.u it! '0. m." 1-. -hdwfiwwmwfir. 3.... .35 a: ? a . .7 i JR...» .1: ‘ . f. I an . .. .v . ‘ 33.1.2.9? .3. 2.. rd‘ , 2 x. :.l. 2. .7 A. v, 1., . a 1.: £33. i... ‘ ...::I: h’!!! :( SI 4: :5: :Vs... 3.. f. :3 .55.). ... ft. cl. )bxz‘v.‘ :T..(.Kiiva: r I‘ll. a}, 1“}... 13.11.333.31 Ii Arsqsav ll... .‘anl 151»! . tit. 23:. g >2 7.! ~33 3;: :..f-., -2 ,... . 41.01: ... : wt. “3 p , .3 . 4 . 5...??? Egg}. ... "Eng .. ‘ : \axs ”i-.. HI-IIIIIII THESIS 3 aOOl LIBRARY Michlgan Stat-3 University "W' ' This is to certify that the dissertation entitled Family Gromov-Witten Invariants for Kahler Surfaces presented by Junho Lee has been accepted towards fulfillment of the requirements for Ph 0 D 0 degree in mathematics H Major professor Date 14 2001 MS U i: an Affirmative Action/Equal Opportunity Institution 0- 1 2771 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:/ClFIC/DateDue.p65-o15 Family Gromov—Witten Invariants for Kahler Surfaces By J unho Lee AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2001 Professor Thomas H. Parker ABSTRACT Family Gromov—Witten Invariants for Kahler Surfaces By J unho Lee The usual Gromov-Witten invariants are zero for Kahler surfaces with pg 2 1. In this paper we use analytic methods to define Family Gromov-Witten Invariants for Kahler surfaces. We prove that these are well-defined invariants of the deformation class of the Kahler structure and develop methods for computing them, including a version of the TRR formula and the symplectic sum formula. Finally, we explicitly compute some of these family GW invariants for elliptic surfaces. To my parents. iii ACKNOWLEDGMENTS I would like to express my sincere gratitude and thanks to my advisor, Professor Thomas H. Parker for his constant and patient help, encouragement, and excellent advice. I would also like to thank Professors Ronald Fintushell, John D. McCarthy, Ivanov Nikolai, and Akbulut Selman for their time and valuable suggestions. iv TABLE OF CONTENTS Introduction 1 (J, a)-holomorphic maps 2 Curves and Canonical Families of (J, 0) Maps 3 Family GW—Invariants 4 Kahler surfaces with pg _>_ 1 5 Virtual Moduli cycles 6 The Invariants of E(n) — Outline 7 The Tapological Recursion Relation (TRR) 8 Ruan-Tian Invariants of E(n) 9 Degeneration of E(n) 10 Relative Invariants of E(n) 11 Gluing Theorem 11 Appendix — Relations with the Behrend-Fantechi Approach BIBLIOGRAPHY 13 17 24 30 49 52 55 71 77 82 93 97 Introduction Gromov-Witten invariants are counts of holomorphic curves in a symplectic manifold X. To define them using the analytic approach one chooses an almost complex structure J compatible with the symplectic structure and considers the set of maps f : 2 —> X from Riemann surfaces 2 which satisfy the (nonlinear elliptic) J -holomorphic map equation 51 f = 0. (0-1) After compactifying the moduli space of such maps, one imposes constraints, requir- ing, for example, that the image of the map passes through specified points. With the right number of constraints and a generic J, the number of such maps is finite. That number is a GW invariant; it depends only on the symplectic structure of X. There are some beautiful conjectures about what the counts of holomorphic curves on Kahler surfaces ought to be ([V],[KP],[YZ],[G]). However, as currently defined, the corresponding GW invariants of Kahler surfaces with pg 2 l are all zero! This discrepancy occurs because GW invariants count curves for generic almost complex structures J, whereas Kiihler structures are very special — Donaldson details this in [D]. They can have whole families of curves which disappear when the Kahler J is perturbed to a generic J. For example, a generic K3 surface (pg = 1) has no holomorphic curves at all, whereas algebraic K3 surfaces do admit holomorphic curves . Clearly a new version of the invariants is needed — one which counts the relevant holomorphic curves. Work in that direction is just beginning. Bryan and Leung ([BLl],[BL2]) defined such invariants for K3 and abelian surfaces by using the Twistor family; they were also able to calculate their invariants in important cases. In a preprint to appear shortly, Behrend-Fantechi [BF] have define invariants for a more general class of algebraic surfaces using algebraic geometry, but have not yet made calculations. We approach the same issues using the geometric analysis approach to GW invariants. Given a Kahler manifold (X, w, J, g) we constructs a 2pg-dimensional family of elements K J( f, a) in flo’l(f*TX), where a is a real part of a holomorphic 2 form. We then modifies the J -holomorphic map equation (1) by considering the pairs (f, a) satisfying EJf = KJ(f: a). (0-2) The solutions of this equation form a moduli space whose dimension is 2199 larger than the dimension of the usual GW moduli space. Because a range over a vector space compactness is an issue. Here things get interesting because there are instances when the moduli space for (0.2) is not compact. In fact, when the map represents a component of a canonical divisor the moduli space is never compact. Nevertheless, there is a simple analytic criterion — the uniform boundedness of the energy of the map and the L2 norm of a — that ensures that the moduli space is compact. Theorem 0.0.1 Let (X, J) be a Kahler surface and fit a genus g and a class A E H2(X, Z). Denote by C(J) the supremum of E( f)+||a|| L2 over all (J, cud-holomorphic maps from genus g curves into X which represent A. If C (J) is finite, then the family G W invariants GWg{£‘(x, A) are well-defined. They are invariant under deformations {Jt} of the Ka'hler structure with C(Jt) bounded. Furthermore, if A is a (1,1) class then all the maps which contribute to these invariants are in fact J -holomorphic. The last sentence of Theorem 0.0.1 means that the invariants for (1, 1) classes are counts of holomorphic curves in (X, J). That is not the same as saying the invariants are enumerative, since there is no claim that each curve is counted with multiplicity one. But it does mean that the family GW invariants, which a priori are counts of maps which are holomorphic with respect to families of almost complex structures on X, are in fact calculable from the complex geometry of (X, J) alone. Theorem 0.0.1 yields well-defined family GW invariants provided there is a finite energy bound C'( J). Following the Kodaira classification of surfaces, we verify the energy bound case-by-case using geometric arguments. That yields the following cases where the family GW invariants are well-defined. Proposition 0.0.2 The moduli space for a class A is compact, and hence the family G W invariants are well-defined, when (X, J) is (a) a K3 or abelian surface with A sé 0, (b) a minimal elliptic surface it : E -—+ C with Kodaira dimension 1 with —A- (fiber class) 7e deg rr.(A) , and (c) a minimal surface of general type and A is in a certain subspace of the ( 1,1 ) classes (see Proposition 4.0.26). The second half of this paper develops computational methods. We extend sev- eral existing techniques for calculating GW invariants to the family GW invariants. 3 In particular, the ‘TRR formula’ applies to the family invariants, and at least some special cases of the symplectic sum formula [1P3] apply, with appropriate minor mod- ifications to the formula. Those formulas enable us to enumerate the curves in the elliptic surfaces E(n) for the class A: section plus multiples of the fiber. Theorem 0.0.3 Let E(n) be a standard elliptic surface with a section 3 of self- intersection —n. Denote by S and F the homology class of the section and the fiber. Then the g = 0 family G W invariants for the classes A = S + dF are well-defined and are given by the generating function 1 127: 2 GW,’,{‘; (E(n), s + dF) td = H (1 _ td) . (0.3) «120 (120 Bryan and Leung used algebraic methods to show (0.3) for K3 surfaces (i.e. n = 2) [BL 1]. This provided a verification of the well-known Yau-Zaslow Conjecture [YZ] for those cases when the homology class A is primitive. On the other hand, the above formula for n = 1 gives the ordinary GW—invariants of rational elliptic surface E (l), which was shown by Ionel and Parker [1P3]. They related TRR formula and their sum formula for the relative invariants to obtain a quasi-modular form as in (0.3). We follow the same argument — relating TRR formula and sum formula — to show Theorem 0.0.3. Chapter 1 gives the definition of a (J, a)-holomorphic map and some of the analytic consequences of that defintion, most notably an expression for the energy in terms of pullbacks of the symplectic form and the form a. Chapter 2 begins by describing the relation between a complete linear system IC' | — or more generally a Severi variety — and the moduli space of (J, a)-holomorphic maps. That leads us to consider the family of (J, a)-holomorphic maps in which a is the real part of holomorphic 2-form; 4 the corresponding family moduli space should be an analytic version of the Severi variety. As partial justification of that view, we prove the last statement of Theorem 0.0.1: any (J, a)-holomorphic map which represents a (1,1) class is in fact holomorphic (theorem 2.0.12). Chapter 3 summarizes the analytic results which lead to the definition of the family GW-invariants. That involves constructing the virtual moduli cycle by adapting the method of Li and Tian [LT]. Thus defined, the family invariants satisfy a Divisor Axiom and a Composition Law analogous to those of ordinary GW-invariants. To keep the exposition flowing the main results are stated in Chapter 3 and their technical proofs are deferred until Chapter 5. Chapter 4 contains examples of Kahler sufaces with pg 2 1 with well-defined family invariants. We focus on minimal surfaces and establish the results summarized in Proposition 0.0.2 above. For the case of K3 and Abelian surfaces we prove that our family GW-invariants agree with the invariants defined by Bryan and Leung. That is done in the course of the proof of Theorem 4.0.23 by relating the holomorphic 2—forms to the twistor family. Chapter 5 contains the analysis which proves that the family GW invariants are well-defined. Slightly modifying the arguments of Li and Tian, we consider the prod- uct of the space of C‘ stable maps and the parameter space ’H for a. The (J, a)- holomorphic map equation defines a section of a generalized bundle E —> B whose zero set is the moduli space of (J, a)-holomorphic maps. In general that space is nei- ther smooth nor compact. For the ordinary GW-invariants Li and Tian showed that after perturbing its zero set becomes smooth and compact and defines a virtual moduli cycle. Our case requires more care because the parameter space ’H is not compact. The construction consists of two main steps. First, using the Fredholm prOperty of the section of E —i B, one can construct a collection of finite dimensional 5 subbundles E,- —+ U,- = ‘1(E,) whose union contains the moduli space and has each restriction ,- = (Fla.- being smooth. Compactness of moduli space, which follows from the energy bound 0' (J) of Theorem 0.0.1, ensures that there is such a finite collection. Second, one can perturb the moduli space locally in each U.- in such a way that the local perturbations fit together to produce a well-defined cycle, the virtual moduli cycle. This second part is very general procedure and is proved in Theorem 1.2 of [LT]. Thus the bulk of Chapter 5 is devoted to working through the first step. The arguments parallel the proofs for J-holomorphic maps in [LT]. The key step is es- tablishing uniform estimates for the linearization of holomorphic map equation and its adjoint operator. Those estimates are still true for (J, a)—holomorphic maps and are locally uniform in a. The exposition ends up being rather lengthy because of the need to recall the extensive notation of [LT] and because we have taken the trouble of filling in missing details and some fixing minor errors in [LT]. At the end of Chapter 5 we prove the two properties of family GW-invariants: the Divisor Axiom and the Composition Law. Turning to the computations, we give an overview of the proof of Theorem 0.0.3 in Chapter 6. This argument is an extension of the elegant argument used by Ionel and Parker to compute the GW—invariants of E (1) [1P3]. It involves computing the generating function for the invariants in two ways, first using the so-called TRR formula, and second using a syplectic sum formula as in [1P3]. Roughly, the only modification needed is a shift in the dimension counts. But to justify the computation we need to extend both the TR formula and the symplectic sum formula to apply to the family GW invariants. The extended TRR formula is proved in Chapter 7 and sum formula is established in the last four Chapters. Chapter 8 gives an alternative definition of the family invariants for E (n) based on the idea of perturbing the (J, a)-holomorphic map equations as in [RTl] and [RT2]. This alternative definition is better suited to adapt the analytic arguments in [1P2] and [1P3] to a family version of sum formula. The proof of the sum formula begins by studying holomorphic maps into a degeneration of E (n) Because E (n) is a Kahler surface we are able to degenerate within a holomorphic family, rather than the symplectic family used in [1P3]. The degeration family Z is constructed in Chapter 9. It is a family A : Z —> D2 whose fiber Z; at A 75 0 is a copy of E(n) and whose center fiber is a union of E(n) with T2 x 5'2 along a fixed elliptic fiber V. As A —+ 0 maps into Z; converge to maps into Zo, and by bumping a to zero along the fiber V we can ensure that the limits satisfy a simple matching condition along V (there is a single matching condition for the classes A that we consider). Conversely, if a map into 20 satisfies the matching condition then it can be smoothed to produce a map into Z; for small A. That smoothing is described in Chapter 11 and then used prove the required sum formula for the family invariants of E(n). The appendix contains a brief discussion of how the family GW invariants defined here relate to those defined by Behrend and Fantachi in [BF]. CHAPTER 1 (J, oz)-holomorphic maps A J-holomorphic map into an almost complex manifold (X, J) is a map f : 2 —-> X from a complex curve 2 (a closed Riemann surface with complex structure j) whose differential is complex linear. Equivalently, f is a solution of the J -holomorphic map equation DJf = 0 where 51f = $(df + dej). In this Chapter we will show that when X is four-dimensional there is natural infinite- dimensional family of almost complex structures parameterized the J -anti-invariant 2-forms on X. Let (X, J) be a 4~dimensional almost Kiihler manifold with the hermitian triple (w, J, 9). Using J, we can decompose a E 92(X) as a = a+ + a- where a(u, v) — a(Ju, Jv) 2 a(u, v) + a(Ju, Jv) 2 a+ (u, v) = a- (u, v) = (0.1) Definition 1.0.4 A 2—form a is called J -anti-invariant if a = a_. Denote the set of all J -anti-invariant 2-forms by 03(X). Each a E 93(X) defines an endomorphism Ka of TX by the equation (u, Kev) = a(u, v). (0.2) It follows that (Kau,v) = —(u,Kav), JKO, = -KaJ, and (Ju,Kau) = 0. (0.3) Definition 1.0.5 For a E 93(X), a map f : E —> X is called (J, a )-holomorphic if 51f = KJ(f, a) - (0.4) where KJ( f, a) = Ka(6 f 0 j) = grew — dej)j. The next proposition and its corollary list some pointwise relations involving the quantities that appear in the (J, a)-holomorphic equation. We state these first for general 0'1 maps, then specialize to (J, a)-holomorphic maps. Proposition 1.0.6 Fix a metric within the conformal class j and let dv be the asso- ciated volume form. Then for any 0'1 map f we have the pointwise equalities (a) |5Jf|2 dv =%Idf|2 dv—rw, (b) «if, KJ(f.a)> dv = so, (c) K: = -lal21. (d) mm)? dv = lalz (gw dv + m) . Proof. Fix a point p E E and an orthonormal basis {e1, e2 = gel} of T923. Setting v1 = df(e1) and v2 = df(e2), we have 251f(e1) = v1+Jv2 and 2KJ(f,a)(e1) = Ka’Ug— JKa’Ul, and similarly 25]f(€2) = ’02 — J’Ul and 2K](f,(1)(62) = — avl — JKavg. Therefore, 4|51f|2 = In + Jvzl2 + I'vz — Jvll2 = 2(I'vll2 + I’v2l2) + 4(v1, Jvz) = 2W -— 4rw(e.,e.). That gives (a), and (b) follows from the similar computation 4(5Jf,K(f, (1)) = (v1 + ng, Kavg — JKav1)+ (v2 — Jv1,—Kav1 — JKaU2> = (v1,Kav2) — (v1,JKav1) + (Jv2,Kav2) — (Jv2,JKav1) -<’U2, Kavl) — (v2, JKa’U2) + (Jul, Kav1)+ (Jv1,JKavg) = 4(v1,Kav2) = 4f*a(€1,82)' Next fix a: E X and an orthonormal basis {w1,w2,w3,w4} of T;X with w2 = —Jw1 and w4 = —Jw3. Then the six forms wlsziw3Aw4, wlAw3iw2Aw4, w1 Aw4iw2Aw3 give an orthonormal basis of A2 (TgX), and two of these span the subspace of J anti-invariant forms. Hence a=a(w1/\w3—w2/\w4)+b(w1/\w4+w2/\w3) for some a and b, and in this basis K0, is the matrix (OOafl 0 0b—a —a—b00 t—baooj Consequently, K 3 = —-(a2+b2)I = —|a|21. Lastly, since K0, is skew-adjoint, (c) shows that |KJ(f,a)|2 = -(3f°j. K§(0f°j)) = |a|2|5f|2- Equation ((1) then follows from (a) because Ide2 = |a f |2 + I51f|2. [3 Corollary 1.0.7 Suppose the map f : E —r X is (J, a)-holomorphic. Then 10 (a) [51f]2 d’” = f‘a, (b) (1 - lalz) lalfl2 dv = 2(1+ |a|2)f*w. and (C) IOII2 ldfl2 = (1+ |a|2)|51f|2- PI'OOf- Since f is (J,a)'h010m0rP1fiC, ngfl2 = (ngIKJUfiI) = IKJU, 01W, 30 (a) follows from Proposition 1.0.6b while (b) and (c) follow from Proposition 1.0.6 (a) and (d). C] There is a second way of writing the (J, a)—holomorphic equation (0.4). For each a E 93(X), I + J K0, is invertible since J K, is skew-adjoint. Hence Jo, = (I + JKa)‘1J(I + JKO) (0.5) is an almost complex structure. A map f : E —r X is (J, a)-holomorphic if and only if f is Ja-holomorphic, i.e. satisfies _ 1 . air = E(df+Jade) = o. (0.6) From this perspective, a solution of the (J, a)-holomorphic equation is a Jo, holo- morphic map with Jo, lying in the family (0.5) parameterized by a E 93(X). In particular, we see from (0.6) that the (J, a)-holomorphic equation is elliptic. Proposition 1.0.8 For any a E 0i (X), the almost complex: structure JO, on X satisfies _l—Iozl2 2 = and Ja——1+|a|2 __1+la|2 K. (0.7) Proof. From (0.3), the endomorphisms A+ = I + JKO, and A- = I -— JKO, are transposes, and A+J = JA- and A+Ka = KaA_. Consequently, A;1 and A21 are 11 transposes, with AZIJ = JA;l and AIIKa = KaA;1 and therefore AIIA+ = A+A:1. Consequently, (Jau, Jav) = (A;1JA+u,A;1JA+v) = (JA:1A+u, JA:1A+v) = (A:1A+u,A:1A+v) = (u, A-A;1A:1A+v) = (u,v). On the other hand, noting that K 2 = —|a|21, it is easy to verify that 1 l Ka"1=——— —— ([+J ) 1+|a|21 1+[al2 J K a. (0.8) With that, the second part of (0.7) follows from the definition of Ja. D In summary, (J, a)-holomorphic maps can be regarded as solutions of the Ja- holomorphic map equation 5.1.. f = 0 for a family of almost complex structures para- meterized by a as in (0.6). We will frequently move between these two vieWpoints. 12 CHAPTER 2 Curves and Canonical Families of (J, a) Maps Given a Kiihler surface X, we would like to use (J, a)-holomorphic curves to solve the following problem in enumerative geometry: Enumerative Problem Give a (1, 1) homology class A, count the curves in X that represent A, have a specified genus g, and pass through the appropriate number of generic points. We begin this Chapter with some dimension counts which show that in order to interpret this problem in terms of holomorphic maps we need to consider families of maps of dimension pg. We then show that there is a very natural family of (J, a)- holomorphic maps with exactly that many parameters. We conclude the Chapter with a theorem showing that such maps do indeed represent holomorphic curves in X. One can phrase the above enumerative problem in terms of the Severi variety 19(0) C [C], which is defined to be the closure of the set of all curves with geo- metric genus 9. Assuming that C — K is ample, ‘it follows from the Riemann-Roch theorem that the dimension of the complete linear system |C | is given in terms of 13 pg = dimcH0'2(X) and q = dimcH0'1(X) by . 02—0-K dlmchl = ——,— + p. — q and the formal dimension of the Severi variety is dichg(C) = —K~C + 9—1 + pg — q. (0.1) The right-hand side of (0.1) is the ‘appropriate number’ of point constraints to impose; the set of curves in V9(C) through that many generic points should be finite, making the enumerative problem well-defined. Now let Mg(X, A) be the moduli space of holomorphic maps from Riemann sur- faces of genus g, which represent homology class A. Then its virtual dimension is given by dim cMg(X,A) = —K-A + 9—1. (0.2) The image of a map in MAX, [0]) might be not a divisor in ICI, instead it is a divisor in some other complete linear system IC’ I with [C’] = [C]. As in [BLl], we define the parameterized Severi variety Vgucv = H mm [C']=ICI Its expected dimension is now given by dichg([C]) = -K - C + g —l + pg. (0.3) We still have pg dimensional difference between (0.3) and (0.2). Therefore, the cut- down moduli space by (0.3) many point constraints is empty when pg 2 1. This implies that the corresponding Gromov-Witten invariants is zero, whenever pg 2 1. We show that there is a natural — in fact obvious — pg-dimensional family of (J, a)-holomorphic maps associated with every Kahler surface. 14 Definition 2.0.9 Given a Ka'hler surface on X, define the parameter space H by H = {a+a [a e H2'°(X)} (0.4) Here H2’°(X) means the set of holomorphic (2, 0) forms on X. Note that all forms a E H 2"’(X ) are closed since do: = 6a + 5a = Ba is a (3,0) form and hence vanishes because X is a complex surface. Thus H C 93(X) is a 2pg-dimensional real vector space of closed forms. We give it the (real) inner product defined by the L2 inner product of forms: (a, s) = Law. (0.5) We can use the forms a E H to parameterize the right-hand side of the (J, a)- holomorphic map equation (1.0.5). Definition 2.0.10 Henceforth the term ‘(J,a)-holomorphic map’ means a map sat- isfying (1.0. 5) for or in the above family H. Lemma 2.0.11 The zero divisor Z (a) of each nonzero a E H represents the canon- ical class. Proof. Write a = 16+? with E E H 2")(X ) Since fl is a section of the canonical bun- dle, this means that Z (a) = Z (fl) represents the canonical divisor with appropriate multiplicities. CI Next, using this 2pg dimensional parameter space H, we define the family moduli space We, [01) = {(f.a) IEJJ = 0. [Imfl = [01, and a e H} 15 Since we just parameterize the D-operator by 2pg dimensional parameter space, the formal dimension of the family moduli space is given by (Formal) dime-M-zi(X,[C]) = -K'C' + 9—1 + Pg On the other hand, we define a component of the canonical class to be a homology class of a component of some canonical divisor. Theorem 2.0.12 If f is a (J, a)-holomorphic map which represents a class A E H1’1(X). Then f is, in fact, holomorphic. Furthermore, if A is not a linear combi- nation of components of the canonical class, then a = 0. Proof. Since a E H2'°(X) EB H2'°(X) is closed and A E H1’1(X), it follows from Corollary 1.0.7a that / IEJfli = a(A) = 0. 2: Thus f is holomorphic, that is, 51f E 0. Consequently, |oz|2|df|2 E 0 by Corol- lary 1.0.7c. Since df has at most finitely many zeros, we can conclude that a = 0 along the image of f. Hence a = 0, otherwise it contradicts to the assumption on A by Lemma 2.0.11. C] 16 CHAPTER 3 Family GW—Invariants Let X be a complex surface with a Kahler structure (w, J, g). In this Chapter we will define the Family Gromov-Witten Invariants associated to (X, J) and the parameter space H of (0.4). We also state some properties of these invariants. To keep the discussion clear we defer the proofs and some technical definitions until later Chapters. Our approach is to extend the analytic arguments of Li and Tian [LT] to show that the moduli space of (J, a)-holomorphic maps carries a virtual fundamental class whenever it is compact. While compactness is automatic for the usual Gromov- Witten invariants, it must be verified caseby-case for the family GW invariants (see Example 3.5 below). Thus compactness appears as a hypothesis in the results of this Chapter. First, we recall the notion of C‘ stable maps as defined in [LT]. Fix an integer l 2 0 and consider pairs (f; 2,221, - - - ,zrk) consisting of 1. a connected nodal curve 2 = U 2,- of arithmetic genus g with distinct smooth i=1 marked points 221, - ~ - ,atk, and 2. a continuous map f : )3 —» X so that each restriction f,- = fl»,- lifts to a Cl-map from the normalization E. of E into X. 17 Definition 3.0.13 A stable 0’ map of genus g with k marked points is a pair (f; 23,331, - ~ - ,zk) as above which satisfies the stability condition: a If the homology class [f;] 6 H2(X,Q) is trivial, then the number of marked points in 2.- plus the arithmetic genus of E.- is at least three. Two stable maps (f,E;a:1,--- ,xk) and (f’,E’;:r’1,~- ,r],) are equivalent if there is a biholomorphic map a : )3 H 2’ such that o(a:,-) = a): for l S i S k and f’ = foo. We denote by f 9,1.(X, A) the space of all equivalence classes [f ; 2, $1, - .. , 2:1,] of C'-stable maps of genus g with k marked points and with total homology class A. The topology of 7-"; k(X, A) is defined by sequential convergence as in Chapter 2 of [LT]. There are two continuous maps from 75‘. First, there is an evaluation map ev: ”(X, A) —> ch (0.1) which records the images of the marked points. Second, for 29 + k 2 3, collapsing the unstable components of the domain gives a stabilization map t _.,.’1(0) is the moduli space of (J, a)-holomorphic maps. The following is a family version of Proposition 2.2 in [LT]. Proposition 3.0.14 Suppose that the set ‘1(0) is compact. Then the section gives rise to a generalized Fredholm orbifold bundle with a natural orientation and with index r = 2c1(X)[A] + 2(g — l) + 2k + dim H. (0.4) We will postpone both the proof of Proposition 3.0.14, and the definitions of the terms in its statement until Chapter 5. Until then we will accept it, and continue following the construction of Li-Tian. By Theorem 1.2 of [LT], the bundle E has a rational homolog “Euler class” in “(X , A) x H; in fact, since H is contractible this Euler class lies in H..(7¥g,k(X,A);Q) where r is the index (0.4). We call this class the virtual fun- damental cycle of the moduli space of family holomorphic maps parameterized by H and denote it by [Mitts A)]"". (0.5) In particular, dim [M;;f(X, Ans = 2c1(X)[A] + 2(g -— 1) + 2k + 2199. (0.6) 19 The next issue is whether the virtual ftmdamental cycle is independent of the Kahler structure on X. The next proposition is analogous to the Proposition 2.3 in [LT]. It shows that the virtual fundamental cycle depends only on certain deformation class of the Kahler structure. Proposition 3.0.15 Let (wt,Jt,gt), 0 S t S 1, be a continuous family of Ka'hler structures on X. Let H, be the corresponding continuous family of finite sub- spaces defined by (0.4) and let in be the corresponding family of sections of Et over it 9 ,k(XaA) X Ht- If;'1(0) is compact for all 0 S t S 1, then warm, AW" = were, AM”. We also postpone the proof of Proposition 3.0.15 to Chapter 5. The family GW invariants can now be defined by pulling back cohomology classes by the evaluation and stabilization maps and integrating over the virtual fundamental cycle. That of course requires that the virtual fundamental cycle exists, so we must assume that we are in a situation where <1),— 1(0) is compact. Definition 3.0.16 Whenever the virtual fundamental cycle [./\/l;’;‘(X,A)]Vir exists, we define the family G W invariants of (X, J) to be the map GW.’;Z*(X. A) : [H*< WOW—gm) H 0 defined on 011, - -- ,ak E H*(X;Q) and fl 6 H*(X/lg,k;Q) by GWMX. Axe; a1, - ~ at) = [M;:Z:(X.A)1V“ n (sew) u ev*(«;a1 A - - . A «ram. We will use the shorter notation GW9{£1(XIA)(01,°“ fire) for the special case when ,6 = 1 E H°(Wl_g,k); this corresponds to imposing no con- straints on the complex structure of the domain. 20 The condition that ’1(0) is compact must be checked “by hand”. In general, ‘1(0) is compact for some choices of A, but not for others. Example 0.17 Let (X, J) be a Kiihler surface with pg > 1. Then there is a non-zero element fl 6 H 2'0 whose zero set Z ([3) is non-empty, represents the canonical class K, and whose irreducible components can be parameterized by holomorphic maps. Fix a parameterization f : E —> X of one such component; this represents a non-zero class A E H2(X, Z). Then a = [3 + 3 lies in the space H of (2.0.9) and (f, Aa) = O for all real A. Thus on any Kahler surface with pg > 1, the set <1>"1(0) is not compact for an component of the canonical class A. On the other hand, in the next Chapter we will give examples of classes A in Kahler surfaces with pg > 1 for which '1(0) is compact. Theorem 3.0.18 If there is a constant C, depending only on (X ,w, J, 9) such that E( f) + “all < C for all (J, a)-holomorphic maps into (X, J), then (II-1(0) is compact and hence the family G W invariants are well-defined. Proof. Consider a sequence (fn, an) of Ja-holomorphic maps. The uniform bound on llanll implies that the Jo, lie in a compact family. Since E(fn) < C the proof of Gromov’s Compactness Theorem (see [PW] and [IS]) shows that {(fn, an)} has a convergent subsequence. Consequently, ‘1(0) is compact as in the hypothesis of Proposition 3.0.14. That means that the virtual flmdamental cycle (0.5) is well- defined. The family GW invariants are then given by Definition 3.0.16. C] We conclude this Chapter by listing two important properties of the family GW invariants. These are analogous to divisor axiom and composition laws of ordinary GW invariants. Again, the proofs appear in Chapter 5. Proposition 3.0.19 (Divisor Axiom) If oak E H2(X, Z) then GW;£1(X,A)(a1,-~ ,ak) = ak(A) GWg{}c7:1(X,A)(a1,-~ ,ak_1). (0.7) 21 The second pr0perty generalizes the composition law of ordinary Gromov-Witten invariants. For that we consider maps from a domain 2 with node p and relate them to maps whose domain is the normalization of E at p. When the node is separating the genus and the number of marked points decompose as g = g1+gg and k = k1 + k2 and is a natural map 0 : Hauler-+1 X Myth-f1 H M9,]; (0'8) defined by gluing (k1 + 1)-th marked point of the first component to the first marked point of the second component. We denote by PD(o) the Poincaré dual of the image of this map 0'. Given any decomposition A = A1 + A2, g = g1 + g2, and k = k1 + k2 let E1 69 E; be the generalized bundle over igl,k1+l(X, Al) x 7'-.<12.Ic2+1(X, A2) X H whose fiber over ([f1,)31; {x,-}], [f2, 22; {yj}],a) is Ag’l EB AO'1 . The formula 2, J. my... ‘I’thh 31; {ital}, lfz. 232; {ll/1}], 0‘) = (de + Jadfljzudfz + Jtadfzjm) (0-9) defines a section of E1 63 E5. On the other hand, for non-separating nodes there is another natural map 9 1 fig-IJC-ifl H mime (0-10) defined by gluing the last two marked points. We also write PD(0) for the Poincaré dual of the image of 6. The composition law is then the following two formulas. Proposition 3.0.20 (Composition Law) Let {H7} be any basis of H‘(X; Z) and {H7} be its dual basis and suppose that GWfifiX, A) is defined. (a) Given any decomposition of (A, g,lc), if the set II!,_1(0) is compact for all OStSl, then 22 Swift‘s. A)(PD(a); a1, - -- ,a.) = EmA=A1+A2 GWQI’Tzl+1(X’ AlXQh ' ° ' ’akl’ H7)GW92J=2+1(X7 A2)(H7r ak1+1a ' ' ' r all?) (b) GWg{£‘(Xr A)(PD(0); a1) ' ' ' talc) = 27 GWng-ik+2(X9 A)(ali ° ' ' talc, H1) H7) That completes our overview of the family GW invariants. We next look at some examples, namely the various types of minimal Kahler surfaces. There we can use the specific geometry of the space to verify that the moduli space is compact and hence the family GW invariants are well-defined. 