Genus Zero Gromov-Witten Invariants

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1 Genus Zero Gromov-Witten Invariants Roland Grinis June 11, 2012 Contents 1 Introduction 1 2 Moduli Spaces of Stable Maps and Curves 3 3 Kuranishi Structure on M 0,k (X, β; J) 9 4 The Virtual Moduli Cycle 17 5 Gromov-Witten Invariants 21 6 Quantum Cohomology 27 7 Mirror Symmetry 39 8 Appendix: The Polyfolds of Hofer-Wysocki-Zehnder 47 1 Introduction This essay presents some aspects of the Gromov-Witten theory from the point of view of symplectic topology. Symplectic manifolds are smooth even dimensional manifolds admitting a symplectic structure: a non-degenerate closed 2-form. There is not much one can say about the global structure of a symplectic manifold by working locally, which is in contrast to Riemannian geometry where we have local invariants, like the curvature for example. In fact, locally, symplectic manifolds are classified by their dimension (Dardoux s theorem). In 1985, seeking global invariants, Gromov has introduced J-holomorphic curves to study the geometry of symplectic manifolds (these are the generalization of holomorphic curves to the almost complex world). This technique is still very popular and is at the heart of much of the research concerning symplectic topology today. Using J-holomorphic curves defined on a compact Riemann surface of a given genus and representing a given homology class one can obtain rational numbers which are global invariants, called Gromov-Witten invariants, of a closed symplectic manifold we are interested in; they are moreover invariant under continuous deformations of the symplectic structure. Defining them rigorously presents some analytical challenges depending on the geometry of the chosen symplectic manifold. In this essay, we have chosen not to restrict ourselves with the class of closed symplectic manifolds we would like to work with, but we have decided to focus on the genus zero case (as a minor simplification). Also, the applications we will describe concern mainly the genus zero case as well. Let us say a few words about the contents of the present essay. In section 2, we discuss some properties of J-holomorphic curves in symplectic manifolds (this is in parallel with the integrable This essay was written as part of the assessment for the course M3R (3rd year undergraduate project) at the Department of Mathematics, Imperial College London. roland.grinis09@imperial.ac.uk 1

2 case), describe the important bubbling phenomenon and introduce the moduli spaces of stable maps stating some of their topological properties. In section 3 we introduce the reader to Kuranishi spaces, these were elaborated by Fukaya and Ono in 1996 [6] and represent today an important tool while working with moduli spaces of J-holomorphic curves (in open Gromov-Witten theory as well [7]). We proceed by studying some properties of the Cauchy-Riemann equation and its linearization. We put then a Kuranishi structure on the moduli space of stable maps (a rather detailed discussion on gluing J-holomorphic curves in presence of an obstruction space is presented). In section 4 we explain how to obtain the virtual moduli cycle using multivalued perturbations in order to get transversality (this technique is widely adopted beyond the Kuranishi set up, although a recent work of Joyce [16] suggests a different approach). In section 5 we finally reach the definition of Gromov-Witten invariants and present their formal properties encoded in the axioms of Kontsevich and Manin introduced in 1994 [19]. In section 6 we introduce the reader to the basics of quantum cohomology (small and big) which is characterized by a new product structure on the singular cohomology: a quantum correction of the classical cup product using Gromov-Witten invariants. This leads to the proof of the beautiful Kontsevich s recursion formula counting the number of degree d rational curves in the projective plane through 3d 1 generic points. We spend more time with this example trying to understand some aspects of these new techniques in enumerative geometry. In the last section, we tell the reader about the rôle played by Gromov-Witten invariants in the mysterious relationship which exists near the compactification point of the (complexified) Kähler moduli space of a Calabi-Yau threefold and some of the compactification points of the moduli space of complex structures on a mirror Calabi-Yau threefold. This relationship, which roughly speaking compares the symplectic geometry of a Calabi-Yau manifold against the complex algebraic geometry of its mirror, is one of the manifestations of Mirror symmetry: a phenomenon which finds its roots in String theory and represents one exciting area of research in geometry today. In the appendix, we come back to the analytical problems involved in the construction of the virtual fundamental class of the moduli space of stable maps and describe the alternative approach due to Hofer, Wysocki and Zehnder [14]. They generalize the classical