose a connection B, on E, such that B0 Q B, = A over X. New define 07,’ E F(E,;U U V,) and 68 91’ e r(E1 e K-1;U e V,) by C1 = ’70 ® d1, (91) C = "/0 8) Ci, (92) Proceed similarly over V0. However, since some of the data is new already defined, more caution is required. Choose a section 7, E F(E,; V0) with 7, = 021’ on (Ufl Vo)\(3V0 X [0,45» (93) |7,| =1 on (V0\U) U (8V0 x [0,26)) We continue by defining 070’ and Co, over V0 by a = (1’70I ® "/1 (94) 9 = 90’ e e. (95) Choose one forms a0 and a, such that over V,, the following two relations held: (81 +ia,)’y, =0 (96) (Bo+ia0)Q(B,+ia,) =A (97) With these preliminaries in place, we are now ready to define (A2, 2,9,2): 070 = X469 do, + (1 — X469)“I'0 3:0 = X469 C0 ~ I C71 = (1 — X469) (it, + X46971 51 = (1 — X469) 51 (98) A6 = 30 + 2X469 00 (‘1’, = Bl + 2X469 01 Lemma 7.2 The (A[,29,’) defined above, satisfy the following properties: a) A[,QA’1 = A on all ofX. 6) d0 = “/0 071. (U (I V0)\(8V0 X [0,46>). 69 c) F13O = 0 on UU V, and FB,+,a~, = 0 on V,,. d) On (U D V0)\(8V0 x [0,46)), |(i,| and |dd',| converge exponentially fast to zero as T'—>OO. Proof. a) This is trivially true everywhere except possibly on the support of (1)0169 which is contained in U 0 V0. However, on U (1 V,, we have A = B0 Q B, and A = (B,, + iao) Q (B, + 2a,) and thus a0 + a, = 0. In particular, A6 Q A’1 = Bo <8) Bl + 2X469 ((10 + (11) = Be (8) Bl = A b) Notice that on (U (I V0)\(0V0 x [0,46)), 7, = ei’l. Thus, a = 70 Q 7, and a = (if, Q 7, imply that 70 = 0:6. The claim now follows from the definition of (10. c) Fellows from the fact that both connections annihilate nowhere vanishing sec- tions on the said regions. (I) On (U H V0)\(8V0 x [0,46)) we have a = 70 Q7, and VA = VBOHOO Q VBI+‘“1. Also, recall that V8070 = 0 and V81+ial7, = 0. Thus Vaa = (VBOHQO ® VBIHOIXVO ® ’71) = ia07o (8) 71 This equation yields [(10] = — (99) The claim follows new for a0 by evoking the bounds (51). The same result holds for a, by the proof of part (a) where it is shown that a0 + a, = 0 on U (1 V0. The statement for da, follows from part (c), the equation F A = F 301,00 + F B,+,a, and the bounds (51) for [EA]. I 70 7 .2 Pointwise bounds on SW(A§, 29;) Proposition 7.3 Let (A[, 29:) be defined as above, then there exists a constant C and an r0 2 1 such that for all r 2 r0 the inequality C ’ ’. ’. . < — ISM/( sz)I-13 — «‘7: holds for all x E X. Proof. We calculate the size of the contribution of each Of the three Seiberg-Witten equations separately. The only nontrivial part of the calculation is in the region of X which contains the support of dxym i.e. in BVO x [46,86]. We will tacitly use the results of lemma 7.2 in the calculations below. a) The Dirac equation To begin with, we calculate the expression D A((oI0 Q uo + 60) Q 7,) in two different ways. On one hand we have: DA((do ® 110 + C0) (8) 71) = 04(01 + X46,0C) = (1 ‘- X469)DA01 + dX469C On the other hand we get: DA((d0®uo+C0)®71)= (100) = 71 <8) DAMCO <8) U0 + Co) + ei-(do ‘31 ”0 ‘1‘ CO) 18’ Allhl) = 71 ‘3) 0123010 ® U0 '1' C0) + z(X469 — 1)0«171 Equating the results of the two calculations we obtain: [CY] ' [DA6(Q~'O®UO + 59)] = [718) 046(d0 ® 110 +,13~0)I S 71 S C(|<1||91|+IDAOI+|51)S SIC? Since over 8V0 x [46,86], |o| ——2 1 exponentially fast as r ——> 00 we obtain that ID.4g,(d0 <8) U0 + 30)] S (101) SIG b) The (1,1)-component of the curvature equation Again, we only calculate for x E dVo x [46,86]: ir (1.1) (1.1) F -FA0 —8 ~ ~ 1'7: .. A6 ([0012 — 1‘ [30]?)01 = X469(dao)(1’1)+ glfiolzw = X469 (dao)(l’1) + 27‘ 2 2 LI.) 8]O[2|X46’0l jIBI Both terms in the last line converge in norm exponentially fast to zero on (9V0 x [46,86] BET—+00. c) The (0, 2)-component of the curvature equation Similar to the calculation for the (1,1)-component of the curvature equation on (9V0 x [46,86], we have for the (0, 2)-component of the same equation: ir 4 lal2 0,2 0.2 7T:— ~ F116 )_ F110 LE 0B0 = X469 (d00)(0’2) — X469 5 I8 Once again, both terms on the right-hand side of the above equation converge in norm exponentially fast to zero as r converges to infinity. The proofs for the case of (A’1,29[) are similar and are left to the reader. 