23 CHAPTER 4 Kiihler surfaces with pg _>_ 1 In this Chapter we will focus on the family GW-invariants for minimal Kahler surfaces X with pg 2 1. The Enriques-Kodaira Classification [BPV] separates such surfaces into the following three types. 1. X is K3 or Abelian surface with canonical class K = 0. In this case, pg = 1. 2. X is an elliptic surface 1r : X —-—> C with Kodaira dimension 1. If the multiple fibers B,- have multiplicity m,, then a canonical divisor is K = rr‘D + 2(m, — 1)B, where deg D = 2g(C) — 2 + x(0x) (0.1) 3. X is a surface of general type with K 2 > 0. We will examine these cases one at a time. For each we will show that the family invariants GW;£‘(X, A) are well-defined. By Theorem 3.0.18 the key issue is bound- ing the energy E ( f ) and the pointwise norm |a| uniformly for all (J, a)-holomorphic maps into X. K3 and Abelian Surfaces Let (X, J) be a K3 or Abelian surface. Since the canonical class is trivial, Yau’s proof of the Calabi conjecture implies that (X, J) has a Kahler structure (w, J, g) whose 24 metric g is Ricci flat. For such a structure all holomorphic (0, 2) forms are parallel, and hence have pointwise constant norm (see [3]). Thus H ’5 (C consists of closed forms a with [a| constant. Furthermore, the structure is also hyperkahler, meaning that there is a three-dimensional space of Kahler structures which is isomorphic as an algebra to the imaginary quaternions. The unit two-sphere in that space is the so-called Twistor Family of complex structures. Consider the set ’13 = {Ja I a E H}. Since a has no zeros, equation (0.7) shows that Ja —+ -J uniformly as |a| —; 00. We can therefore compactify 76 to T ’_—‘i P1 by adding —J at infinity. Proposition 4.0.21 T is the Twistor Family induced from the hyperkdhler metric 9. Proof. Let a E H with |a| = 1. It then follows from Proposition 1.0.8 that Jo, = —K., and (a, Ja, g) is a Kiihler structure on X. On the other hand, we define a’ by a’ (u, v) = a(u, Jv). Then |a’| = l and a’ E H since fl’ is holomorphic for each holomorphic 2-form fl. Moreover, by definition we have Jar = —Kar = —JKO, = JJa. Since (a’, Jar, g) is also Kahler and J JaJa: = —I d, the Kahler structures {J, Ja, Jar} multiply as unit imaginary quaternions. It follows that ’I' is the Twistor Family induced from the hyperkhler metric g. D Lemma 4.0.22 Let A be a nontrivial homology class with w(A) 2 0. Then there exits a constant CA such that every (J, a)-hol0morphic map f : C —> X representing A with a E H satisfies E(f) = é/Zldflz < w(A) +CA and |a| g 1. 25 Proof. Since [al is a constant, we can integrate Corollary 1.0.7b to conclude that la] 5 1. Let CA be an upper bound for the function a H |a(A)| on the set of a E H with |a| S 1. Because a is closed, Proposition 1.0.6a and Corollary 1.0.7a imply that 1 E(f) = 5 / ldflz = / f*(w + a) = w(A) + a s w + C... a c )3 Theorem 4.0.23 Let (X, J) be a K3 or Abelian surface. For each non-trivial A E H2(X,Z), the invariants CW; E(X , A) are well-defined and independent of J. Furthermore, if A = m8 and A’ = mB’ where B and B’ are primitive with the same square, then ijmx, A) = GWg{£‘(X, A’). Proof. For any nontrivial homology class A, we can choose a Ricci fiat Ka'hler structure (w, J, g) such that w(A) 2 0 ( if w(A) < 0, then we choose (-w, —J, g) ). It then follows from Lemma 4.0.22 and Theorem 3.0.18 that GWg‘f,:’(X, A) is well- defined. Bryan and Leung have applied the machinery of Li and Tian to define family GW invariants associated to the Twistor Family T [BL1, BL2]. Their invariants, which we denote by ‘I’LAX, A): are actually independent of the Twistor Family since the moduli space of complex structures on X is connected. On the other hand, if A = m8 and A’ = mB’ where B and B’ are primitive with the same square, then there is an orientation preserving difieomorphism of X which sends the class B to the class B’. That implies that (PLAX, A) =3:k(X, A’). To complete the proof it suffices to show that Gngf‘(X, A) = eggx, A). (0.2) 26 For that, recall from Theorem 1.2 of [LT] that the moduli cycle is defined from a section 3 of a generalized Fredholrn orbifold bundle E —> B and is represented by a cycle that lies in an arbitrarily small neighborhood of s‘1(0). Both sides of (0.2) are defined in that way using the same hedholm bundle E over the space of Kahler structures. In the first case B is {J0 | a E H} and s‘1(0) is the set of of all (f, a) where f is a Ja-holomorphic map, and in the second case B = T is the Twistor Family and s‘1(0) is the set of Ja-holomorphic maps for JO, 6 T . By Proposition 4.0.21 {Ja | a E H} parameterizes‘the Twistor Family after adding a point at infinity to H. But since w(A) Z 0, Lemma 4.0.22 shows that |a| S l for all JO holomorphic maps representing the homology class A with a E H. Thus the moduli cycle is bounded away from the point at infinity, so the two definitions of the moduli cycle are exactly equal. That gives (0.2) C] Elliptic Surfaces First, we recall the well-known facts about minimal elliptic surfaces X with Kodaira dimension 1 [FM]. 1. X is elliptic in a unique way. 2. Every deformation equivalence is through elliptic surfaces. Therefore, there is a unique elliptic structure 7r : (X, J) ——> C. Moreover, for the fiber class F and any homology class A E H2(X; Z), the integer F - A + deg(rr.A) (0.3) is well-defined for each complex structure J and it is invariant under the deformation of complex structure J. Let (w, J, g) be a Kahler structure on X and H be as in (0.4). For a E H, let ”all denote the L2 norm as in (0.5). 27 Lemma 4.0.24 Let A E H2(X; Z) such that the integer ( 0. 3) is positive. Then, there exit uniform constants E0 and N such that for any Ja-holomorphic map f : 2 —> X, representing homology class A, with a E H, we have 153(f)=-;-/2|dfl2 s E... Hall 3 N. Proof. It follows from (0.1) and Lemma 2.0.11 that for any nonzero a E H, the zero set of a lies in the union of fibers F,. Let N (a) be a (non-empty) union of e~tubular neighborhoods of the E. Denote by S the unit sphere in H and set . . 2 m“) - 13.1? first.) '0" and N - m- We can always choose a smooth fiber F C X \ N (a) such that f is transversal to F. Let f‘1(F) = {p1, - - . ,pn} and for each i fix a small holomorphic disk D,- normal to F at f (p,) We can further assume that f is transversal to each D,- at f (p,) Define sgn(r) to be the sign of a real number r if r aé 0, and 0 if r = 0. Denote by I (S, f )p the local intersection number of the map f and a submanifold S H X at f(p). In terms of bases {e1, e2 = jel} of TIRE, {v1, v2 = jvl} of Tf(Pi)F’ and {v3, v4 = j v3} of Tfm.)D,- we have [(Fr f)Pi = sgn ((1)1 A '02 A ’03 A U4)(’Ul,’l}2, f*81:f*62)) = sgn ((1)3 A v4)(f*el) f¢e2))r I(D,~, flp. = sgn ((01 A ”2 A ”3 A v4)(f*€1,f:€2,1’3,v4)) = sgn ((01 A v2)(f.e1, 1382)) - Comparing with sgn f*w(e1,e2) = sgn((vl /\ v2)(f.el, fiez) + (v3 A v4)(f*€1,f*82)) shows that 1(Fr f)Pi + [(013 f)Pi = sgn (f*w)(81, 62). (0'4) Now suppose m(J)||a|| Z 2. Then |a| 2 2 along each F,-, so by (0.4) and Corol- lary 1.0.7b ZUU’Flm +I(f!Di)p,-) <0. I 28 This contradicts to our assumption A - f + deg(rr...A) > 0 since by definition 2,1(f,F)p, = A - f and 2,1(f,D,-)p, = deg(rr,.A). Therefore ”all < N with N as above. The energy bound follows exactly same arguments as in the proof of Lemma 4.0.22. El Proposition 4.0.25 For any homology class A with ( 0.3) positive, the invariants GW9{£{(X, A) are well-defined and depend only on the deformation class of (X, J). Proof. It follows from Lemma 4.0.24 and Theorem 3.0.18 that the invariants GWg{f’(X, A) are well-defined. On the other hand, (0.3) is invariant under the de formation of J. Therefore, applying Proposition 3.0.15, we can conclude that the invariants only depends on the deformation equivalence class of J. D Surfaces of General Type Let (X, J) be a surface of general type. Proposition 4.0.26 If A is of type (1,1 ) and is not a linear combination of compo- nents of the canonical class, then we can define the invariant GWg{£’(X , A). They are invariant under the deformations of complex structures which preserve (1,1)-type of A. Proof. Lemma 2.0.12 and Theorem 3.0.18 imply that the invariants GWJI‘AX, A) are well-defined under the assumption that A is type (1,1). On the other hand, Proposition 3.0.15 also implies that the invariants GWg’fk(X, A) are invariant under deformations of the complex structure which preserve the (1, 1) type of A. D 29 CHAPTER 5 Virtual Moduli cycles This Chapter is devoted to the technical proofs of the analysis results stated in Chap- ter 3. Specially, we will prove Proposition 3.0.14, 3.0.15, 3.0.19, and 3.0.20. We first recall the notion of orbifold bundle. Definition 5.0.27 A topological fibration 7r : E —+ B is an orbifold bundle if there is an open covering {U,-} of B such that (a) each U,- is of the form I7,/P,~, where F,- is a finite group acting on (7,- (b) for each i, there is a topological bundle E,- —-> (7,, such that Ely,- = Ei/I‘, (c) For any i, j, there is a bundle map (pi-l : Ejlxj—lwmuj) —> E’lwflwmuj) which is compatible with actions P,- and P,- and descends to the identity map of Elana,- , where rr;c : (I), --> U), is the natural projection (d) for each x E rr‘1(U,- flUj), there is a small neighborhood U3, such that (1).-flint, ) is an isomorphism fmm each connected component of 1r; 1(Um) onto its image Any such (1 7r,- : U,- —i U,- is called a local uniformization of B. We denote ¢ij by the induced map from rrJ-"1(U, fl Uj) to rr,‘1(U,- n Uj). An orbifold section 3 : B —r E is a continuous map such that for each i, sly, lifts to a section 3,- of E,- over U,. I 30 We now define a generalized Hedholm orbifold bundle following [LT]. Definition 5.0.28 An orbifold bundle E -> B is called a generalized Hedholm orbifold bundle of index r if there is an orbifold section 3 : B -—> E satisfying (a) s‘1(0) is compact with a finite covering {U,-} (W for 806,1 31' I (I; —> 133,-, there is a topological subbundle E0,- of finite rank over (7,- such that (i) 3:1(Em) C (J,- is smooth of dimension r + rk(Eo,-) (ii) Eml-ylwo.) is a smooth bundle over 3:1(Em) with Silsflwo.) smooth (c) for each i, there is a finite dimensional vector space F,-, on which F,- acts, and a Pi-equivariant bundle homomorphism w,- : (I, x F, —i E,, satisfying (i) 1,0,] “(is ) F : 3:1(Em) x F, —» E0, is smooth and transverse to 3,- along or X r 8:1(0) 0 0i (ii) if diij S dimF,, then there is an injective bundle homomorphism rr;1(U,~FI Uj) X Fj —> 77:1(Ui fl Uj) X R such that p,- o 0,-1- = (12,-,- o 15,-, where p’, : U,- x F,- —+ (J,- is a natural projection, and v, o 9,,- = <1», 0 v,- on 1r;1(U.- n U,) x F, (iii) dika S diij S dimF,-, then 0,; = 0,,- o 03-), over rr;1(U,- fl Uj) (iv) for any x E U,- D U], 0,, is I‘m-equivariant near r“1 (x), where I‘._., is the uniformization group of B at x For each i, (F,, E0.) is called a resolution of s,- : II,- —* Bi. Proof of Proposition 3.0.14 Following Chapter 3 in [LT], we will show that <1>: 71“, ,( (X A) )x H -» E 31 satisfies Definition 5.0.28b. Namely, we will show a family version of Proposition 3.4 in [LT]. Then Proposition 3.0.14 follows from the proof of Proposition 2.2 in [LT]. The proof of above assertion consists of 4 steps. First, we recall local uniformizations of PLAX, A) in [LT]. Then these give the local uniformizations of TEAX, A) x H in an obvious way. Second, we recall the definition of approximated maps and weighted norms as in [LT]. Next, we show the main estimates for the linearization of the Cauchy-Riemann operator 5].: and its adjoint operator. These are family versions of Lemma 3.9 and 3.10 in [LT]. Finally, we use the main estimates of the previous step and the Inverse Function Theorem to conclude our assertion. Step 1 In this step, we recall the local uniformizations of PLAX, A) in [LT]. In the following, we will denote by C a stable map (f, 2; x1, - -- ,xk). We fix l 2 2. Let [C] = [ f, 2; x1, - -- ,xk] 6 PLAX, A). A component of E is called a bubble component if it collapses to a point under the stabilization of 2. We add one or two marked points to each bubble component, to obtain a stable curve (2; x1, - -- ,xk, 21, - - - ,21) 6 HM“. Let W be a small neighborhood of (2; x1, - - - ,xk, 21, - - - , 2;) in WIN,“ and W be the uniformization of W, i.e. W = W/I‘, where F = Aut(2; {x,-}; {z,~}). Let L7 be the universal family of curves over W. We fix a metric h on LI. Define the distance of two maps f1 and f2 from fibers of LI over W as follows: dW (f1: f2) = suprDom(f1)Supdh(y,x)=dh (x,Dom(f2))dX (f1 (:11) ) f2 (3’)) Since the homology class of any non-stable component under the map f is non- trivial, there is at least one regular value of f on each unstable component. Therefore, we can assume that f ‘1( f (23)) consists of finitely many immersed points. We choose local hypersurfaces H1, - -- ,Hl such that H,- intersects Im( f ) transversally at f (2,). 32 Fix a small 6 > 0, and define Map5(W) ={ (f,i;{fé.-},{2j}) | (f3;{:t,-},{2,~}) 6 W, dw(f,f) < 6. f is C0 on E and C’ on Reg(E), and f(z",-) 6 Hj }. Let K be any compact subset in Ll\Sing(Ll) of the form : there exits a diffeomorphism 112K : (K n E) x W —» K such that wK((K n E) x {t}) lies in the fiber of LI over t = (E; {53,}, {i,-}) E W. Then we define MapAW. K) ={ (f, is; {a}, (is) e Map(W) l Ilfwx.....,,.,., -— usual < 6, where t = (E; {55,-}, {Z,-}) 6 W } By forgetting added marked points, each point in Map5(I/V, K) give rise to a stable map C and consequently, an equivalence class [C] E PLAX, A). Let pwx : Map6(W, K) —> —g,k(X, A) be such a projection map and let MaP5(Wo, K) = PW( Map6(W, K) )- Let Aut (C) be the automorphism group of the stable map C. It is a subgroup of I‘ = Aut(E; {x,}; {z,-}), so it is finite and acts on LI. Denote by m(C) its order. LFYom now on, K always denotes a compact set in L7 \ Sing(Ll) containing an open neighborhood of Uj f ‘1 (f (z,-)). Moreover, we may assume that K is invariant under the action of Aut (C) There is an action of Aut(C) on Map6(I/V, K) with Map5(Wo, K) as its quotient: For 7' E Aut(C) and C’ = (f’,E’; {x2}, {z;}) E Map5(W, K) we define 7*(0') = (f’T‘l, 7(2’); {7132)}, {7(Z§)})- Then Map5(Wo, K) = Map5(W, K)/Aut(C), see Lemma 3.1 [LT]. Let V C H be a small neighborhood of a. The topological bundle EIMamWo. over Map5(Wo, K) x V lifts to the bundle K)xv —-) Map6(VV, K) X V. (0.1) [Map5(W,K)XV 33 Infact,E—rfl 9 ,k(X, A) x H is a topological orbifold bundle with the local uni- formization PW,K X I I Map5(W, K) x V —> Map6(Wo,K) x V. Without further confusion, we simply denote the lifted bundle ElMap5(W,K)xv by E. On the other hand, the section Q defined as in (0.3) lifts to a section, still denoted by (I), of E over Map6(W, K) x V. Step 2 In this step, we recall the definition of approximated maps and weighted norms in [LT]. In the following, we assume (C,oz) = 0, i.e. f is Ja-holomorphic. Denote by q1,--- ,q, the nodes in E. For any q,- (1 S i S s), by shrinking W if necessary, we may choose coordinates w,1,w,-2, as well as t in W, near C, such that the fiber (2:; {Mil}. {zj(t)}) of LI over t is locally given by the equation wilwiZ = 62'“), [Wu] < 1, [we] < 1, where e,- is a C°°-function of t. For any y in 2;, if |w,1| > Lm or [wig] > L\/|—e,-(T)| for all i, where L is a large number, then there is a unique m(y) in E = 20 such that dh(y,rrt(y)) = dh(y, 2). Note that if y is not in the coordinate chart given by w,1,w,-2, then simply set wi1(y) = w,2(y) = 00. Introduce a complex structure Jo. = ju + Jo, on ll x X and let F : E —> Ll x X be the graph of f. Put p,- = F(q,-). We may assume 1. F({w,1w,-2 = 0 | |w,—1| < 1, |w,-2| < 1}) is contained in a coordinate chart (u1,---.,u2N) ofbl x X near p,-. 