Fredholm theory (which fails in presence of bubbling) by changing the notion of smoothness in infinite dimensions such that a version of the implicit function theorem still holds and introduce their new spaces: polyfolds (using the rather recent approach to orbifolds via groupoids due to Moerdijk, see [1]). The moduli space is then presented as the zero set of the suitable generalization of a Fredholm section of a (strong) polyfold bundle. However, multivalued perturbations are still needed to achieve transversality. This theory is very powerful and conceptual and it is hoped that it will generalize beyond the problems involving moduli spaces of J-holomorphic curves. We have tried, in this essay, to focus on explaining the constructions and give precise definitions as well as provide the reader with some intuition behind the problems encountered (to the extend of my understanding of the latter). It was not our aim to give full proofs and work out all the details: works in the literature doing this (references to a small subset of which are available in the bibliography) constitute several hundreds of pages! I would like to thank my supervisor Richard Thomas who has spent a lot of time trying to explain to me the geometrical intuition behind the various analytical and algebraic constructions arising in Gromov-Witten theory and helping me to sort out all my numerous confusions. It was a great experience for me to learn from him the way he thinks and the way one should tackle problems in geometry. I would like as well to thank Dominic Joyce for giving a very inspirational course on symplectic geometry in autumn 2011 on TCC which has influenced me a lot while I was working on this project. I want to express my deep gratitude to Tom Coates who has initially introduced me to symplectic geometry as well as to some ideas in Mirror symmetry while I was working on a summer project under his supervision. Also, many thanks to Oana Irina Pocovnicu and Hans-Joachim Hein for helping me with some analysis involved in this project. I dedicate this project to my parents. Their infinite love, support and care are the most marvelous things I have. They have made the three years of my undergraduate education at Imperial the most enjoyable period of my life which I am closing now with this essay. 2

3 2 Moduli Spaces of Stable Maps and Curves This section is devoted to the study of the compactification of the moduli space of J-holomorphic spheres representing a given homology class. We start first with a few general remarks. Let (X, ω) be a closed symplectic manifold of dimension 2n and J J (X, ω) a compatible almost complex structure. We also set H (X) := H (X; Z)/Tor. Given a Riemann surface (Σ, j), j a complex structure, a smooth map u : Σ X is called a J-holomorphic curve if its differential satisfies the nonlinear Cauchy-Riemann equation: J (u) := 1 2 (du + J du j) = 0 Γ(Σ, u T X J 0,1T Σ) (2.1) In the next section we will discuss in more details the properties of this elliptic equation (which means that its linearization will be Fredholm) while constructing Kuranishi neighborhoods. The case when Σ = S 2 = C { } is the Riemann sphere will be the one mainly considered. We say that u is multiply covered if there exist a J-holomorphic u : S 2 X and a holomorphic branched covering ϕ : S 2 S 2, deg ϕ 2, such that u = u ϕ. It is called simple otherwise. Denote by z := (z 1,..., z k ) a vector of k distinct points on S 2 and fix β H 2 (X). If β = 0 we assume k 3 unless otherwise stated. Consider the following space of parametrized J-spheres: M 0,k (X, β; J) := { (u, z) : J (u) = 0, u simple, u ( [ S 2] ) = β } The group G of Möbius transformations (G = PSL(2, C) if we view S 2 = CP 1 ) acts 3-transitively on S 2 and on M 0,k (X, β; J) by reparametrizations: for φ G we have (u, z) = φ(u, z ) if u = u φ and φ(z ) = z. We write [u, z] for the orbit and denote the quotient space: M 0,k (X, β; J) := M 0,k (X, β; J)/G This is the moduli space of simple J-holomorphic spheres with k marked points representing the class β and it carries a natural evaluation maps that we will use to define Gromov-Witten invariants: ev : M 0,k (X, β; J) X k, [u, z] (u(z 1 ),..., u(z k )) If we set the problem (2.1) in terms of Banach manifolds (see next section), using elliptic regularity, the infinite dimensional version of the implicit function and Sard-Smale theorems we can show: Theorem 2.1. For a generic choice of J J (X, ω) the space M 0,k (X, β; J) is an orientable smooth manifold of dimension dim M 0,k (X, β; J) = µ(β, k) := 2n + 2 c 1 (T X) + 2k 6 Moreover for two choices J 0, J 1 J (X, ω) the manifolds M 0,k (X, β; J 0 ) and M 0,k (X, β; J 1 ) are cobordant. For the proof we refer the reader to the excellent book by McDuff and Salamon [22], chapter 3. To define Gromov-Witten invariants we want the evaluation map to represent a homology class in H µ(β,k) (X k ; Q) independent of the almost complex structure. In view of the cobordism result above this is possible if M 0,k (X, β; J) happens to be