72 7.3 Surjectivity of L( A2414) and deforming (A;,¢f) to an exact solution The strategy employed here is very similar to the one used in section 5.4 and we only spell out part of the details. We start by showing that LMM’él is surjective, the case LUV; AM) is identical. We begin by asking ourselves when the equation L(A;,,wg,)§0 = 90 (102) has a solution {0 E L1’2(iA1 €19 WEO) for a given go 6 L2('iAO @1A2’+ EB VVEO). Define the analogues of the isomorphisms T,- from section 5.3 to be To : (CX(U U V1) ——+ F(E0; U U V1) given by T0(A,:I:) = A - 70(03) and T1 1C X V0 —* F(E1;V0) given by I‘d/V517): A ‘ 71(55) Let 7 E L2(z'/\O $279+ 69 W5; UUVl) be determined by the equation x2559 g0 = Tab) on U U V1 and < E L2(z'A0 EB iA2'+ EB W§;V0) be given by the equation Tf1(§) = (1 — X2559) go on V0. Thus we can write go as 90 = X10069 T0(7) + (1 — X46,O)T1—1(<) (103) This last form suggests that, in order to split equation (102) into two components involving L(A.w) and S, one should search for {0 in the form 50 = X10069 TC(77) + (1 — X46,O)T1—1(’{) (104) with 77 E Ll‘2(i1‘\1 <8 W5“; U U V1) and K E L1'2(i/\1 69 WE; V0). Using relations (103) 73 and (104) in (102) one obtains the analogue of equation (70): X1006.0T0(T("7) — TalTl—lpw X469, 5) - ’7’)+ +(1— X45,0)T1_1(]l/I(I‘£) + T1T0p(dX1005,0,7]) — C) = 0 (105) The operators T’ and M’ are defined over U U V1 and V0 respectively, through the relations 1104394,) To = TOT -1 —1 , We use these operators, defined only over portions of X, to define the operators T’ and 111’ defined on all of X by T, =(1 — X69)T + X695 1W, =X200691W + (1 — X2006,0)L(A,¢) Split equation (105) into the following two equations: T’(n) — TJITI1P(dX46,o, n) = 7 111,06) + T1 Torp(dX1005,o, 7]) = C (106) It is easy to see that solutions to the system of equations (106) provide solutions to (105) by multiplying the two lines with X1006,0T0 and (1 — X45,0Tf1) respectively and adding them. The following lemma is the analogue of lemma 3.6, its proof is identical to that of lemma 3.6 and will be skipped here. 74 Lemma 7.4 For every 6 > 0 there exists on re 2 1 such that for r 2 7“6 the following hold: ”(111' - Low») IE||2 S€||17||2 ||(T' - S)yllz Sells/H2 Here :1: E L1’2(z'/\1 EB l/Vg) and y E L1*2(z'./\1 EB W0+). The lemma allows us to replace the system (106) by the system 507) — TalT;1P(dx4.s,o, 6) = 7 L(A,d))(’{-) + T1T0’P(d)(10069,77) = C (107) The process of solving (107) is now step by step the analogue of solving (74). In particular, we solve the first of the two equations in (107) for 77 in terms of K: n = no) = S‘1(T61T;1P(dx4.s,o, n) + 7) Use this in the second equation of (107) and rewrite it as K = 1133),)“ - T1Tor/3(dX10069777(6)» To solve this last equation is the same as to find a fixed point of the map Y : L2(2'/\l {9 WE) —> L2(2'A1 69 WE) (the analogue of the map described by (78)) given by: Y(K) = Law) (C — T1 T0P(dX1006,0, 77(5)» The proof of the existence of a unique fixed point of Y follows from a word by word analogue of the proof of theorem 5.7 together with the discussion preceding the theorem. 75 With the surjectivity of L( A2611) proved, the process of deforming (141,161.) to an honest solution (Aura) is accomplished by the same method as used in section 5.4 and will be skipped here. To finish the proof theorem 7.1, we need to show that (Ami/90) ' (141,131) = (14,110) This follows from the fact that as 7‘ —> 00, the distance dist((A,-,1l'.-), (A1¢'1)) converges to zero, together with the following relations which follow directly from the definitions: d0®071=a d0®l§1+d1®fio=fi 3®A1=A 76 REFERENCES [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] T. Aubin, Nonlinear Analysis on Manifolds. Mange-Ampere Equations, Springer Verlag, New York, 1982. [3] H. B. Lawson, Jr., M-L. Michelson, Spin Geometry, Princeton Math. Series Vol. 39, Princeton University Press, Princeton NJ, 1989. [4] S. Donaldson, I. Smith, http://axrr.lanl.gov/abs/math.5670012067 [5] D. McDuff, The local behavior of holomorphic curves in almost complex: 4-manifolds, J. Diff. Geom. 34, (1991), 143-164 [6] D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1995. [7] J. W. 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