2. as.) = 6.3,, + 0(Iul) Jami“) = me, + 0(Iul) where Iul = ME,- MI”- 34 3. In complex coordinates Uj + V—IUN+j, F(w,-1,w,-2) = (w,1,w,-2,0,- ' ° ,0) + 0(lwiII2 + |w,-2|2). Then we can extend F to a neighborhood of q,- using the formula in 3. Let 5 be a cut-off function such that E(x) = 0 for Ix] S 1, E(x) = 1 for [x] > 2, and |dfi(x)| S 2. Definition 5.0.29 We define ft(y), where y 6 2,, as follows: f(rrt(y)) if either |w,-1(y)| > 1 or |w,-2(y)| > 1 for alli ft = «2(F(y)) if]w,-1(y)| < % and ]w,2(y)| < % for somei 91(9) if% S [7011(9)] S 1 07‘ l S [midi/ll S 1 where 9,-(y) = expf(q,) (.B( 2dh(ya (Ii) )eXPj—r(lq,)f(7rt(y)) + (1 — fi( 2d”(y’ q” ))eXp;(1qi)7r2(F(y))) Next, we will define weighted norms as in [LT]. Let r be the distance function to the singular set Sing(bl) with respect to the metric h on Ll. In the below, all norms and covariant derivatives over 23; are taken with respect to the induced metric his. . Definition 5.0.30 For any smooth section 5 E I‘°(Et, f,*TX), we define ”5“" = (fr ('5'? + 'Vi'p’d’“); + (is ”ELF—23mm) 5 Lina, em = { as was”) I IIrIII. < oo } L” = { (use I e e LWE. :TX), te W, H e v } where p Z 2 and P°(E¢, ffTX) is the space of continuous sections of f,*TX over 2,. If E; has more than one components, then 5 consists of continuous sections of components which have the some value at each node. 35 Definition 5.0.31 For n 6 Hom(2t, ft‘TX), we define 1 1 P _2(2_2) 2 ”’7”? = ([2 I’llpdflt) + (L 7" ” [film/Ix) LP (Arms) = { n e nemesis”) I 773” = -an. ”77”.: < oo } LP> = { (tfim) I n e If (A3:(f:TX)) , te W. A e v } Lemma 5.0.32 For any p > 2, there exits a uniform constant c such that for any t E W and B 6 V l ll‘P(ft.fl)llp S C( Ill2 + Ila - fill) Proof. It follows from Lemma 3.8 in [LT] that ||(ft,oz)||p S c |t|i. It also follows from Lemma 3.7 in [LT] that Idftl is uniformly bounded. On the other hand, we have (2—2) / r—2 ? dut S C(p). 2t Therefore, we can conclude that ll‘1>(ft.fi)||p S ||¢(ft.a)||p + ||(Js - Jaldftjllp S C(ltP + Ila - fill ). 0 Step 3 In this step, we will show main estimates. The linearization Lm of (I) at (ft, B) with respect to f, is an elliptic operator LN; : LINE“ fg'TX) —» LP(A3;31(f,*TX)) given by Lug“) = Vé + Jpvé jt + g (Vng) (dftjt + Jgdft) . (0.2) where j; is the complex structure on 2,. Its adjoint operator L22), with respect to the L2-inner product is given as follows: for any 17 E 93% {TX) LEM”) = -2V81(’71) - 2vez (772) + Bt,fi(’l) (O3) 36 where {e1,e2 = jtel} be an orthonormal basis on 2,, 171 = r)(e1),172 = 17(e2), and Bt,,3(r)) is defined by (if, Bt,s(77)) = ((Ver) faez, 771) - ((VeJA)fa.61, 772 )- For the proof of (0.2) and (0.3), see lemma 6.3 and 6.4 in [RTl]. The next lemma follows from Lemma 3.9 in [LT]. This shows the uniform elliptic estimates for L”. Lemma 5.0.33 For any fixed p 2 2, there is a uniform constant c such that for any t, fl, andfi E LL42), (TX), we have ||€||1,p S C(||L(I,s)€||p + |l€||1,2)- (04) Proof. It follows from Lemma 3.9 in [LT] that (0.4) holds with some constant c(fl) which might depend on lVJpl. However, by shrinking V, if necessary, we can choose cwithc>c(fl)foranyflEV. CI Fix a node q, of 2: and choose t e W with 6,-(t) 7e 0. Let w,-1 = pe‘/'—1”. Then wig = E¥Me¢—_I(90-0) and r2 = p2 + 15%;”: where 6,-(t) = |e,-(t)|e‘/"—19°. We define the neck region by N,., = 2,n{ (w,1,w,2,t)€ L? | r g l/k} and ,e = r—_(_Hp;2. (0.5) Denote by ht the induced metric on N“ and let h6 = r‘2 p2 ht. Then we have h6 = dp2 + p2 d02 and due = r‘2p2d/it = idpdfl Lemma 5.0.34 For any fixed p > 2, there exist kg and a uniform constant c, which is independent of t and B, such that if k > k0, then each 17 E (23’; (f,"TX) satisfies [N scavenge] cplLi,e(n)|2due+c/ «p(ln|’+lV°n|3)d0 (0.6) In! Ne: 0Nk,t 37 Proof. Using the metric hc on 2,, for any n 6 03;“ ,*TX), we can write it as r) = m dp — Jg’lh pd0. In terms of p and 6, we also have 617121 3771 2 2_ 2_ 2 d c 2 = 2 _ _ 2 . |n|= |n|= (;) lnl an Ian (lap! +|p Bel For fixed p > 2, we can choose k0 which satisfies 2 (39’ < 3 ko 10 For k > k0, we first show the following : / vlnl’dm S 4/ cplnl’d0+/ le°n|§dIue (0-7) Nlc,t 6N)“: Nle,t Let w(p) = fpchpdp, where p0 S p on N”. Then w S dpd9 k,t 6N,” S 2/ wlnl2d0+f wlml’dpdil + 2/ vpp29|V°n|§dpdfl 6N,” NM! Nk-t Using p S r S l/ko, we can rearrange the above to conclude (0.7). Next, we will show the following : C * T2 C / wIanidMeS e/ cp(lLe,e(n)|’+-3|m|’)due+cf e(IVnI2)d6 Nk,t Nu P 3N“ ' (0.8) Since [Vng and ldftl are uniformly bounded, r r IBr.,ra(77)| S Cldft lclm l S Czldft | lm | S Czlm l- Consequently, we have 1 t _ 6’71 l pa__’ll 38 This implies that 1 c a 16 a 16 -/ ‘PIVUIZd/Jc:/ (0.11) We have 9—771 1 6771 6171 1 6771 t/Nm’p [<(Jfio_ Jfl)_ 6p p 80>— < ap1(<]flo Jfl) P 60 d/Jac l —<- 4— «prCnIde, (0.12) Nk,t On the other hand, using integration by parts, we also have em 1 6771 0m l 6_171 , J J dc [11..’0<0p Spa—6» <fi°0p’ Pa_9 6771 > r 6171 go 17 a, J d6 — / <77 a, J 01> dp d9 AN,” < 1— 50— 06 N,” (p 1_ [30— 69 S C/ 90 ([771 - “[2 + lV‘nlg) d9 ON)“; 19-2 2 13171 2 —— — Jfi, ,. 0.13 + p ~.,.’0(Il 2|171 al +| pa_e’ (1,1 ( ) where a(p) = 2%. f 771(1), 0) do and Ip’ = —2;72 90% gal—If]; Now, apply Wirtinger’s 39 inequality [GT] and then use (0.9) to derive the following : 277-0’ dl‘cS/ _ dc [We I1 I N cplpagl It k,t g 1 + ) — + c L“ + c d c. ./Nk,t so (( 2(1) _ 2) Ip 0p I I t’fil [771' 'u (0.14) Combining (0.10), (0.11), (0.12) (0.13), and (0.14), we can deduce (0.8). On the other hand, E satisfies Definition 5.0.28b. Consider the vector bundle Ev —> Li X X x V whose fiber over (q, p, [1) consists of all n 6 Hom(Tqu, TPX) with n jg = — on. We denote by Pl)’l(a1TX)V the set of all sections of Ev —1 Ll x X x V, which are C’ smooth and vanish near Sing(Ll). For 6 = (i,f3;{e,-},{z,-}), s e v and n e P?”(Ll,TX)v, we define the restriction 17' ( c. p) as follows: for any x E E n.,.,,,(e~) = no, f(a=).fi)- Lemma 5.0.35 There is a finite subspace S C F?’1(Ll , TX )v such that Sl(c.a) is trans- verse to Lfm i.e., ifn1,- . - ,nn span S, then 171k“), - - - ’"nlrcm and Im(Lm) generate I)" (A3:(f‘TX)), and dim(S) = dim(S|(C,a)), where S|(C.a) = { "kc-,0) l 77 E S } 4O Proof. Denote by Coker(Lf,a) the space of all 17 E L2(/\3;1(f*TX)) such that Liam) = 0. Then it is a finite dimensional subspace of If (A3:(f‘TX)) and for any n E LP(/\3’a1(f*TX)), there are 5 E L1’2(E,f*TX) and 770 6 Coker(Lf,a) such that Lf,a(€) = n — 770- Moreover, E E LINE, f‘TX). On the other hand, the set Fla...) = {m(cmln 6 F?’1(LI,TX)V} is dense in LP (Ag’:(f‘TX)). Therefore, we can always find 01 e P?'1(LI,TX)V such that the restriction "Ila... is not in Im(Lfia) UCoker(Lf,a), if Coker(Lf,a) 7i {0}. Then, n1 Ic... = Lf’a(€) + 171 for some 5 in L11P(E,f*TX) and n1 in Coker(Lf,a). Let Coker1(Lf,a) be the orthogonal complement on (171) in Coker(Lf,a). Its dimension is one less than those of Coker(Lf,a). If Coker’(Lf,a) 74 {0}, then we can also find r72 6 I‘?’1(LI,TX)V such that 172 la... = L ”(5’ ) +am [6.0 +172 for some 6’ in L11P(E, f *TX ), some nonzero 172 in Coker1 (L La)’ and some constant a. In this way, we can use the induction on the di- mension of Coker(Lf,a) to find 171, - - - ,nn 6 I‘?’1(LI, TX )v such that 171' ( c.a)’ - - - ,1an ( c...) and Im(LLa) generate LP (A3:(f*TX)). El Denote by C6+1(LI,TX) the set of all CHI-smooth sections of TX —-» Ll x X, which vanish near Sing(Ll). Let S be as in Lemma 5.0.35 and rrs be the orthogonal projection onto the orthogonal complement of SIM) in L2 (Agfl f,*TX )) . Let P be a finite dimensional subspace in Cé+1(11 , TX ) such that dim(P)=dim(Pf) and q5( Ker(rrSU'O)L(f,a))) = Pf, where qs : L1'2(E, ffTX) -—> Pf, is the projection with respect to the L2-inner product. Lemma 5.0.36 Let P and S be as above, and t and “3 - all be sufl‘iciently small. Then for any p > 2, £0 E Pf, and r) 6 LP (A3“ ,‘TX)), there are unique g 6 L11"( f,*TX ) and 770 E Slurry satisfying: (15(0 = $0, Lt.fi(€) = 77 - 770, (0-16) max{ llflllan ll’lOllp } S cmax{ lléOlllmr llllllp} 41 where c is a uniform constant. Proof. Its proof is similar to the proof of Lemma 3.10 in [LT]. We first show that there is 6 and no such that Lt,p(§) = n — no for sufficiently small |t| and Ila — fill. Suppose not. Then we can find a sequence {(tn, ,Bn)} with (tn, fin) —* (0, a) and 17,, in Coker(Ltmo,,) with ”null? = 1 such that 177, is perpendicular to S with respect to the L2-metric on LP (A3}; (f,;TX)). It follows from the standard elliptic estimates that nn converges to some n in L? (A31: ( f ’TX )) outside of nodes of 2. Since n is perpendicular to S and Lia (n) = 0, we have n = 0. This implies that n" -—> 0 on the compliment of Neck region as in (0.5) and thus ||nn||p —+ 0 by Lemma 5.0.34. This contradicts to ||nn||p = 1. Next, we show that there is a unique 6 and no satisfying (0.16). First, choose 5 and 77 With Lt,fi(§) +770 = 77 and SBt 5' = §+qS(§o - E) and 776 = 770 + 14.3015“ — 50))- Then €’ and n’ satisfy (0.16). One can prove the uniqueness by the similar argument as above. Finally, we show the estimate by contradiction. Suppose not. Then there is a sequence {(tn, fin)} with (tmfln) —-> (t, ,6) and g, in Ll'P(f,’;TX) and non in S such that (i) max{|l€nl|1.pll|770nllp} = 1, and (ii) max{ll€o:zll1.p,||77nllp} —’ 0 where q5(£,,) = {on and among“) + no" = n... By Sobolev Embedding Theorem, we may assume that 5,, converges to some 5 and £071 to 0 both in Ll’z-norm. We may further assume that non converges to some no. Note that Ltfi (5) + no = 0 and qs (g) = 0. Therefore, by uniqueness we have 5 = 0 and no = 0. On the other hand, by (i) ||§||1,p —-> 1. It then follows from Lemma 5.0.33 that ”any. are uniformly bounded away from zero. This contradicts to {n —> g = 0 in L112. D Let S be as in Lemma 5.0.35. We define E5 over Map5(W, K) x V as follows : for 42 any (If, A) in Map5(W, K) x v, ESde) = Sl(C’.A)' The following shows that Q: 77’ k( (X A) x H —> E satisfies Definition 5. 0. 28b. Proposition 5.0.37 By shrinking V and W if necessary, if 6 is sufliciently small and K is sufl‘iciently large, Q'1(E5) is a smooth submanifold, which contains (C,a), in Map6(W', K) x V and of dimension 2c1(A) + 2(g — l) + 2k + 2 dimH + dimS. (0.17) Moreover, E5 —> Q’1(ES) is a smooth bundle with Q|g_1( Es) Proof. This proof is similar to the proof of Pr0position 3.4 in [LT]. By shrinking V and W if necessary, we can assume that for any (t, S) E W x V, Lemma 5.0.36 holds. We first show that there exists an co > 0 such that the subset { 0,5,5) 6 131"" | 7Ts ‘1’(t.fi,€) = 0, lléllle < 60} (0-18) is smooth of dimension dim(S) + 2c1(A) + 2(g — 1) + 2k + 2l + dimH. Let Vf’: 1/2 (J); — JgVJp) and r is the parallel translation with respect to V". Define a map ‘1' : L11? —> If(/\°'1(TX)) by \Il(t,fi,€) = rQ( expf,(£),fl). Then the linearization of \II at (t, S, 0) is [10.5) as in (0.2). Now, consider the following expansion \p(t1161€) : $611810) + L(t,3)(€) + H(t,fl)(£) where H(t.fi) is the higher order term satisfying ||H(t,o)(§)||p S Clléllmlllélllp for some uniform constant c; this constant may depend on Jp, but we can still assume it is uniform on S by. shrinking V, if necessary. It also follows from the Sobolev Embedding Theorem that ||H(I,A)(€)llp S Clléllie (0-19) 43 Let Ep be the bundle induced by P over W x V with fibers Pom = Pf, and define a map E : L149 x E5 —* LP(/\0’1TX) x Ep by (t1 181$) 770) —" ( t1 :6) m(ta 1816) + ’70) 43(6)) Then the linearization of E at (t,fl, 0,0) is the map DE:L1*P()3,, f,TX) x SW.) —» LP(/\3’plf,*TX) x PM given by (5,770) "* (L(1,A)(§) + 770.43%» By Lemma 5.0.36, it is an isomorphism with uniformly bounded inverse. Therefore, by the Inverse Fimction Theorem there exists an co > 0 such that E is a diffeomorphism from the region {(trfiréa’lol 6 L1,? X ES l max{ll§ll1.pall770llp} < 60} onto its image. Furthermore, by Lemma 5.0.36, if [tl and Ila - fill are small, then for any éo 6 130.3) with ||£o||1,p < so, there is a unique (t, B,€, no) satisfying E(t,fi,€,770) = 0.5.0.60). On the other hand, it also follows from (0.19) that ||n||o S c||£||1,p when \Il(t,fi,£) + no = 0. Therefore, we can conclude that the subset (0.18) is parameterized by W, V, and some open set of P. Note that by our choice of P and S, dimP—dimS = ind(L¢,o). The subset (0.18) is thus smooth manifold of dimension ind(Lt,o) + dimW + dimS+ dimV. Next, we will show that if 6 is sufficiently small and K is sufficiently large, then Q‘1(ES) is an open subset of the following set { (MM) E 131’” l 7rs ‘1’(t,fi,€) = 0, ||€|l1.p < €01 eXPf,E(Zj) 6 H1 } (020) 44 where z,- for 1 S j S l is the added marked points and H; for 1 S j S l are local hypersurfaces given in Step 1. Note that it is a smooth manifold of dimension (0.17). Let (f,E;{x,-},{zj},fi) in Q‘1(ES). Denote by t the corresponding point (f3;{e,-},{z,-} in W. Since d( f, f) g 5, there is some 5 in ma, f,*TX) with f = expftfi. It follows from Lemma 3.12 in [LT] that “fill”, < so. This implies that Q‘1(E5) is an open set of (0.20) and thus Q‘1(E5) is a smooth manifold of dimension (0.17). Finally, it follows from the smooth dependence of solutions of rrSQ(f, fl) = 0 that E5 —+ Q‘1(ES) is a smooth bundle with Q|¢_,( Es) smooth. CI Proof of Proposition 3.0.15 This proof is similar to the proof of Proposition 2.3 1n [LT]. Let <1, z?’ ,,(X, A) x H, —» E. be the generalized Fredholm orbifold induced 9, by the Kiihler structure (wt, Jt, gt). We define H={(t,a) I (16H; }. Similarly as in Chapter 3, we then define a generalized bundle E over 77:”, (X, A) x H and consider \II: ?,(X,A) xH—>E definedby ((ffl. {173), 0.01)) -+ df + Ja(t) dfj where Ja(t) is the almost complex structure on X defines by J, and a E H; as in (0.5). By definitions, we have \IllfJg .IAX’ 21)ch = Q. Since all Q,‘1(0) are compact, it follows from the same argument as above that Q: .7: k(X, A) x H —> E is a generalized Fredholm orbifold bundle. Moreover,\II gives homotopy between Qo and Q1 as generalized Fredholm orbifold bundles. Now, this proposition follows from 45 Theorem 1.2 in [LT]. D The following lemma gives two facts about the setup used by Li and Tian which are used in the course of several proofs in [LT]. Lemma 5.0.38 Let s : B —> E be a generalized Fredholm orbifold bundle. (a) pr : B —-> V be a continuous map and K be a cycle in V with PD( [K]) = '7 Then 3’ : p‘1(K) —i E’ is also a generalized Fredholm orbifold bundle, where I _ I _ s — SIP-1m andE — Elp-lm' Moreover, i.e(s' :p—1(K) —» E’) = c(s : B —* E) 01917) where i :p'1(K) H B. (b) If s’ : B’ —+ E’ is a generalized Fredholm orbifold bundle with a continuous onto map 7r : B —-> B’ and an injective bundle map r : rr'E’ —-> E such that s‘1(0) = (1r‘s’)'1(0), then e(s:B—-+E)=e(s':B'—>E') Proof of Proposition 3.0.19 Let Q: .7" k(,X A) —> E be a generalized Hedholm orbifold bundle as in (0.3) and a). 