compact, which in general is not true. Nevertheless if one impose some conditions on X, for example if we assume X is Fano, then one can prove that the boundary M 0,k (X, β; J) \ M 0,k (X, β; J) of the compactification (see later) of M 0,k (X, β; J) will be of codimension 2 and hence the evaluation map will represent a pseudocycle of expected dimension (see [22], chapter 6) for a generic almost complex structure. Unfortunately this technique does not generalize well to the cases when we have a class β H 2 (X) with J-holomorphic representatives and β c 1(T X) 0; this causes in general an excess of dimension for the boundary. To overcome this problem we will explain in sections 3 and 4, following Fukaya and Ono [6], how to put a Kuranishi structure on M 0,k (X, β; J) and construct, using multivalued perturbations, a virtual moduli cycle representing the homology class we seek. In the appendix we describe an alternative approach using more sophisticated functional analytic methods. β 3

4 To begin, we would like to understand the compactification of the moduli spaces of J-holomorphic spheres. We recall that J J (X, ω) gives rise to a Riemannian metric, J = ω(, J ) which induces a Hermitian structure on Γ(S 2, u T X J T S 2 ). Hence for our map u : S 2 X we define the energy: E(u) := 1 du 2 J 2 1 S 2 The volume form 1 depends only on the complex structure j on the Riemann sphere as du = du j. The important point is that we can bound the energy from above and from below. The upper bound comes from the symplectic structure: in local coordinates z = x + iy equation (2.1) becomes x u = J y u which implies that x u, y u J = 0 and x u J = y u J. This gives us locally du 2 J 1 = 2 xu 2 J dx dy = 2ω( xu, J x u)dx dy and hence we obtain: E(u) = u ω (2.2) S 2 This expresses the L 2 -norm of the derivative in terms of the topological data ω since we assume β u ( [ S 2] ) = β. The lower bound comes from the following apriori estimate (which depends this time on the almost complex structure): Lemma 2.2. Let (X, J) be a compact almost complex Riemannian manifold, r > 0 and u : B r (0) C X a J-holomorphic curve. Then δ > 0 such that: du 1 < δ du(0) 2 8 B r πr 2 du 1 B r For the proof see [22] page 80. This result tells us that curves with energy below some positive constant > 0 should collapse to a point. Indeed writing S 2 = C { }, then given any point z C and r > 0 the apriori estimate above gives us a positive δ > 0 such that E(u) < δ du(z) 2 const/r 2 0 as r hence u must be constant. We let then := inf { E(u) > 0 : J (u) = 0 } and obtain that for any non-constant J-holomorphic sphere: E(u) > 0 (2.3) The following two theorems, proved in [22] on pages 22 and 73 respectively, draw a parallel with the integrable case: Theorem 2.3. (Unique Continuation). Suppose u, v C 1 (B ε, R 2n ) satisfy equation (2.1) for an almost complex structure J of class C 1. We conclude that u v if u v = O(r k ), k 1 and 0 < r < ε B r This result allows us to conclude that if u, v : S 2 X agree to infinite order at some point z S 2 then they must agree on a set which is both open and closed and as S 2 is connected we can conclude that u v. Theorem 2.4. (Removal of Singularities). If u : B ε \ {0} X is a J-holomorphic curve with E(u) < then u extends to a smooth map B ε X. This theorem suggests us that we should consider a J-sphere u : C { } X as a J-holomorphic curve u : C X with the map z u(1/z) extending smoothly over zero. Now we turn ourselves to the study of limits of sequences u ν : C { } X of J-spheres. We say that such a sequence converges if both z u ν (z) and z u ν (1/z) converge in the C -topology (i.e. uniform convergence of all derivatives on compact subsets). We have mainly two cases. The first one is when the derivative is uniformly bounded, then we have the following useful result: Theorem 2.5. Let u ν : C X be a sequence of J-holomorphic curves with sup ν du ν L (K) < for all compact K C. Then {u ν } has a subsequence converging in C to a J-holomorphic curve u, moreover E(u ν ) = E(u). 4

5 We refer the reader to the appendix B.4 of [22] for the proof of this fact via elliptic bootstrapping where actually only an L p, p > 2, uniform bound is assumed. We should point out at this stage that if our sequences {u ν } M 0,k (X, β; J) were uniformly bounded in W 1,p -Sobolev norm with p > 2 then the discussion above would have enabled us to conclude on the compactness of M 0,k (X, β; J). Unfortunately the energy identity (2.2) gives only a bound on the W 1,2 -norm and the elliptic estimates do not hold. Hence a phenomenon of bubbling off can occur: we have du ν L with sup ν E(u ν ) <. By reparametrizing with Möbius transformations in the isometries subgroup SO(3) PSL(2, C) so that we do not loose energy, see [22] chapter 4.2, we may assume that we have a sequence of J-holomorphic curves u ν : C X with a sequence of points {z ν } C converging to 0 such that: Choose an arbitrary ε > 0 and let v ν : B εcν with 1 = dv ν (0) J = dv ν L, E(v ν ) = c ν := du ν (z ν ) J = du ν L X, z u ν (z ν + zc 1 ν