6 H2(X; Z). Choose a cycle K which represents a Poincaré dual of a. Then by Lemma 5.0.38a, [J\/l;f,,(X,A)]Vir fl ev;(ak) can be regarded as a class in H.(ev;1(K); Q), where evk is the evaluation map of the k—th marked points. On the other hand, there is a continuous surjective map 7r : ev—1(K) —+ _g,k_1(X, A) x H 46 which forgets the k-th marked points. 7r satiefies the condition of Lemma 5.0.38b and hence we have e. (was, A)]"” 0 ev;(ae)) = [Mane AW”- That implies Pr0position 3.0.19 since 7r is a finite branched covering of order ak(A). D Proof of Proposition 3.0.20 (a) For III, as in (0.9), the set \II{1(0) is compact for all 0 S t S l by assumption. Hence, the arguments used in the proof of Proposition 3.0.14 show that for each OStS 1 thebundle E1 EB E; —> f9,,k,+1(X,A1) x P9,,k2+1(X,A1) x H with a section Q, is a generalized fiedholm orbifold bundle. Denote by )]vir vir [Mt] = [Mllnh+1>.(eake+1)(x , A11 A21 ’ the corresponding virtual moduli cycle. As in the proof of Proposition 3.0.15, it also follows that [MIIVir = [Mor’ir = [Mime A101” e [Meo,ke+1(X,A2)IVir (0.21) as homology classes in H. (P9,,k,+1(X, Al) x Tm,k,+1(X, A1);Q) . Note that [M9,,k,+1(X, A2)]Vir is the cycle which defines ordinary GW-invariants. On the other hand, there is a natural map p:?91,k1+1(X1A1) X ?gg,k2+1(X1 A1) X H -+ X X X defined by(lf1,21;{$1}lIlf2,22;{yj}lea)—* (f1(1‘k1+1),f2(311)) Thereis also a sur- jective map rr : U p‘1(A) —> st’1(Im 0) obtained by identifying xk,“ and y], where 47 the union is over all decompositions of (A, g, k), A is the diagonal in X x X, a is the gluing map in (0.8), and 3t is the stabilization map on it k( (X A) )x ’H. It follows from Lemma 5.0.38a that the classes vir vir Z [Mal.k1+1),(92,k2+1)(x’ Al’ A2’ t)] and [M:k(X’ A)] 0 PD(”) can be regarded as a class in H.(U p‘1(A); Q) and H.(st‘1(1ma); Q), respectively. Moreover, by Lemma 5.0.38b we have [M:k(X, A)]m fl PD(a) vir . t = 7'“I (Z [Mg1,k1+l),(92,k2+1)(X’ A1, A2, 1)] Fl(e'vk1+1H1 A 8le7 )) (0.22) where evk,“ and em are evaluation maps of 1:le and y1, respectively. Combining (0.21) and (0.22), we have [M2,(x, A)]Vir n PD(a) = 1r. 2 ([M;,k1+l(xa 141)] Vll‘ ® [Mgg,k2+1(X, A2)]Vir) fl (BU;1+1H.’ /\ BUIH7) = 7r. z ( [M3,k1+1(X, [41)]V1r n evzl+1H7) ® ( [M92.k2+1(X, A2)]Vir n evl H7). That implies the first Composition Law. (b) Similarly, we have an evaluation map of last two marked points 1) I ~1.9.1,]¢+2()(,A) X H -> X X X ([f, 2; {all ,a) -* (f(:vk+1), f(:vk+2) ). There is also a surjective map 1r : p‘1(A) —+ st‘1(Im6l). It also follows from Lemma 5.0.38 that [Mgk(x, A)]Vir 0 190(9) = 7r. 5: ([My_,,k+,(x,) A )]Vlr n (ev;+,H /\ evg+2m )) . That implies the second Composition Law. [3 48 CHAPTER 6 The Invariants of E(n) — Outline Let 7r : E (n) ——> P1 be a standard elliptic surface with a section 3 of self-intersection number —n. Denote by S and F the homology class of the section and the fiber. We will compute family GW-invariants for the class S + dF with 2139 = 2(n — 1) dimensional parameter space 71,. defined as in (0.4). These invariants G Wit; (3 + dF) are unchanged under deformations of Kahler structure. For convenience we assemble these into the generating fimction F(t) = Z GWg}; (s + dF) td. (0.1) ago In the this and the following four Chapters we will calculate the invariants Gng; (S + (1F) by deriving the formula for F(t) stated in Theorem 0.3. Thus our aim it to prove: PrOposition 6.0.39 For n _>_ 1, F(t) = H(l—:—i§)l2n (0.2) dZO As mentioned in the introduction, the cases n = 1, 2 have been proven by Bryan- Leung and Ionel-Parker. This Chapter shows how Proposition 6.0.39 follows from two formulas, equations (0.4) and (0.5) below, that are proved in later Chapters. Our proof parallels the proof 49 of Ionel and Parker [IP3] with two changes. First, we replace the use of the 7' class by 1/) class; that makes the argument conceptually a bit easier. Second, we must extend the TRR formula and the Symplectic gluing formula of [IP3] to family invariants. Here is the outline the proof of (0.2). Let G (t) be the generating fimction for the function for the sum of divisors flmction 0(n) = Zdln: d dtd G(t) = 2m” = :14, dZO dZO Following [IP3] we also consider the generating fimction for a genus 1 invariant, namely Hm = ZGWff." (3+ dF) (¢(1.4>;4;F‘>t“ (03) (120 where 10(9),)”- denotes the first Chem class of the line bundle L(g,k);,- -—» My). whose geometric fiber over (0; 3:1, - - - ,xn) is TgiC'. We can compute H (t) in two different ways. In Chapter 7, we show how to combine the composition law together with the relation between 2p class and the divisors in Ill—1,4 to obtain the formula H(t) = $tF’(t) — le-F(t) + (2 — n)F(t)G(t) (0.4) Then, in Chapters 8—11 we develop a family version of the Gluing Theorem in [IP3] to obtain the sum formula H(t) = —112-F(t) + 2F(t)G(t) (0.5) (see Proposition 11.0.62). Equations (0.4) and (0.5) give rise to the ODE with tF’(t) = 12nG(t)F(t) (0.6) 50 and we show in Proposition 8.0.47 that the initial condition is F(O) = 1. It is well- known that the solution of this ODE is given by F(t)=H(1_ltd)l2n. ago That completes the proof of Proposition 6.0.39 and hence of the main Theorem 0.3 of the introduction. 51 CHAPTER 7 The Topological Recursion Relation (TRR) A pinched torus can be regarded as a two-sphere with two points identified. Conse- quently, maps from a pinched torus are a special class of maps from the two-sphere. That observation allows one to express certain g = 1 GW invariants in terms of g = 0 invariants, and more generally express certain genus g invariants in terms of genus g invariants. Such formulas are called t0pological recursion relations or TRR formulas. In this Chapter we will prove formula (0.4), which is a TRR formula for the family CW invariants. We begin by recalling the notion of the dual graph associated with a stable curve. Given a stable genus 9 curve with n marked points (C; 33,-, - -- ,mk), its dual graph is defined as follows. Let 7r : C —+ C be the normalization of C. The dual graph G has one vertex for each component of C, and the edges of G correspond to nodal points of C; if two points on C maps to a node, then the edge, corresponding to that node, are attached to the vertices associated to the components of C on which the two points lie. The legs (half-edge) of G correspond to marked points of C, and these are indexed in an obvious way. 52 We denote by M(G) the moduli space of all genus g curves with n marked points whose dual graph is C. We also denote by 60 the orbifold fimdamental class of m, that is, the fundamental class divided by the order of the automorphisms of a general element of M(G). Graphs with one edge correspond to degree two classes. There are two types of such graphs. One is the graph G," with one vertex of genus g -— l. The other types are the graphs G0,], which have two vertices, one of genus a, with attached the legs indexed by I, and one of genus g —- a, with attached the legs indexed by {1,--- ,k}\I. For any 2' E {l,--- ,k}, we have ¢(1,k);i = $50.... + 213500,, in H 2(—M1,k;Q)- (0-1) ”:22 For the proof of (0.1), see [AC] and [G]. Proposition 7.0.40 The generating function (0.3) satisfies H(t) = étF’U) — 112.51» + (2 — n)F(t)G(t) Proof. It follows from (0.1) that the coefficients GWfiflS + dF)(’l/J(1,4);4; F4) of H (t) is GW,,;(S + dF)(6G,,,; F4) + Z GWI’jE,"(S + dF)(6GO.,; F4). (0.2) i6! l1l_>.2 i 12 We will apply the first Composition Law to 0W3," (S + dF)(6Go,,; F4) and the Second . F4). irr ’ composition Law to 0W3," (S + dF)(6G Recalling Proposition 3.0.20, the only possible decompositions of the class S + dF, which can appear when we apply the first Composition Law, are S + le and d2F with d1 + (12 = d. It then follows from a dimension count and the first composition 53 law that ij;(s + dF)(6Go.,; F4) 2 ZGWJI';I+1(S+le)(F”',H.)GWI,5_m(d2F)(F4—'”,H"). (0.3) di +d2 =d '7 where {H,} and {H 7 } are bases of H * (E(n)) dual by the intersection form. It also follows from Proposition 3.0.19 that, if I = {1, - .. ,4}, then (0.3) becomes 2 2 (H, . (s + W» (H7 . sz)GWg:,n(s + M) GW1,o(d2F) dl+d2=d ’7 (12 >0 + 201, (S + dF) )GW,3*‘,,n(S + dF)GW1,1(0)(H") (0.4) Otherwise, (0.3) vanishes. Since 27(H7A)(HVB) = AB and kGW1,o(kF) = (2 — n)a(k) (see [IP1]), the first sum in (0.4) becomes (2 — n) 2,21 GWJI,"(S + (d — k)F) 0(k). On the other hand, GW1,1(0)(H") = 5%(KH7) ( see [IP3]), where K = (n — 2)F is the canonical class. This implies that the second sum in (0.4) becomes n — 2 —2—4— W3}; (S + dF). In summary, we have 2GW$¥21 Note that PD(Im(0)) = 260," where 0 : Ho’s -> _MM as in (0.10). It then follows from the second Composition Law and Pr0position 3.0.19 that wafgw + dF)(6G,"; F4)- — 1: Gngn(s + dF)(F4, H,, H”) =21:(H, (s + dF)) (m(s + dF)) GWij,"(S + dF)(F4) 2d: ”a The proof follows from (0.2), (0.5), (0.6) and the definition of F(t) and H (t) 54 CHAPTER 8 Ruan-Tian Invariants of E(n) Instead of constructing virtual moduli cycle directly from the moduli space of stable J-holomorphic maps, Ruan and Tian [RT1, RT2] perturbed the equation (0.6) to 51f = V where the inhomogeneous term V can be chosen generically. For generic (J, V), the moduli space of stable (J, V)-holomorphic maps is then a compact smooth orbifold with all lower strata having codimension at least two. Ruan and Tian defined GW-invariants from this (perturbed) moduli space. We can follow as similar procedure for the family invariants by introducing an inhomogeneous term into the (J, a)-holomorphic equation and vary (J, u) and corre- sponding parameter space H. In taking that approach, we immediately face two main problems: compactness and the dimension of lower strata. In general, it is difficult to show the compactness of a perturbed moduli space, even if M is small and the moduli space without perturbation is compact. It is also difficult to determine the dimension of lower strata which contain bubble components. However, for the moduli space of perturbed (J, a)-holomorphic maps representing a homology class S + dF in E(n) with fixed complex structure J, the moduli space of (J, a)-holomorphic maps with generic perturbation is still compact and the image of lower strata under stabilization and evaluation map is contained in a set of codimension at least two. Therefore, we can define invariants from the moduli space with fixed Kahler structure and generic 55 perturbation in the same way as for ordinary GW-invariants. This alternative defin- ition of invariants is more geometric. In particular, using this definition of invariants we can follow the analytic arguments of Ionel and Parker in [IP2, 1P3] to show sum formula (0.5) for the case at hand: the class S + dF in E(n). To simplify notation in this Chapter we will set X = E(n) and A = S + dF. The construction of invariants starts from the perturbed equation 51f = V. Using Prym structures defined as in [L], we can lift Deligne-Mumford space WM to a finite cover pl! I Vigil, —’ Hg’k. (0.1) This finite cover is now a smooth manifold and has a universal family 71’" Z 17:”: "" Hg’k which is projective. Moreover, for each I) 6 H3,“ 7r;1(b) is a stable curve isomorphic to p,,(b). We fix, once and for all, an embedding of a; into some I?” . An inhomogeneous term 11 is then defined as a section of the bundle Hom(7rf(TlPN),7r;TX) which is anti-J-linear : V(jp(v)) = —J(V(v)) for any 1) E TlP’N (0.2) where jp is the complex structure on P” . For each stable map f : 2 —-> X, we can specify one element j E p;1(st(2)). Then 7r;1( j ) is isomorphic to the stable curve st(2). In this way, we can define a map 4) : z: _. 52(2) :2 7r;1(b) c: 1.7;, e» P”. (0.3) Definition 8.0.41 A stable (J, V, a)-holom01phic map is a stable map f : (2, (b) —-» X satisfying (df + ‘10de!) )(P) = Va(¢(P). f (17)) 56 where (15 is defined as in (0.3), and Va = (I + JKa)_ll/. Cl Two stable (J, u, a)-holomorphic maps (f, (¢,E);$1,--- ,xk) and (f', (¢',2');$'1,- -- ,xjc) are equivalent if dams), ¢'(2’))) + dH(f(E), f’(2’)) + Zd(f(m.-), m2» = 0 where dH is the Hausdorff distance. We then define the moduli space M9.k(Xi A, V7 Hi (1') as the set ofall pairs ([f, (¢, 2);:121, . - . ,xk],a) , where a E H and [f, (q), 2); 2:1, - -- ,xk] is the equivalence class of (J, V, a)-holomorphic maps with [f (2)] = A E H2(X; Z). We denote by Mg,k(X, A, u,’H,p) the set of ([f, ((15, 2); $1, - -- ,xk], a) with a smooth domain 2. We will often abuse notation by writing (f, j, a) or simply (f, or), instead Of (f, (¢’2)3 Cx). There is a stratification of M g”. parameterized by the automorphism group of Riemann surfaces MM = 2,, T2,, where each strata T3,, is smooth and consists of the Riemann surfaces with a fixed automorphism group n. We can also assume that M5,]. = X T53; K where T5,): = p;l(T;k) is smooth. Let Mg,k(X, A, 11,7141); consist of all (f, j, a) with j in T51, where I denote the trivial automorphism group. Consider the following stabilization and evaluation maps st" X en“ : Mg,k(X, A, 1471,11); —+ Rik x X". (0.4) Its Hontier is defined to be the set {7' 6 fig, x xk | 7' =lim(st“ x ev")(f,,, a.) and (fn, an) has no convergent subsequences }. 57 We denote by yo the space of all V with |V|°° is sufficiently small. Now, we are ready to state ”Structure Theorem” for the moduli space. Theorem 8.0.42 (Structure Theorem) For generic V 6 yo, (a) Mg,k(X, A, V,H,[.t)1 is an oriented smooth manifold of dimension —2 KA + 2(g — 1) + 2k + dim(H) = 2(g + k) (0.5) (b) the Frontier of the smooth map st" x ev" : M9,).(X, A, V,'H,p)1 —> 17;), x X’c lies in dimension 2 less that 2(g + k). Proof. This proof is similar to the proof of Proposition 2.3 in [RT]. We will sketch proof, without specifying Sobolev norms. (a) For each ’6, define W = U Map..(>3.,X) x {j} jag)? where M apA(Eg,X ) = {f : 29 —+ X ] f.[29] = A}. Consider the vector bundle 8M“ —* Xp’n X H X yo whose fiber over (f, j, a, V) is 522’), ( f *TX ). Obviously, the (J, V, a)-holomorphic equa- tion defines a section (D of 8"” by (f,j,oz,V) = df + Jadfj — Va. The differential D of (I) at (f, j, a, V) is then an elliptic operator D : o°(f*TX) 39 7:35;; ea TQH s HomJ(TlPN, TX) —. {23,14 f*TX) 58 given by 02a. w, v) = 14(5) + Jadf k + aw) — A-lv where (0.6) L,(g) = V5 + JaVEj + % (v.1...) (dfj + Jadf) — (7.12,, L(1JK,3J., + JaA-lmp) dfj, {7 (A m—é-(v JMVJ), andA= (I+JKa) Consider the universal moduli space Liam) = {(f,j.a.V) 6 x97." x H x we I (f 2a v) =0} At any (f, j, a, V) E Ll;,;‘(X, A), the differential Dd) is surjective because of the term A’lv. This implies that the universal moduli space is smooth. Let 7r : u;,;‘(X, A) —2 yo be the projection. Its difierential at (f, j, a, V) ”1771’ :(X, A) ——+ T37“, is just the projection (5, k, 6, v) —-> o. It then follows that the kernel of d7r is isomor- phic to the kernel of L f 63 Jadf EB La. Moreover, its image consists of all 1) with A‘lv E Im(Lf EB Jadf EB La). Note that L f EB Jadf EB L, is Piedholm, and hence Im(dvr) is a closed subspace of T370. On the other hand, the map p: T370 -» 93,14 f‘X) /Im(L, ea Jadf ea La) defined by v —. 21-12; is onto since D is onto. Therefore, dim(Coker(Lf EB Jadf GB La)) = dim(Tyo / Ker(p)) = dim(Coker(d1r)) and hence d7r lS Fredholm of the same index as the index of (Lf EB Jadf EB La). Applying Sard—Smale Theorem, we can conclude that for generic V 6 yo, the moduli space 7r‘1(V) = M9,),(X, A, V, Hm)” 59 is a smooth manifold. The dimension formula follows from the Index Theorem. For generic (J, V), the tangent space Tf’j,aMg,k-(X, A, V, H, #)n = Ker(Lf 6 Jadf 6 La), so we have det (Tf,j’aMg’k(X, A, V, H, u),,) = det(Lf 6 Jadf 6 La) On the other hand, there is a decomposition L f = L f“ + Z, where L?