B εcν v ν ω = ). This is a J-holomorphic curve B ε(z ν ) u ν ω E(u ν ) Hence by theorem 2.5 we obtain a subsequence, still denoted {v ν }, converging in C to a non-constant J-holomorphic curve v : C X with finite energy which we compute as follows: fix any R > 0 large, then: v ω = lim v ν ω = lim u ν ω lim inf u ν ω (2.4) B ν R B ν R B R/cν (z ν ) ν B ε the last inequality comes from the fact that c ν ; the energy E(v) is obtained by letting R. We see that the energy of v is approximated by the energy of u ν Bε for an arbitrary ε > 0. Now we have to consider the map v(1/z) on C \ {0}, this is a J-holomorphic curve with finite energy since the Möbius transformation φ(z) = 1/z SO(3) is an isometry and hence by the Removal of Singularities theorem it extends through zero to a J-holomorphic curve on C. Hence v is a J-sphere and it is called a bubble. The main point is that v ν (1/z) is not converging in C to v(1/z) and we are unable to conclude that {u ν } M 0,k (X, β; J) converges to a J-sphere. In fact a more careful rescaling argument will show that the limit is a nodal curve with several components (bubbles). Nevertheless if we consider curves with minimal energy only, then bubbles will not have enough energy to appear. Hence we can obtain compactness in the following special case: Theorem 2.6. Fix any J J (X, ω) and pick β H 2 (X) with J-holomorphic representatives such that β ω =. Then the moduli space M 0,0(X, β; J) is compact with the quotient C -topology. Proof. Consider a sequence {u ν } in M 0,0 (X, β; J). Then as in the above argument construct the reparametrized sequence { v ν (z) = u ν (z ν + zc 1 ν ) } (we don t assume c ν ) with a subsequence converging in C to a non-constant J-holomorphic curve v : C X of finite energy and dv(0) J = 1. We note that on C \ {0} we have v ν (1/z) v(1/z) in C. We need to show that this is the case near 0 as well. We proceed by contradiction: assume w ν (z) := v ν (1/z) does not converge uniformly in an arbitrary neighborhood of 0 and hence repeating the above argument with z ν = 0 we construct a non-constant bubble and this time c ν so we can use the estimates (2.4) to obtain that for an arbitrary large R > 0: lim inf v ν ω lim w ν ω ν C\B ν R B 1/R which implies that the limit υ is constant by the energy identity (2.3), and we get a contradiction. We finally describe the elements of the moduli space M 0,k (X, β; J). From the above discussion we expect limits of J-spheres to be nodal spheres which might be multiply covered. This is made precise by the definition of a stable map (due to Kontsevich [18]). Definition 2.7. A stable map of genus zero with k marked points representing a homology class β H 2 (X) is a simply-connected compact marked curve (C, z) with a map u : C X such that we have inclusions ı i : C i = S 2 C with u i := u ı i a J-sphere, C has only ordinary double singularities, the marked points are smooth, if u i const. then this component must have at least 3 special (i.e. nodal or marked) points (this is called the stability condition) and finally u ([C]) = i u i ([C i ]) = β. 5

6 We also call the tuple (C, z) alone a semistable curve, if moreover each of its components bear at least 3 special points then we call it a stable curve (this is just a stable map with u constant). It is often also convenient to think of a semistable curve (C, z) as a k-labeled connected graph T with the nodal points z sing = {z ij C i C j } being the edges {e ij } connecting the vertices {(i, labels)} representing the components C i labeled by the marked points. Such a graph is called a k-labeled tree. A tree homomorphism f : T T is a graph homomorphism such that f 1 (i) is a tree for each vertex i T, it is an isomorphism if it has an inverse, see [22] appendix D for more details. We then say that the stable map (u, C, z) is modeled over this tree T. On the other hand, by the above construction, an isomorphism class of (u, C, z) (see below) gives rise to a unique k-labeled tree (up to tree isomorphism). Remark 2.8. By the stability condition a stable map can have at most k 2 constant (ghost) components, and since each non constant component should have energy greater than we obtain that only a finite number of combinatorial configurations of stable maps are possible in a given homology class β H 2 (X) or if we bound the energy from above (the labeling i of the components of C = C i and hence their assignment to vertices of a tree is not part of the data of a stable map which we consider as a topological space). We say that two semistable curves (C, z) and (C, z ) are isomorphic if there exist a homeomorphism γ : C C which restricts to a biholomorphism γ ı i : C i C j for all i and some j and γ(z) = z. So we say that two stable maps (u, C, z) and (u, C, z ) are isomorphic if there exist an isomorphism γ : (C, z) (C, z ) of the associated semistable curves such that u γ = u. Equivalence of J-spheres with k-marked points (example of stable maps) defined earlier in this section is precisely the notion of isomorphism introduced