“ is Ja-linear and Z is the zero order term. It follows that det(Lf 6 Jadf 6 L0,) is isomorphic to det(L}° 6 Jadf 6 La). Since both kernel and cokemel of Lf" 6 Jadf 6 La are complex vector spaces, there is a canonical non-vanishing section of det(L}°' 6 Jadf 6 La). Therefore, there is a nonvanishing section of det(Lf 6 Jadf 6 La) which orients Mg,k(X, A, V,H,p),,. D (b) This proof consists of 5 steps. In step 1, we show that the stabilization and evaluation map as in (0.4) extends continuously to the moduli space of stable maps. That follows from the compactness. In step 2, we show all possible homology classes, which can be represented by the components of stable maps. In step 3, we reduce the moduli space. The resulting reduced moduli space will have the same image as that of the moduli space under the stabilization and evaluation maps. In step 4, we stratify the reduced moduli space. The Frontier is then contained in the image of all lower strata. In the final step, we show that each image of the lower strata is contained dimension 2 less than 2(g + k). Step 1 There are well-defined stabilization and evaluation map st“ x ev" :VQJJX, A, V,H,H) —-> 7W; X Xk (0.7) where we still use the same notation for the map as in (0.4), without further confusion. It follows from Gromove Compactness Theorem, Theorem 3.0.18, and the following lemma that (0.7) extends (0.4) continuously. 60 Lemma 8.0.43 There exit uniform constants E0 and N such that for any (f, 2, a) E Hg,k(Xa A: V371) l E(f)=§/ Idflsto and HaIIsN. 2 Proof. Similarly to Corollary 1.0.7, we have _ 2 - ’a — V . AIM —/Ef +2/2(31f. > (08) (1 + (amrw do = %(1 — |a|2)|df|2 do — 4(5) f, u) do + 4M” do (0.9) Note that f represent homology class A = S + dF which is of type (1,1) with respect to the complex structure J. Therefore, it follows from (0.8) and Pr0position 1.0.6a g f |de2 s M) + 2 (I. ldfl2)% ([2 Iul2)% We then have a uniform energy bound by using the inequality 2ab S s a2 + 5'1 b2 on that the last term and absorbing the [de term on the left-hand side. Next, we will show uniform bound of Hall. This proof is similar to those of Lemma 4.0.24 except for using (0.9) instead of Corollary 1.0.7b. Let 1r : X —> (IF1 be the elliptic structure for J on X and N (a), m(J), and N be as in the proof of Lemma 4.0.24. If there is a holomorphic fiber F C X \ N (a) such that (i) f is transversal to F, (ii) at each p E f ’1(F), f is transversal to a holomorphic disk D MD) normal to F at f (p), and (iii) 4|de IVI + 4|V|2 S %|de2 on f‘1(F) then the proof follows exactly as in the proof of Lemma 4.0.24. We can clearly find fibers satisfying (i) and (ii), so we need only verify that we can also obtain (iii). For that we consider the set 20 of all points in 23 where 4|de |V| + 4|V|2 > % [de2. Then 61 |df|2 g 16|V| on 20, since both Ide and [Vloo are less than 1. Therefore |d1rodf|2 g 16Area(st()3))|d7r goluloo (0.10) 20 We can thus assume that (0.10) S §Area(CP1) for sufficiently small |V|°°. On the other hand, from the definition of N (a), we can also assume that Area(1r(N(a)) ) S §Area(CP1). Therefore, we can always choose a holomorphic fiber F = 7r‘1(q) as in the above claim with q 6 (CF1 \ ( 7r(N(a)) U 7r 0 f(20) ). El Step 2 Let (f, E) be a stable map. A stable bubble component which maps to a point by f is called ghost bubble. Now, we reduce the moduli space as follows : for a stable (J, V, a) holomorphic map, (i) we collapse all ghost bubbles, (ii) we replace each multiple map from a bubble by its reduced map (iii) we identify those bubble components which have the same image. Denote by MAX, A, V, H, p) the quotient of RELAX, A, V, H, p) by this reduction. We define the topology on it as the quotient topology. The reduced moduli space is still compact. On the other hand, the map (0.7) descends to the reduced moduli space and by definition the image of reduced moduli space is same as that of stable moduli space under stabilization and evaluation maps. Step 3 Let (f, 2, a) e E,,(X, A, 1471,11) with 23 = 11,23. and [f,-(2.)) = 21,-. The following lemma shows all possible homologt classes for A,. Lemma 8.0.44 If [Vloo is sufliciently small, then A.- is one of the following homology classes S, S + le, d2F With 0 < d1,d2 S d. (0.11) and each bubble component 2,- represents either S or dgF. 62 Proof. Suppose not. Then there exits a sequence of (J, an, Vn)-holomorphic maps (fn, 2,.) and a homology class C which is not one of classes in (0.11) such that (i) Vn —+ 0 as n —2 00, (ii) for each n, there is some component En, with [fm(2n,)] = C. By Lemma 8.0.43 and Gromov Convergence Theorem, we may assume that Jan —» Jan uniformly for some a0 6 H and fa converges to f0, where f0 is (J, ao)-holomorphic and f0 represent the class S + dF. Since S + dF is of type (1,1) and do is closed J-anti invariant 2-form, do = 0. This implies that the homology classes C and D = S + dF — C are both represented by holomorphic map with possibly reducible domain. Note that C - F is either 0 or 1. Assume C - F = 0. Then C = d3F with d3 > d by the assumption on C and D = S + (d — d3)F. Let (f1), ED) be the holomorphic map representing homology class D. Then for some component ED, , the restriction map f1).- represent a homology class S — d4F with d4 > 0. This is impossible since (S — d4F) - S < 0 and ED, is irreducible. Similarly, we also have the contradiction when C - F = 1. When 2.- is a bubble component, V vanishes on 2,- and hence f.- is holomorphic. Therefore, A,- cannot be S + d2F since 2,- is irreducible. [II Step 4 In this step, we stratify the reduced moduli space. For each (f, a) E —r Mg,k(X, A, V,H, u), the normalization of the domain of the map f, without spec- ified complex structure, is a disjoint union of smooth Riemann surfaces 2=P1U---UPnUB1U---Bm (0.12) where B,- is the bubble component. We will call R a principal component. Each component of 2 has points corresponding to the marked points of the do- main, which we also call marked points. Each component also have points corre- sponding to the singular points of the domain. We call these points intersection 63 points. Together with the normalization map, the intersection points carries the in- tersection data of the domain. Each component is also associated to a homology class in H2(X; Z), which we denote by [P.-] and [8;]. Note that [13,-] is one of the class in (0.11) and [BI] is either 3 or f by definition of reduction. Moreover, if some [8;] = f, then the image of the restriction map lez is some nodal fiber F} and oz 6 { fl 6 H I F; C Z(fi) } by Lemma 2.0.12. Similarly, if some [8;] = s, then the image of the restriction map fla, is a section S and a = 0 by Lemma 2.0.12. In this way, the set of bubble components determines a subspace H3 C H. If there is no bubble component, we simply set H3 = H. Since there’s no fixed component in the complete linear system [KI of a canonical divisor, Lemma 2.0.11 implies that this subspace H B has at least 2 real codimension whenever it is proper. We denote by DV( f, a) the set of following four data, (i) E with marked points, (ii) the intersection points with intersection data, (iii) the set of homology classes each of which is associated to a component of E, and (iv) the subspace H3 determined by bubble components. Let 1),, be the set of all Du( f, a)’s. It then follows from the uniform energy bound and Lemma 8.0.44 that D” is finite. Fix D 6 ’13., such that ED has more than one component. If ED has just one component, proof follows from proof of (a) above. Let g,-, IQ, and d,-, be the genus of P,-, the number of marked points on P,-, and the number of intersection points on P,- with principal components, respectively. Similarly, let k‘ be the number of marked points on 8;. Note that k' might be 2 by reduction. Let it = (n1, - - - , mp), where It,- is an automorphism group of some j E 791.9%..de and p is the number of principal component. We will use (D, R) to label each stratum of the reduced moduli space. We denote by M(D, Ft) c MAX, A, u, H, ,1) 64 the stratum labeled by (D, R). Step 5 Finally, we will show that the image of each strata is contained dimension 2 less than 2(g + 1:). Consider the universal moduli space “0,75 = {( (f11j1)1' ' ° ,(fmjnlaV1a)} Where dfi + Jadfiji = Vi With ji 6 Tgi’:‘+d£a Vi = leia a E HUB: and ldfi] 2 [Bl we have an evaluation map 6110,, 2110;, —> Xd which records the image of intersection points, where d = 2 d;. We denote by A DP the multi-diagonal in X ‘1, which is determined by the intersection pattern of principal components. Now, the inhomogeneous term V ensure that 110,-, is smooth and e120,, is transversal to App. Therefore, um. fl ev5;(A DP) is smooth. Finally, the natural projection 7r 2 Up"? n evBL(ADP) —’ yo is Fredholm. By Sard-Smale Theorem, we can then conclude that for generic V, the moduli space M(Dp, V, HDB, K.) = 7r‘1(V) is smooth. Its dimension is obtained from the routine count and is less than or equal to 29 + 2 Z k.- + 2 — 2p — codim(7‘iDB) if H0, 75 {0} (0.13) 29 + 2 Z k,- — 2 if Hp, = {0} (0.14) where p is the number of principal component. Note that 2 — 2p — codim(HDB) S —2 since ED has more than one component. 65 Recall that for each bubble component B), the homology class [8,] is either 3 or f. We identify each B, with a section S or nodal fiber F according to [3;], and then fix, once and for all, a holomorphic map h; : (CP,1 —» B; C X. Let b is the number of bubble components and set 8(1),.) = («JPN x --~(CP§)"". Clearly, it is smooth of dimension 2 Z: k‘. Let k1 = 2k,- and k2 = Zk‘. Note that 1171 + k2 = k. Each element (j1,--- ,jp) determines an element in 7W“, by gluing intersection points by intersection data. Obviously, we have an evaluation map which records In marked points on principal components. We also have an obvious map from 8(DB) into X k“ by hl’s. Combing those three maps to obtain a continuous map 6 x evp x hB : M(Dp,V,HDB,It) x 8(DB) —+ Rik x X" It then follows that its image Im(6’ X 611;: x hB) lies in dimension 2(g + k) — 2. Now, it remains to show the following : st“ x eo"(M(D, 173)) C Im(6 x evp x hB) (0.15) There is a decomposition of the evaluation map ev" on M(D, F.) as ea” = eo‘I‘, x evg, where eoflevg) records marked points on principal(bubble) components. Note that Im(ev’g) C Im(hB) since for all (f,2,a) E M(D,Ft), the image of bubbles are all 881116. On the other hand, for each (f, 2,01) 6 M(D, R), if we forget all bubbles, then we obtain an element (fp, 2;», a) in M(Dp, V, HDB , It). In this way, we can define a map 1rp : M(D,Fc) ——> M(Dp, V, HDmn) 66 Then (0 x evp) 0 HP = st“ x e127,. Together with Im(ev’g) C Im(hB), this implies (0.15). [:1 Now, we are ready to define invariant. Instead of using intersection theory as in [RT] , we will follow the approach in [1P2]. The above Structure Theorem implies that Fr( st" x eo") C st" x ev" MAX, A, V,H,/1.) \Mg,k(X, A, V,H,p)1) In particular, the Frontier lies in dimension 2(g + k) — 2. It then follows Proposition 4.2 of [KM] that the image Sty X 6’1)” (Mg,k(X1A1V1H1#)I) give rise to a rational homology class. We denote it by [Hg,k(XaA1V1H1/1)l E H*(m:,kiQ) ® H*(Xk;Q)' (0'16) Definition 8.0.45 For 2g + k 2 3, fl 6 H‘(_J\79,k;Q), and a1, - -- ,ak E H‘(X";Q), we define invariants by (pg,k(X1A1H)(fl;ala°” yak) = (IB®(al AmAak)) n [mg,k(X1A1V1H1/J)] 1 Au where A” is the order of the finite cover in (0.1). By repeating the arguments used in [RT] for the ordinary GW—invariants, we can prove that these invariants g,k(X, A, H) are independent of the inhomogeneous term V, the finite cover pp, and the projective embedding UL, H P”. Alternatively, we can simply observe that those three facts emerge as corollaries of the following proposition. 67 Proposition 8.0.46 g,k(X, A,H) = GW:,,(X, A) Proof. As in Chapter 3, we define f g’,,(X,A,/1) to be the set of all equivalence classes of the stable maps of the form ( f, ()3, 05) ), where 43 is defined as in (0.3); two stable maps ( f, (2, 45)) and ( f’, (2’, ((3’)) are equivalent if there is a marked points preserving biholomorphic map a : 2 ——> 2’ such that f = f’ o a and (b = ¢’ 0 a. Note that FLAX, A, p) is a finite cover of T-j,’k(X, A). Similarly, we define a generalized bundle E'" over H,,,(X, A, n) x ’H and a section <1»: by ( f, (2, d),a) -. df + Jadf j. It follows from Lemma 8.0.43 that the zero set of (1)6 is compact. Therefore, by Proposition 3.0.14 there is a virtual moduli cycle which satisfying mlMZuX. A. 11)] = Aelequ. A. )1 (0.17) where 7r : fiAX, A, p.) —-> g’k(X, A) and A” is the order of p,,. Now, fix a generic V as in Theorem 8.0.42. It follows from Proposition 3.0.15 and Lemma 8.0.43 that we still have the same moduli cycle as in (0.17) when we change the section “ by adding —V. We still use the same notation <1)“ for this new section. Note that U1 = M(X. A, 11,71, II)! C (Ml—1(0) It follows from the proof of Proposition 3.0.14 that U, is one of the open sets of the finite cover of (")’1(0) as in Definition 5.0.28. Let n = dim(-NI“,c x X") and d = 2(g + k) = dim(M(X, A, V,H,/1.)] ). Since the Frontier of st“ x ev" lies in dimension d — 2, there is an arbitrary small neighborhood V of Fr(st" x ev") such that every homology class in H,,..AW/IZ,‘ x X k;Q) has a representative disjoint from 7. We can assume that for any open set U, in the finite cover of (4)”)‘1(0) as above with U,- aé U), the intersection U,- n U, lies in (st“ x ev“)‘1 (M5,, x X" \7). It then 68 follows from the proof of Theorem 1.2 in [LT] that the cycle Z which represents the virtual moduli cycle satisfies st" x ev“(Z) fl (WA—L, x X'c \V) 2 st” x ev"(U1) fl (fig,c x X"\V) This implies that (st“ x ev").[M;f,'; (X, A, u)] = {7179,}.(X, A, V,H, u)] Therefore, by Definition 8.0.45 and (0.17) we can conclude that two invariants are same. 1:] In the below, we will not distinguish two invariants and use the same notation 0W3? (X, A) for them. The following proposition shows F(0) = 1 which provides the initial condition for (0.6). Pr0position 8.0.47 GWJS‘ (X, S)(F3) = 1. Proof. Fix V = 0. Since the section class S is of type (1, 1), Theorem 2.0.12 implies that for any (J, a)-holomorphic map (f, a) with [f] = S, f is holomorphic and a = 0. In fact, there is a unique such f since 82 = —n. Now, consider the linearization as in (0.6). Propositions A.63 and A64 of the appendix show, quite generally, that L f is a 51 operator and Lo defines a map L0 2 H —* Coker(Lf) which is injective if and only if the family moduli space ”SAX, A) is compact. But we just showed the moduli space is a single point, and hence compact. On the other hand, Ker(Lf) is same as the Dolbeault cohomology group H§(f“TX). It is trivial since c1( f *TX )[S2] = —n + 1 < 0. Therefore, dim( Coker Lf) = —Index(Lf) = —2(c1(f*TX) + 1) = 2(n — 1) 69 Since L0 is injective and dim(H) = 2(n — 1), Lf 6 L0 is onto. That implies V = 0 is generic in the sense of Theorem 8.0.42. Consequently, the invariant is :l:l. In this case, the sign is determined by L f and L, is BJ-operator, the invariant is 1. D 70 CHAPTER 9 Degeneration of E (n) Throughout this Chapter, X always denotes the standard elliptic surface E (n) -> (IF1 and Y always denotes T2 x S2 with a product complex structure. In this Chapter, we construct a degeneration of X into a singular surface which is a union of X and Y with V = T2 intersection. The sum formula (0.5) will be then formulated from this degeneration. We also define the parameter space and inhomogeneous terms corresponding to this degeneration. We fix a small constant e > 0 and let D(€) C C be a disk of radius 6. Choose a smooth fiber V in X. We then define p : Z —2 X x D(e) to be the blow-up of X x D(e) along V x {0} and let /\:Z—>D(e) be the composition Z@ > p >> X x D(e) —> D(e), where the second map is the projection of the second factor. The central fiber Z0 = A‘1(0) is a singular surface X UV Y and the fiber Z A with A aé 0 is isomorphic to X as a complex surface. Since Z is a blow-up of a Kahler manifold, it is also Kahler. Denote by (tag, Jz, 92) the Kahler structure on Z induced from the blow-up. We also denote by (tax, J A, g,\) the induced Kahler structure on each Z ,\ with A 79 0. 