here. From now on we identify the stable map (u, C, z) with its isomorphism class. The main point of the definition of a stable map is that (u, C, z) has a finite automorphism group Aut(u, C, z) = {γ isomorphism : γ(z) = z}. In our case (genus zero), we can describe the group more precisely: if two components with nodal points only have the same image then γ might permute them or if a given u i is multiply covered then the action might be by deck transformations. The crucial fact is that precisely because of the stability condition Aut(u, C, z) cannot act by reparametrizations on stable components since it must fix all the special points and, for the unstable components, it must preserve the map, so it has to be finite. We now come to the main results of this section. We compactify the moduli space M 0,k (X, β; J) by (isomorphism classes of) genus zero stable maps with k marked points in the homology class β. We denote this new space, to which we will usually refer as the moduli space of stable maps, by M 0,k (X, β; J). Fixing a labeled tree T we denote by M T 0,k(X, β; J) the moduli space of stable maps modeled over this tree. We obtain then a coarse stratification: M 0,k (X, β; J) = T M T 0,k(X, β; J) where T runs over the isomorphism classes of k-labeled trees so that the union is finite by remark 2.8. The automorphism group Aut(u, C, z) is usually non trivial so that the nicer geometric structure we might be able to put on M 0,k (X, β; J) is the one of an orbifold. Nevertheless in the special (but important) case of stable curves (X = pt.) we are able to obtain a structure of a smooth manifold of dimension 2k 6 (in genus 0 since the automorphism groups are trivial, a smooth orbifold for higher genera). These moduli spaces are known in literature as the Deligne-Mumford compactification of the moduli space of k-marked spheres and are denoted M 0,k. In [22] appendix D, the approach via cross ratios is adopted to prove: Theorem 2.9. The Deligne-Mumford (or Grothendieck-Knusden) compactification of the moduli space of stable curves of genus zero with k 3 marked points is a smooth compact manifold. However we will sketch the construction of Fukaya and Ono [6], section 9, as this will be useful for the construction of the Kuranishi structure in the general case. We first note that if k 2 then M 0,k is set to be empty. If k = 3 then, since the group of Möbius transformations acts 3-transitively on S 2, M 0,3 consists of one point: a Riemann sphere with 3 marked points. Now assume k 4. We note as before that our moduli space carry a stratification induced by the combinatorial type of the curves: M 0,k = T M T 0,k 6

7 Given a stable curve in some strata we describe its neighborhood in the moduli space. For, we start with the lowest stratum: the trees associated to the stable curves which has only 3 special points on each component and so k 3 singular points. After fixing such a tree, we note that there is only one stable curve in the induced stratum (C, z) M T 0,k. We assume that it comes with a Kähler metric which is flat in a neighborhood around each singular point. Hence for x C i C j the exponential maps exp i x : T x C i C i and exp j x : T x C j C j are isometries on some neighborhoods of 0 with respect to the standard Hermitian metric dz dz on C. Fix a gluing parameter: α T x C i T x C j = C with α = R 2, R large enough so that we are in the domain of isometry for the exponential maps. Consider the inversion φ α (z) = α/z. We use the diffeomorphism: ( ) exp j x φα ( exp i ) 1 x : exp i x (B R 1/2 \ B R 3/2) C i exp j x (B R 1/2 \ B R 3/2) C j to glue the curves C i \exp i x(b R 3/2) and C j \exp j x(b R 3/2) (this is just the connected sum construction). We construct a Riemannian metric on the new curve (which gives rise to an almost complex structure automatically integrable since the Nijenhuis tensor vanish in dimension 2). For, we note that φ α maps the circle z = R 1 to itself so we can find a smooth function χ α : R >0 R >0 (fix it once for all time) such that χ α (r) = 1 for r > R 1 + ε, ε > 0 small, and φ α(χ α ( z )dz dz) = χ α ( z )dz dz. Using the exponential maps we obtain two new metrics, one on each component we want to glue, which agree on the gluing domain and extends consistently to the whole curve via the original metric (if we choose ε small enough). Performing this local construction at each singular point we obtain a family of curves parametrized by an open neighborhood of 0 in: x ij z sing T xij C i T xij C j = C k 3 If we start with a curve belonging to a higher stratum, i.e. with less nodal points, say k 3 m, to obtain a parametrization by an open set in C k 3 we note that the remaining degrees of freedom are described by neighborhoods (small enough) of any collection of m marked points belonging to components with more than three special points. The nontrivial result that we are not going to prove is that this is diffeomorphic to an open neighborhood of (C, z) in M 0,k, in other words this construction gives us a chart centered at (C, z). A similar construction is carried by McDuff [21]. Example For k = 4 we can see that M 0,4 = S 2 \ {0, 1, } by fixing z 1 = 0, z 2 = 1 and z 3 = and letting z 4 move. The lowest strata of M 0,4 consists of 3 nodal curves with two components corresponding to the 3 splittings of the set of marked points {1, 2 3, 4}, {1, 3 2, 4} and {1, 4 2, 3}. The description via charts of the neighborhoods of this curves leads to: M 0,4 = S 2 For k = 5, fixing again z 1 = 0, z 2 = 1 and z 3 = and letting z 4, z 5 to move we obtain that the top stratum is diffeomorphic to M 0,5 = ( S 2 S 2) \ ( {z 4, z 5 0, 1, }), where is the diagonal z 4 = z 5. The open sets not containing the points (0, 0), (1, 1) and (, ) are diffeomorphic to neighborhoods of curves with two components as in the first example. Now suppose z 4, z 5 z 1 = 0, then we move to the stratum corresponding to the splitting {2, 3 1, 4, 5} together with the lower strata concerned. The parametrization by the marked points (z 4, z 5 ) S 2 S 2 is now redundant and so we have to quotient by the involution (z 4, z 5 ) (z 5, z 4 ) obtaining this way a construction diffeomorphic to CP 2. Therefore, back to M 0,5, a more careful argument, taking in account orientations, will tell us that we should replace the point (0, 0) by a copy of CP 2, and similarly for (1, 1) and (, ). So finally we obtain that M 0,5 is diffeomorphic to S 2 S 2 blown up at the above three points on the diagonal: M 0,5 = ( S 2 S 2) #3CP 2 7

8 We end this section by going back to the general case when (X, ω) is a closed symplectic manifold. We state the results about the topology of M 0,k (X, β; J): for more details and proofs we refer the reader to the book by McDuff and Salamon [22], chapter 5 and the paper by Fukaya and Ono [6], section 10 and 11. We will put more structure on the moduli space in section 3. We introduce some useful notation. The energy of a stable map is given by E(u) = i E(u i), but the energy of stable map on some open set U i C i will be the energy of u i restricted to this set to which we add the energy of all branches of C intersecting U i, which we denote: E i (u; U i ) = u i ω + E ij (u), with E ij (u) = E(u l ) U i z ij z sing U i where we say that the component l is related to i via j, writing l j i, if any path connecting them contains the nodal point z ij, i.e. it is just the branch of the tree T obtained by cutting at the edge e ij (note i j i). Now we define the C -convergence for stable maps: Definition (Gromov Convergence). A sequence of stable maps (u ν, C ν, z ν ) modeled over trees T ν is said to Gromov converge to a stable map (u, C, z) modeled over T if for large enough ν there exist a sequence of surjective tree homomorphisms f ν : T T ν and Möbius transformations φ ν i PSL(2, C) acting on C = C i such that we have convergence for the marked points (φ ν i ) 1 (zj ν) z j, where z j C i and f ν (z j ) = zj ν, the J-holomorphic maps converge in the C -topology outside the nodal points: u ν f ν (i) φν i C u i, i T and the rescaling has been chosen such that, for all i and j concerned, (φ ν i ) 1 φ ν j converges in C0 to z ij C i on C j \ {z ij } and no energy is lost in this process: E ij (u) = lim ε 0 lim ν E f ν (i)(u; φ ν i (B ε (z ij )) We have then the following fundamental result establishing the compactness of M 0,k (X, β; J): Theorem (Gromov Compactness). Let (u ν, C ν, z ν ) be a sequence of stable maps with k marked points such that sup ν E(u ν ) <. Then there exists a Gromov convergent subsequence (u νj, C νj, z νj ) converging to a stable map (u, C, z). Moreover for ν j large enough the connected sum # i u νj i is homotopic to # i u i and the limit is unique (up to isomorphism). The uniqueness of the limit suggests us that we should be able to put a Hausdorff topology on our moduli space. We define indeed the Gromov topology on M 0,k (X, β; J) by declaring closed sets to be the ones containing all the limit points in the sense of Gromov convergence. We have then the following important result: Theorem The moduli space M 0,k (X, β; J) is a compact Hausdorff space with respect to the Gromov topology. Moreover a sequence in this topology converges if and only if it Gromov converges. Convergence in homotopy in the Gromov compactness theorem is already suggested by our construction of the charts for the Deligne-Mumford spaces and will be also clear once we have constructed the Kuranishi neighborhoods in the next section. We note that any homology class β H 2 (X) with representatives by J-holomorphic spheres must be in the image of the Hurewicz homomorphism π 2 (X) H 2 (X) and satisfy ω 0 because of the energy identities. But we can say actually more β about these classes, the following useful corollary will appear in our discussion of quantum cohomology: Corollary Fix a constant c 0. Then there exist only finitely many homology classes β H 2 (X) that can be represented by a stable map with ω c. β Proof. Because H 2 (X) is a discrete lattice, an infinite collection of such classes will give rise to a sequence of J-spheres with uniformly bounded energy with no Gromov convergent subsequence contradicting Gromov compactness. Since homology classes represented by stable maps are finite sums of the latter the corollary follows from remark 2.8. Using convergence in homotopy, we obtain the same conclusion for homotopy classes from π 2 (X). l ji 8