71 We can describe the Z locally along V C Z as follows : fix a normal neighborhood N of V in X. It is then a product V x D, where D C C is some disk. Let a: be the holomorphic coordinate of D. Then, Z is given locally along V C Z as {(v,:c,A,[lo;ll]) Iv E V, xll = Alo} C N x D2 x (CP1 where [l0; II] are the homogeneous coordinates of S2 = CPI. It is covered by two patches U0 = (lo sé 0) and U1 = (ll 51$ 0). On U0, we set y = l1/ lo. Z is then locally given as {(v,:1:,y) | v E V} with A(p,:t,y) = my. Clearly, the fiber Z ,\ is given locally by the equation xy = A. Note that we can also think of y as a holomorphic normal coordinate of the normal neighborhood of V in Y. We now decompose Z as a union of three pieces, two ends and a neck. These are defined as follows : Let N x(e) (Ny(6)) be the normal neighborhood of V in X (Y) of the form V x D(e). We then set Endx = p—1((){\.IVX(¢E))XI)2)flr()(\IVx(tE))XI)2 (0.1) Endv = Harm/)6 U1 I y’=lo/li, ly’l S 1/6 } 39-5 (Y\Ny(6)) X 02 (02) U = {(v.$,y)€Uol|$|SZe, lyls26} (03) where the map (px ((,Oy) is the isomorphism which extends the holomorphic map (may) -: (11.22.9031) ((v,w.y) -> (v,y.xy)) for (v,a:,y) E U and e S [7:] 3 26(6 3 lg] S 26). Next, we define the parameter space on Z as follows : choose a bumf fimction 6 on the neck region U which satisfies fl(|x|) = 1 if lit] 2 (3/2)e and E(lxl) = 0 if 72 [ar| g e. We then extend fl on the whole Z such that B = l on Endx \ Supp(1 - fl) and ,6 = 0 on Endy. Let H be the parameter space of X = E(n) defined as in (0.4). We consider each a E H as a 2-form on X x D(e). Then each p‘a is closed and J z-anti-invariant. Definition 9.0.48 We define the parameter space of the fibration A : Z —-> D(e) by Hz = {fip‘a|a€H} and H,\ = {ai=a|zA|a€Hz} when A7é0 We can consider X as a Kahler submanifold of Z. It then follows from the above definition that Hx={)6a|aEH}={alx|oz€Hz}. (0.4) Lemma 9.0.49 There erm't uniform constants E0 and N, which does not depend on A, such that l E(f) = 5 f 1de s. E. and Maine s N E for any (f,2,a,\) E XII—g,k(ZA,A, V, HA), where |V|oo is sufficiently small and A = S + dF. Proof. The proof of the uniform bound of a; is similar to the proof of Lemma 8.0.43. We define N (a A) as an Open neighborhood of zero set of a) and define m(JA) as in the proof of Lemma 4.0.24. Since each a; is supported on the End x, there is a constant c > 0 such that m(JA) > c for any A. Then, the argument in Lemma 8.0.43 shows that N = 2/c > 2/m(J,\). It remains to show the uniform energy bound. Note that a; = Bp‘a for some a E H. For each p E 2, let {e1(p), e2(p) = je1(p) } be an orthonormal basis of TPE. We set 2— = { p E 53 | f’p*a(ei(p).ez(p)) < 0} 73 Since [5.],[fl2 do = f‘flp‘a+2(51,f, V) du, we have ngnfl S 2|V| on 2-. This implies -f'P'a(61(P),82(P)) S MldfllVl where p E E- and M = max{ |p‘a| | ||p*a||2 g N }. Therefore, we can conclude that 1 . . 5 / Idflz s / f (sp a) + / Idf||V| + WM) 2: 2 z: s - / rpm + j W”!!! + WM) 2- z: s (1 + M) ([2 WY ((3 ldf|2)% +wz(A) This implies the uniform energy b01md independent of A. [3 Finally, following [1P2], we define inhomogeneous terms on the fibration A : Z —+ D(e). As in Chapter 8, we fix a finite cover 173*, universal family Hg; over it, and a projective embedding 22-13,, H P” . We denote by the orthogonal projection onto the normal bundle Nx (Ny) of V by 5 —r 5". Definition 9.0.50 We define an inhomogeneous term V of the fibration A : Z -> D(e) to be a section of the bundle Hom(TlPN,TZ) over I?” x Z which satisfies (i) V is anti-JZ-linear, i.e. Vij = -Jz V, (ii) the restriction of V to P” x Z,\, we simply denote it by V,\, is a section of Hom(TlPN,TZ,\) over I?” x Z,\ when A aé 0, (iii) Vx (Vy) is also a section of Hom(TlPN,TX) (Hom(TlPN,TY)) such that V5: (V5) =0, and (iv) for all§ E Nx(Nx) andv 6 TV [Val/X + JVJtVXIN = [(JVuxJ)€]N ([Vew + JVthlN = [(JVwJ)€]N) 74 We denote by y" the set of all inhomogeneous terms on Z. Proposition 9.0.51 For generic V E 37V and generic A 76 0, (a) Mg,k(Z)(,A,H,\,VA)1 is an orientable smooth manifold of dimension 2(g + k), and (b) the Frontier of the smooth map 8t X en 1 Mg,k(Z,\,A,H;(,VA)1 —-> n.9,): X Zf. lies in dimension 2 less than g + 10. Proof. We can consider Z \ Z0 as a fixed smooth manifold E(n) with a family of Kahler structures parameterized by D(e) \ {0}, namely, for each A 94 0, (Jiiwxigx) = (Jziwzigzllz, It then follows that the universal moduli space a = { (frji JAiaArl/A) l f is (JAraAiuA)'h010m0rphiC1 [f] =s+df, Aut(j) :1, a; 6 HA} is smooth. On the other hand, we have a canonical projection 7r : LI -2 yV. By Sard—Smale Theorem, rr‘1(V) is smooth of dimension 2(g + k + 1) for generic V. Again, applying Sard-Smale Theorem to the projection rr"1(V) —> D(e) defined by (f, JA,V,\, 0;) -—> A, we can conclude (a). In order to prove (b), we first consider the stable compactification WWJZA, A, HA, V,\) as in Chapter 8. It follows from Lemma 9.0.49 that this is com- pact. We also reduce this moduli space and stratify the reduced moduli space by the same way as in the proof of Theorem 8.0.42. Note that Lemma 8.0.44 still holds for 75 this reduced moduli space since each J; is a fixed complex structure. On the other hand, we consider the bump function [3 in the Definition 9.0.48 as a flmction on Z ;. We can then assume that all singular fibers of Z; = (E(n), J;) -2 CCP1 lie in the support of l — fl. That implies that if (f, oz) has a bubble component, then the bubble component maps into either a singular fiber or a section. It follows that zero set of a should contain a singular fiber or a section. Therefore, we can conclude (b) using the same argument as in the proof of Theorem 8.0.42. [3 We end this Chapter with the splitting argument as in [IP3]. This shows how maps into X = E(n) split along the degeneration of E (n) It is also a key observation for gluing of maps into X and Y, which leads to the sum formula (0.5). Lemma 9.0.52 Let (fn,2n,an) be any sequence of (Jz,V,an)-holomorphic maps such that (i) each fn maps into Z ;n, (ii) each fn represent the homology class S +dF, and (iii) An -> 0 as n -—> 00. Then we have (i) ffl converges to a limit f : E —> Z0 = X UV Y and an converges to a, after passing to some subsequence. (ii) the limit map f can be decomposed as f1121—>X,f2:22—>Y, andf3:E3—+V where fl is a stable (J X, Vx,a)-holomorphic and f2 ( f3) is a stable (Jy, w) ( (JV, Vv) )-h010m0mhic (iii) fori = 1,2, each f,- tmnsverse to V with f,"1(V) = {pi}, where each p,- is a node of 2. Proof. (i) follows from Gromov Convergence Theorem and Lemma 9.0.49. Note that a = 0 near V C Z when a 6 Hz. Hence, Jan = J near V C Z. Therefore, we can apply Contact Lemma in [1P2] to conclude (ii). Lastly, (iii) follows from Contact lemma in [1P2] and lemma 3.3 in [IP3]. Cl 76 CHAPTER 10 Relative Invariants of E(n) In this Chapter, following [1P2], we define relative invariants for (X=E(n), V=T2, 93 1, A=S+dF, Vx, Hx) (Y=T2XS2, V=T2, g_<_1, A=S+dF, Vy) where Vx (W) is the restriction of V on Z to X (Y), H x is the parameter space in (0.4) and g is the genus of Riemann surfaces. As in Chapter 8, we fix the complex structure on X (Y) and we only vary the inhomogeneous term Vx (W) to define perturbed relative moduli space. In the below, we will not specify complex structures on X and Y in the notation of moduli spaces. We also assume that we always work with a finite good cover pp as in (0.1) without specifying it. For each V E y" we define the relative moduli space as Mzk+l (X, A, 71x. VX)! ={(f1jra)e Hg,k+1(X1A1HXrl/X) l f($k+l) E V) Allt(j) = I} Proposition 10.0.53 For generic V E y", (a) Mgk+1(X, A, Hx, Vx)1 is an oriented smooth manifold of dimension 2(g + k), and 77 (b) The Frontier of the map st x ev x h : MXHJX, A, Hx,l/x)1 —> .A—d-g,k+1 x X" x V (0.1) is contained in dimension 2 less than 2(g+k), where ev is the evaluation map of the first k marked points and h is the evaluation map of the last marked point. Proof. Since for each a E Hx, a = 0 in some neighborhood of V C X, J0, = J on that neighborhood. Therefore, (3) follows from Lemma 4.2 in [IP2]. On the other hand, the Hontier of (0.1) is the image of CMXk+1(X1A1HX1l/X)I C 7‘7.q,t+1(X . A. Hx. Vx) (02) under stabilization and evaluation maps, where (0.2) is the closure of Mgk+1(X, A, Hx,Vx) in M9,k+1(X, A, Hx,VX). In order to prove (b), we first re- duce the closure (0.2) under the reduction as in Chapter 8 and stratify the reduced moduli space by the same way as in the proof of Theorem 8.0.42. Similarly as in the proof of PrOposition 9.0.51, (i) Lemma 8.0.44 still holds, and (ii) if (f, a) has a bubble component, then the bubble component maps into either a singular fiber or the section 3. Each strata corresponds to one of the following types of stable maps : (i) f has some bubble components, (ii) f has two principle components 21 U 22 such that the image of 22 maps entirely into V, and (iii) f is neither of type (i) nor (ii). First, consider (f, a) which is of type (i). In this case, the zero set of or should contains singular fibers or a section. This reduces the dimension of possible parameter space for the corresponding strata at least 2. Next, it follows from Lemma 6.6 in [IP2] that those strata corresponding to (ii) is empty. Lastly, note that if f is of type (iii), then it has at least 3 principle components. Therefore, (b) follows from the similar dimension count as in the proof of Theorem 8.0.42. [:1 It follows from the above proposition and PrOposition 4.2 of [KM] that the image 78 of (0.1) gives rise to a rational homology class. We denote it by [Minds A, Hid] e H‘(‘M.,...; o) s H‘(X"; o) s WM 0) Definition 10.0.54 For 29 + k 2 2, 2 in H‘(K/l_g,k+1;Q), a1, - -- ,ak in H*(X’°;Q), and '7 in H...(V; Q), we define relative invariants by GWng+1(X, A,HX)(161 01, ° ' °1ak;C(7)) = (259011 /\'“Aakl®C(7))O[M;t+1(XiA,Hx)l where C(7) is the Poincaré dual of 7. Similarly as above, we set M;k+1(Y1A1VY)I = { (fij) 6 fig,k+l(Y1A1VY) I f(yl) E V? Aut(j) = I } (0'3) st x h x ev : Mgk+1(Y, A,Vy)1 —> VIA—9,1.“ x V x Y"c (0.4) where h is the evaluation map of the first marked point and ev is the evaluation map of the last k marked points. Remark 0.55 Since pg(Y) = 0, the relative moduli space (0.3) is the one in [IP2]. Here, we fixed the product complex structure on Y and we only vary inhomogeneous terms. However, for a given stable map after contracting all ghost bubbles, there is at most one bubble component which maps to some holomorphic section. Using the same argument as in Chapter 6 of [IP2], we can thus show that for generic V E yz, the Hontier of (0.4) is contained in dimension 2 less than the dimension of (0.3). On the other hand, for generic V 6 3’2. (0.3) is an orientable smooth manifold of dimension 2(g + k) + 2. Therefore, we can define relative invariants as in the Definition 10.0.54. In fact, this invariants is less refined than the relative invariants in [IP2], see also Appendix in [IP3]. 79 Finally, we set up some notations which will be used in the next Chapter. We set MV(X)1>:MV(Y), = UM;,k,+,(x,s + dlfflix), if M;,k2+,(Y,s + def), = (h x h)‘1(A) where the union is over all g1 + 92 == 1, k1 + k2 = 4, and d1 + d2 = d. This moduli space comes with the following maps : st' x ev : MV(X)1 x MV(Y)1 ——> U 7W“ x X"1 X Y"2 (0.5) h k1+k2=4 where st’ = a 0 st and a is the gluing map of the domain. For generic V, this moduli space is also smooth and the Hontier of (0.5) lies in codimension 2. On the other hand, we set Map1’4(Z,A)={(f,j,a)| [fl = A136 MIA, 0 E HZ} Note that MV(X)1 l: MV(Y)1 and M1,4(Z;,A,H;)1 are subsets of Map1,4(Z, A). Moreover, there are following commutative diagrams : MV(X)1:MV(Y)1 ——+ Map1,4(Z, A) t——— M1,4(Z;,A,H;)1 nee] em] .tx..[ , (0.6) U WM x X’c1 x Y"2 —i—+ WM x Z4 <—j— WA—M x Zfi where A 31$ 0 and the union is over all k1 + k2 = 4. Remark 0.56 Recall that 7W“ is a smooth finite cover of the DeligneMumford space defined by Prym structures and hence it has a universal farmly 171,4. The metric on 171,4 provides a smooth family of metrics on the domain of maps in Map1,4(Z, A). Therefore, we can define a weighted norm as in Definition 5.0.30 on Map1’4(Z, A) to make it Banach space. There is another way to define a topology on Mapl,4(Z, A) [IP2]. We can identify each j 6 M1,.) with (b : B -> 171,4, where B is a fixed smooth torus. The map 45 80 defines a complex structure on B by pulling back the complex structure on U1 ,4. In this way, we can identify Map1,4(Z, A) with the following space Map(B,Z XU1,4) = { (f,¢,a) I f X 43 I B —> Z XII—(1,4, [f] = 14,0 G Hz }. For C0 close maps C1 = (f1,¢1,a1) and 62 = (f2,¢2,a2), we can write C2 = expel (61 h) :8) and set dist(Cl,C2) = ||€||1,p+ llhll + llflll Taking the inf of the lenghts over all paths picewise of the above type, we can defines a distance and hence a topology on Map1,4(Z, A). 81 CHAPTER 11 Gluing Theorem In this Chapter, we will establish a family version of Gluing Theorem as in [IP3]. Using this, we will show the sum formula (0.5). We fix a generic V E yz as in Chapter 9 and 10. In the below, we will not distinguish a 6 Hz and its restriction a; to Z; and use the same notation for them. Theorem 11.0.57 (Gluing Theorem) Let Co = (fo,20,ao) be in MV(X)1 )1: MV(Y)1. Then there are A0, 60 > 0, and a small neighborhood W of Co such that we have a continuous family of maps T; : W -1 M1,4(Z;, A,H;)1 for [AI < A0, which satisfies (i) T; is an injective smooth map from W into M1,4(Z;, A, A) 1 (ii) T;(Co) converges to Co as A -—2 0 (iii) if (f, )3,a) in M1,4(Z;,A,H;)1 and d(Co, (f,2,a)) < so, then (f,)3,oz) is in T;(W), where d is the distance of Map1,4(Z, A) defined as in Remark 10. 82 Proof. The proof of this theorem consists of 3 steps. In the first step, following [IP3] we construct approximated maps into Z ;, each of which is associated with an element of W. These are nearly (J ;, V;, a)-holomorphic. In the second step, we use the Inverse Emotion Theorem to perturb these approximated maps to truly (J ;, V;, a’ )- holomorphic maps. This process defines the map T;. The required analysis in this step is same as those in the proof of Proposition 5.0.37. In the last step, we show that the map T; has the desired properties as we stated. Step 1 Let (f1, 21) and (f2, 22) be the two components of (f0, 20). Then 230 lie in the image of a : Mg.,t.