9 3 Kuranishi Structure on M 0,k (X, β; J) In this section we will describe how to put a Kuranishi structure on the moduli space of stable maps (we do not really need to restrict ourselves to the genus 0 case, but we will for consistency) following the construction of Fukaya and Ono [6]. We start with some general remarks about Kuranishi spaces, some parts of the definitions are taken from the appendix A of the book by Fukaya et al. [7]. We note that we will present a stronger version of a Kuranishi space that the one used in [6]. See remark 3.6 for the issues involved! Let M be a compact Hausdorff topological space. In what follows, we allow our orbifolds to have a boundary (unless we state otherwise), but the generalization to corners is not required for our main application: genus zero stable maps. Definition 3.1. A Kuranishi neighborhood of p M is a system (U p, F p, ψ p, s p ) (sometimes called a chart) with: (i) U p = V p /Γ p is an orbifold with V p a manifold (with boundary) and Γ p a finite group acting effectively and smoothly on it; (ii) F p U p a trivial orbibundle called the obstruction bundle, i.e. we have F p = (E p V p ) /Γ p where E p is a vector space carrying a representation of Γ p ; (iii) s p : U p F p is a smooth section (or at least C 1 ) called the Kuranishi map, usually considered as an Γ p -equivariant map s p : V p E p ; (iv) ψ p : s 1 p (0) M is a homeomorphism onto a neighborhood of p M. Roughly speaking M is locally modeled on a zero set of a smooth section of an orbibundle. In what follows we will mean by an embedding (resp. isomorphism) between two orbifolds U p = V p /Γ p and U q = V q /Γ q an injective group homomorphism h pq : Γ p Γ q (resp. isomorphism) and a map ϕ pq : U p U q admitting a lift φ pq : V p V q which is a smooth h pq -equivariant embedding (resp. diffeomorphism). A similar remark holds for orbibundles as well. We will say that two Kuranishi neighborhoods of p M, (U p, F p, ψ p, s p ) and (Ûp, F p, ψ p, ŝ p ), are equivalent if after shrinking, if necessary, U p and Ûp to neighborhoods of ψ 1 p (p) and ψ 1 p (p) respectively there exist a diffeomorphism of orbifolds φ : U p Ûp and an isomorphism of orbibundles φ : F p φ Fp preserving the Kuranishi map: φ s p = ŝ p φ and ψ p φ = ψ p. This gives us the notion of a germ of Kuranishi neighborhoods at p M. Requiring a compatibility condition on the Kuranishi neighborhoods we obtain: Definition 3.2. A Kuranishi structure κ (with boundary) of virtual dimension µ on M assigns germs of Kuranishi neighborhoods for each p M such that if q ψ p (s 1 p (0)) there exist a coordinate change ( ϕ pq, ϕ pq ) between the two Kuranishi neighborhoods (U p, F p, ψ p, s p ) and (U q, F q, ψ q, s q ), assuming that imψ q imψ p, which satisfies: (i) ϕ pq : U q U p and ϕ pq : F q ϕ pqf p are embeddings of orbifolds and orbibundles respectively; (ii) s p ϕ pq ϕ pq s q and ψ q ψ p ϕ pq ; (iii) if r ψ q (s 1 q (0)), then ϕ pq ϕ qr ϕ pr and ϕ pq ϕ qr ϕ pr ; (iv) dim V p dim E p = vdimm =: µ independently of p; The tuple (M, κ) is called a Kuranishi space. One also needs a notion of orientability of Kuranishi spaces in order to construct a homology theory or use cobordism type arguments with virtual cycles. In the situation when q ψ p (s 1 p (0)), choose an open neighborhood T pq ϕ pq (U q ) in U p and denote by F pq the subbundle of F p over T pq such that ϕ pqf pq = ϕ pq (F q ) and define the map ŝ p : T pq F p /F pq as the composition: s p Fp (F p /F pq ) Vpq T pq Definition 3.3. A Kuranishi structure has a tangent bundle if for each coordinate change ( ϕ pq, ϕ pq ) the differential dŝ p of the above map, well-defined on the normal orbibundle N ϕpq(u q)u p, induces an isomorphism on s 1 q (0): dŝ p : N ϕpq(u q)u p = ϕ pqt U p dϕ pq (T U q ) = ϕ pqf p ϕ pq (F q ) = F p F pq ϕpq(u q) 9

10 The equalities above denote canonical isomorphisms obtained by pulling back via ϕ pq. We get then an isomorphism det F p ϕpq(u q) det T U p = det F q det T U q. If moreover there exist a trivializations of the latter bundles compatible with this isomorphism, we say the Kuranishi structure is oriented. Another important concept for our applications is the one of a boundary of a Kuranishi space. Definition 3.4. We say that p M if there exists a germ of Kuranishi neighborhoods of p such that p U p for any representative (U p, F p, ψ p, s p ). The boundary M of a Kuranishi space is therefore in a natural way a Kuranishi space without boundary with charts ( U p, F p Up, ψ p Up, s p Up ) of virtual codimension 1 in (M, κ). If our Kuranishi space (M, κ) was oriented then we obtain an orientation on the boundary in a natural way. The most important result in this