+1 >< Mm.k2+l —* MIA where a is defined as in Chapter 3 and 2; 6 fight,“ for i = 1, 2. Let U be an open neighborhood of 20 in 7171),. We may assume that the intersection W = U flIm(0‘) is smooth. Let N be the tubular neighborhood of W in V”. There is a trivialization N z W x D, where D C C is some disk. Let N = fil’kln’ where 31,1: —1 W”. is the universal family. Denote by N the set of nodes in the fiber of N and let V(N) be some fixed neighborhood of N in N. We can choose local coordinates z, w, as well as (t, p) E N, on V(.N) such that the fiber of N over (t, u) is given by (t,z,w) with zw = )1. Now, we set W = st‘1(W). This is an open neighborhood of Co. By shrinking V(N) and W, if necessary, for any C = (f, E, a) E W, we can assume the followings : 1. f(23 fl V(./V)) C U, where U is the neck region (0.3) of Z. 2. Let f1 : 21 —1 X and f2 : 22 —> Y be the components of the reducible map f. 83 It then follows from Contact Lemma in [IP2] that f1(t,z) = ( v(z), a2: + 0(Izl2) ) and f2(t,w) = ( v(w), bw + 0(le2) ) with 1/2 < [a], [b] < 2. 3. For p E 21\V(./V) (q E 22 \ V(JV)) with f1(p) (f2(q)) in the neck region of Z, we have f1(p)=(v(P),$(p),0)(f2(Q)=(v(Q).0.y(q))) with Im(P)|(ly(Q)l)Zeo where so is a uniform constant which doesn’t defend on C E W. For each (t,p) E N, denote by 2; = 2:1 U 2‘2 the fiber of (t,0) in N. We also denote by East) the fiber of (t, p) in N. We define X -side 25;, (Y-side 22””) of 2(n“) be the set of all points p E 2“,“) which satisfies (109,211) < d(p)2t2) ( d(P,2t2) < (1(1), 2:1) ) When p is in X ~side (Y-side), denote by rrx (p) (rry (p) ) the unique point in En (2,2) such that €109,211) = d(p. rx(p)) ( d(p. 2.2) = 61(1). 7w(p)) )- Let r be the distance function to N in N. We define a bump frmction C. with C¢(r) = 0 ifr 2 2e, B€(r) =1 ifr g e, and [dfl£| S 2. Definition 11.0.58 For each C = (f,2,oz) E W and A aé 0, we define C; = (F;,E;,oz) as follows : 1. 2; = (t,/.t), where u = A/ab and 2 = (t,0), and 84 2. let u be a local coordinate of V centered on the image of the node of 2 and we define F; : 2; —-> Z; by [ ((1 ‘5u)v(2), 53(2), A/flzll 0” { (t,z,w) 6 WM I M Z lwl } (firm 0 f 1 0 77x on Bf O Supp(flp) FA = i ((1 —fl,.)v(w), A/fllwla I?) 0" { (t,z,w) 6 WM l M Z M } \ ¢Y,.\ 0 f2 0 7W on 23), fl Supp(fiy) when: 57(2) = a3(1+(1--5u)0(|15|)), 31(10): bw(1+(1—flu)0(|w|)): flu is the bump function defined as above with e = [p], and (bx; ((by;) is the holomorphic map which extends the following map (0,3,0) -* (ma/Va?) (01.0.3!) -* (UM/31.31))- on the ncek region (0.3) of Z As in Chapter 5, we use the metric on N 1,4 and metric g z on Z to define pointwise norm, weighted norms || ~ II”, and Il - H,, as in Chapter 5. Note that the pointwise norm ldF;| is uniformly bounded. Recall that Va, = (I + JKa)‘1V. Lemma 11.0.59 For some A0 > 0, there is a uniform constant c such that for any CEWandlAl(C;) = 5.1.. F;-Va. Since rrx is holomorphic on the region 2f flSupp(fl,,) and (bx; is also holomorphic, we have (C;) =d¢x,;(U-Jaf1 -— Va )d’l'rx + (Jo, — J)dF; + d¢x,,\(J — Ja)df1 dfl’x + d¢x,; Va Cl’frx -- Va 85 Therefore, we have |(C;)| 5 CIA] on the region 2? fl Supp(B”). On the region { (t,z,w) E V(./\() | [z] 2 |w| } we can assume F; maps into the region at which a = 0. Using zw = p = A/ ab, we have 29') F; = 61(1- flu)”: 025.1[0 - 600201611151 (1 + (1 - 2100020)"2 51[(1 - 600201) All terms in 51F; are bounded by |A|, except for the term involving 51 v. On the other hand, 5] v is the V-component of Cy f1 and hence UK, where Vf1(Z) = V(f1(z), z). Therefore, we have |¢(CA)| S C( IV}: — VXI + lugl + W) S CIAI since the normal component VN vanishes along V. Using the same arguments on the other regions, we can conclude that |(C;)| g c|A| on 2;. This implies the lemma. II] Step 2 As above, we set ( F,Z,a) = EJGF — Va. In this step, we perturb C; = (F;, 2;,01) to C; = (F;, ((,a’) such that (Cf\) = 0. For doing that, we consider the linearization Dc; of at C; Dc, : Ll'P(F;TZ;) 6 T2,‘Mi,4 e Tan —» U”( 1133,1632» ) (0.1) where L19 and L10 are defined by weighted norms as in Chapter 5. Using the Inverse Function Theorem, we will show that there exits a unique (6, k, B) such that (i) ( expcx(£, k, fl)) = 0 and (ii) the projection of (§,k,fi) to the kernel of De A T).(C) = eXPc;(§ikim- with respect to L2—inner product is zero. We then define a gluing map by Let q : LI’P(F;TZ;) 6Tg,m1,4 6TaH —> Ker(Dc,) be the projection with respect to L2-inner product. The following lemma is similar to Lemma 5.0.36. 86 Lemma 11.0.60 There edits A0 > 0 such that if |A| < A0, then for any p > 2, C; with C in W, (50,ko,,60) in Ker(Dc,) and r) in LP(A3310(F;{TZ;) ), there is a unique (5, k, B) which satisfies q(€, Its) = (50. 1901,60): Dc,(£, kifl) = n (0.2) |l€|l1,p+ llkll + llflll S cmax{ ||€o||1,p+ ”’60“ + llfioll, llnllp} (03) where c is a uniform constant. Proof. This proof is similar to that of Lemma 5.0.36. We first show that for sum- ciently small A and any C E W, Coker(Dc,) = 0. Suppose not. Then there exits a sequence { (Cm An, 7771)} such that An —2 0 and DICn.A..)(7l") = 0 with “17“,, = 1, where (Cu, An) denotes the approximated map determined by Ca and An. We can assume C7, —> C and (Cm An) —+ C. Let C = (f, E,a). It follows from the standard elliptic estimates that 17,, converges to some 17 outside of a node of )3. Since DEM) = 0 and Coker(Dc) = 0, we have n = 0. This implies that 17,. —> 0 on the complement of neck region defined as in (0.5). On the other hand, note that L3.“ (17,.) = 0, where Fn = F(c....\..)- It follows from Lemma 5.0.34 that llnnllp —+ 0. This contradicts to our assumption [Infillp = 1. Therefore, for sufficiently small A and any C E W, Coker(Dc,) = 0. Consequently, there exits (5, k,fi) with Dc,(5,k,fi) = 17. Let 5 = (5, 19,5) and 50 = (50, kg, 30). Then 5’ = 5 — q(5) + (0 satisfies (0.2). Uniqueness is obvious. Next, we show the estimate by contradiction. Suppose not. Then there exits (in, kmfln) such that (i) ||§n||1.p+ llknll + llflnll = 1, (ii) IIDn(€n, kn.fin)llp —’ 0, where Dn = D(c..,A..)i and (iii) ||5no||1,p + Ilknoll + Ilfinoll —> 0. We can assume that the approximate map (Cu, An) converges to (C, A) = (F, a). By the Sobolev Embedding Theorem, we can also assume 5,, converges to some 5 and 5,,0 -—> 0 both in Ll'z-norm. We can further assume that (kmfin) converges to some (kfl) and (humflno) —-1 0. 87 Note that D0,;(5, k,fi) = q(5,k,B) = 0, and hence we have (5, k,fi) = 0 by unique- ness. Together with (i), (k,fl) = 0 implies that ||5,,||1,p —+ 1. It then follows from Lemma 5.0.33 that ||5n||1,2 is uniformly bounded away from zero. This is impossible since 5,, —2 5 in Ll'z-norm and 5 = 0. Therefore, we have a contradiction. D As in Chapter 5, we set L1,? ={ (C1 Aaérkr 18) IC 6 W1 (£1 k1 fl) 6 Ll,p(F;TZ/\) 69 TE;-A—4-l,4 G; TaH } P ={ (C’A1601k03fi0) [C 6 W1 (€01k01160) E Ker(DC;) } On the other hand, let Map1,4(Z ;, A) be a subspace of Map1,4(Z, A) which contains all maps into Z; and set If = { (c', M) | c’ e Map1,.(Z.\,A) with C’ = (Eta), n e H(A?§,F*TZ.\)-} and define a map E : L11? —+ LP x P by E(CiA. (6.19.6) = ( ¢(expc;(£.k,fi) ).q(€.k.fl)) The linearization of E at (C, A, 0) is the map DE : L11P(F;TZ;) e T2,M1,4 ea TaH -» H(Agirflm x Ker(Dc,) (5.19.15) —’ (Dc.(€ikifi),Q(E,kifl)) By Lemma 11.0.60, it is an isomorphism with uniformly bounded inverse. Therefore, by the Inverse thction Theorem there exits e > 0 such that E is a local diffeomor- phism from the open set {(C.A.§.k.fi) 6 L1""l l|€||1,p+||k||+llfill< 6} (04) onto its image. It then follows from Lemma 11.0.59 that for any (C,A) with |A| sufficiently small, there is a unique (5 , k, H) such that 3(5, 16,3) = (C, A, 0,0). (0.5) 88 Definition 11.0.61 Fix A with |A| < |A0|. For each C E W, let (5,k,fl) be given as in (0.5). We then define T; : W —-+ M1,4(Z;)1 by C —> expCA(5, k,B). Similarly as in Chapter 5, we have the following expansion ‘1’( eXPc;(€. 16, 5)) = (I)(CA) + Dc.(€. 16. B) + Hc.(€. k. H) where ||Hc,(5,k,fi)||p S c( ||5||1,p + “k“ + Hfll] )2 for some uniform constant. Using this expansion and the estimate in Lemma 11.0.60, we can conclude that (5 , k, B) in (0.5) satisfies ||€||1.p + llkll + llfill S CW (06) for some uniform constant. Step 3 As a consequence of the Inverse Function Theorem, the map T; is smooth. It also follows from (0.6) that T;(C) —+ C as A —1 0. In the below, we will show the injectivity of T; and (iii) of Theorem 11.0.57. Denote by Q the orthogonal complement of P withe respect to L2 norm. For each fixed A, we also denote by A; the set of all appoximated maps into Z ;. As in the proof of Lemma 11.0.60, each Coker(Dc,) = 0 when [AI is small. Hence we can deduce that TciLl’p = TciAA G9 Qle, Let Q; = Uc Qlc; and denote by exp : Q; —-+ Map1,4(Z;,A) the exponential map defined by (C, Aiéo, k0. 50) —* eXP(c,A)(§oi koifio). (0-7) 89 Now, fix a path Q starting C and let (5 , k, fl) be the tangent vector at t = 0 of the corresponding approximated maps (C, A). Using parallel translation we can calculate d a; exp(ch’\)(€0’ ko’flo)lt=0 =(gik1fi)+(€01k0130). Therefore, there exits so > 0 such that for all small [Al the exponential map (0.7) is a difieomorphism from some neighborhood of zero section in Q; onto (so-neighborhood of A; in Map1’4(Z;, A). Together with definition of T; and (0.6), we can conclude that T; is injective. On the other hand, if (f, 23,01) in M1,4(Z;, A, H;) is close to Co, then it is in the «so-neighborhood of A;, which implies (f, 23, a) in T;(W). C] Now, we are ready to prove the sum formula (0.5). Proposition 11.0.62 1 H(t) = ——1§F(t) + 2F(t)C(t) Proof. By definition of generating functions H (t), F (t), and G (t), it suffices to show that GWfiWS + dF)(1/1(1.4);4; F4) = 2: GWJSTS + le) (20%) - 115) d1+d2=d where 0(d2) = 214012 as in Chapter 6. We can choose a submanifold F,- C Z for i = 1, - ~ - , 4 which is in general position with respect to evaluation maps such that for i = 1, 2 each FflX ( anY ) represents a fiber class in X (Y), and each F,- flZ; represents a fiber class in Z; = E(n). On the other hand, without loss of generality, we may assume there is a submanifold K in HM representing Poincaré dual of 1[)(1’4);4. We may also assume that K is in general position with respect to stabilization map. Let A 74 0 be generic as in Proposition 9.0.51. Now, consider the cut-down moduli space M; which consists ofall (f, (j; {xi}), a) in m1,4(Z;, A, H;) with f(a:,-) in F,- and 90 ’fl#, .. -. st( j ) E K. It then follows from Proposition 9.0.51 that M; is finite and Aut( j ) = I. In fact, by definition for generic A we have [St X €‘U(M,\)] = GWE?(ZA, A)(1/J(1’4);4; F4) = CW3," (S + dF)('lp(1,4);4; F4) (0.8) where the second equality follows from Lemma 9.0.49 and Proposition 3.0.15. Similarly as above, let Mo be the cut-down moduli space which consists of all ( (f1. 0., {z.}).a). (f2, 0., {t})) ) 6 Moo. 3; MV(Y)I such that f1(a:,-) E F.- and f2(y.-+1) E Fi+2 for i = 1,2, and 0(j1,j2) E K, where a is the gluing map of the domain as in (0.8). It follows from Pr0position 10.0.53 and Remark 10 that M0 is finite. Moreover, (j1,j2) is an element of either 7W”; x 370,3 or 1703 x M7113, since there are two marked points at each X -side and Y-side. Note that o‘(i/J(1,4),4) = 0, where a : 171,3 x 11710;; -> III-1,4. Therefore, we have Mo c U Mg,(x, s + le, 21..) >5 M{3(Y, s + dgF) (0.9) d1+d2=d Together with routine dimension count, (0.9) implies that [st’ x ev (M0)] = Z Gngfns + le)(F2;C(V))GW1‘,’3(S + sz)(1/1(1,3);3;C(Pt) = F2) d1+d2=d d1+d2=d = Z Gng;(s + diF) (20(d2) — 112) (0.10) dr+d2=d where the second equality follows from Lemma 9.0.49, Proposition 3.0.15, and defin- ition of relative invariants, while the third equality follows from TRR for T2 x 32. It remains to show that i.[st' x ev (M0)] = j...[st x ev (M;)] in Ho(-./\711,4 X Z4) (0.11) 91 where i and j are inclusions as in the diagram (0.6). By Lema 90°52, as ’\ _’ 0 any sequence (fxijxiax) E M; converges to a limit (fij, a). AS above, since there are two marked points on each X -side and Y-side, j lies on one of the following images of the gluing maps : 01:M0,3 X Mp3 —* Mp4, Uzi/V113, X M03 —* Mp4, 01‘ 0'3 1 M03 X M13 X Mo’s —* Mp4 Since j also lies on K , and both 03(zp(1,4);4) and ugh/«1,4,0 are trivial, (f, j, a) 6 M0. Hence, there is a bijective map between M; and M0 for small |A| by Gluing Theo- rem. Moreover, they are homotopic in Mapl,4( Z, A). Therefore, by the commutative diagram (0.6), we can conclude (0.11). [I] 92 CHAPTER 11 Appendix — Relations with the Behrend-Fantechi Approach Behrend and Fantechi [BF] have defined modified GW invariants for Kahler surfaces using algebraic geometry. While their techniques are completely different from ours, the definitions seem to be, at their core, equivalent. In this appendix we make several observations which relate their approach to ours. This is necessarily tentative because the paper [BF] is not yet available; we are relying on the terse description given in [BL3] In algebraic geometry, the virtual frmdamental class [M9,],(X, A)]Vir is obtained from the relative tangent-obstruction spaces together with the tangent-obstruction spaces of Deligne-Mumford space MM. Behrend and Fantechi modified their ma- chinery, intrinsic normal cone and obstruction complex, by replacing the relative obstruction space H1(f"TX) by the kernel of the map H1(f*TX) —» H2(X, 0) (A.1) defined by dualizing of the composition H°(X,S22) —-+ H°(f‘§22) —» IIO(f"‘Q1 <8) f‘fll) ——> H0(f"fl1 ® 91). (A2) 93 1%....2.‘ .~ - In order for their machinery to work, the map (A. 1) is of constant rank — in particular surjective — for every f in M9,)..(X, A) [BL3]. Composing (A2) with the Kodaira.- Serre dual map, we have H°(X, n) —+ H°(f"ol ® 521) —» H1(f"‘TX). (A.3) This map is given by B —> K5 dfj. Proposition A.63 Let (X, J) be a Ka'hler surface and A 6 H1'1(X,Z). Then the family moduli space M2,,(X, A) is compact if and only if the map (AI) is surjective for every f in _M—g,k(X , A). Proof. By Theorem 2.0.12 M2,,(X, A) consists of pairs (f, a) with f E M9,),(X, A) and with the image of f contained in the zero set of a; the latter condition means that Ka = 0 along the image, so KO, dfj = 0 for all (f, a). As usual, M9,),(X, A) is compact by the Gromov Compactness Theorem. Now, suppose (Al) is surjective. Then by duality (A.3) is injective. This implies or = 0 and hence MZAX, A) = Mg,k(X, A) is compact. Conversely, suppose for some f E M9,),(X, A) there is a B in the kernel of (A.3). Then setting a = ,6 + B we have 51f = tKadfj = 0 — and hence (f, ta) 6 MZAX, A) — for all real t. That means that MZAX, A) is compact only when (A.3) is injective or equivalently when (A1) is surjective. C1 The map (A.3) is directly related to the linearization Operator of the (J, a)- holomorphic map equation. Suppose that A is (1, 1) and that the family moduli space MZAX, A) is compact as in Pr0position A.63. Consider the linearization of the (J, a)—holomorphic map equation the (J, a)-holomorphic map equation as given in (0.6). Since J is Kahler, 94 the linearization reduces to 111(5) = V5 + JVEJ' Lf6Lo : 9°(f*TX)6H—>Qo'l(f*TX) where 110(5) = '2Kfldfj In fact, this L f is exactly (twice) the Dolbeault derivative 5. Therefore, Ker(Lf) are Coker(Lf) are identified with the Dolbeault cohomology groups H0 ( f *TX ) and H 0'1 ( f *TX ) , respectively. Proposition A.64 Under either of the two equivalent conditions of Proposition A.63 there are natural identifications H1(f*TX) 2 Ho’1(f'TX) and H°(X, (22) c: H under which identification the map is identified with (A.3) with Lo : H —1 Coker(Lf). By Proposition A.63 this map is injective if and only if the family moduli space MZAX , A) is compact. Proof. This follows directly by comparing the formulas for L0 and (A.3). Altema— tively, we can compute the linearization from scratch as follows. Given 5 6 9°( f ‘TX ) , there is a family of maps ft with f0 = f and M = 5. It follows from Proposi- dt [1:0 tion 1.0.6b and (B, A) = 0 that o: # _ d — . — _ t=o/2;ft (fl) — a tzoLla-lfhKBfttJ) —/2;