section, proved by Fukaya and Ono [6], is: Theorem 3.5. The moduli space of stable maps M 0,k (X, β; J) carry an oriented Kuranishi structure without boundary of virtual dimension µ(β, k) := 2n + 2 c 1 (T X) + 2k 6 A similar analysis is carried by Chen and Ruan [3], who put a Kuranishi structure on the moduli space of stable maps to a symplectic orbifold. Remark 3.6. We should point out here that in definition 3.1 part (iii) we require the Kuranishi map to be smooth, as in [7]. Nevertheless in the definition of a Kuranishi space in [6] only continuity is required. But for the definition of orientability to make sense we need the Kuranishi map to be of class at least C 1. In fact a different notion of tangent bundle is used in [6], weaker than in definition 3.3. As pointed out in [7], appendix A, this notion is not strong enough to establish the cobordism invariance of the virtual cycle: the main application of Kuranishi structures in this essay. Although we will see below that it is clear from the construction that the Kuranishi map will be continuous, unfortunately, for the time being, I do not understand why the map will be differentiable at all points... I apologize for this. We start by considering the elliptic properties of the Cauchy-Riemann equation (2.1). First we formulate the problem in the appropriate Sobolev framework. Let σ = (u, C, z) be a stable map modeled over some tree T. Consider the following Banach spaces: { W 1,p (C, u T X) := (ξ i ) } W 1,p (C i, u i T X) : ξ i (x) = ξ j (x), x z sing i β L p (C, u T X J 0,1T C) := i L p (C i, u i T X J 0,1T C i ) where we fix p > 2. This condition is necessary in order to use the Sobolev embedding theorem to obtain a continuous representative for ξ i W 1,p (C i, u i T X) and hence make sense of the pointwise condition. It is also required if we want the space B := W 1,p (S 2, X) to be a Banach manifold; it will hence be modeled, locally at a map u, on W 1,p (S 2, u T X). We then have the following important result, proved in appendix B.4 of [22]: Theorem 3.7. (Elliptic Regularity). Given an almost complex structure J of class C l, l 1 and u : Σ X a J-holomorphic curve of class W 1,p, p > 2, then u is of class W l+1,p and so, in particular, if J is smooth then so is u. Denote by E B the Banach vector bundle with fibres E u := L p (S 2, u T X J 0,1 T S 2 ). Then we have a section of this bundle, S : B E, given by S(u) = (u, J (u)). The linearization of the equation (2.1), denoted D u, is defined to be the vertical differential of S at u S 1 (0), i.e. the map obtained by the composition: T u B ds T (u,0) E = T u B E u E u, so that D u : W 1,p (S 2, u T X) L p (S 2, u T X J 0,1T S 2 ) 10

11 We can work out a formula for D u. Fix ξ W 1,p (S 2, u T X), by we denote the Levi-Civita connection on X with respect to the Riemannian metric, J = ω(, J ). Let := 1 2J J, this connection commutes with the almost complex structure, hence the parallel transport with respect to it along the geodesics γ(s) = exp u(z) (sξ(z)), 0 s 1, gives an isomorphism Φ γ (ξ) : u T X exp u (ξ) T X of complex vector bundles if we are inside the injectivity radius. Then we compute using equation (2.1), J-linearity of and the fact is torsion free: D u ξ = s 0 (Φ γ (sξ) 1 J (exp u (sξ))) = sγ J (exp u (sξ)) 0 = 1 ( sγd(exp 2 u (sξ)) + J ) sγd(exp u (sξ)) j 0 D u ξ = 1 2 ( ξ + J ξ j) 1 2 (J( ξj) J (u), where J (u) := 1 (du J du j) 2 Note that if the map u is not J-holomorphic, D u will depend on the connection chosen. We see that D u is a real linear Cauchy-Riemann operator, i.e. it can be written as: D u = 0,1 + A where 0,1 is a smooth complex linear Cauchy-Riemann operator and A is a real linear endomorphism of the bundle u T X with values in 0,1 T S 2 the regularity of which depends on u: if u W l,p (S 2, X) then A W l 1,p (S 2, End R (u T X) R 0,1 T S 2 ), and in that case we say that D u is of class W l 1,p. The proof of the following theorem, which is usually referred to as Riemann-Roch, or simply the index theorem, can be found in [22] appendix C. Theorem 3.8. Let u W l,p (S 2, X), then D u is a Fredholm operator of class W l 1,p with index: index(d u ) = 2n + 2 c 1 (T X) u ([S 2 ]) The above theorem tell us that if u W l,p then we can consider D u as a Fredholm operator from W l,p to W l 1,p. Therefore if J is smooth and u is a J-holomorphic curve, then by elliptic regularity the class of D u is independent of the Sobolev completion we work with, hence we can choose smooth representatives for the spanning sets of the kernel and the cokernel (closed finite dimensional) of D u. In particular it is enough to check the surjectivity (we will say usually regularity) of D u on smooth sections only. In general we cannot assume that the cokernel of D u is 0. But if we assume that (X, ω, J) is Kähler (and it will be so for all examples in this essay) then the Levi-Civita connection becomes the Chern connection on the holomorphic tangent bundle, that we denote T X. Hence J = 0 and from the above formula we get D u = 0,1 = when restricted to smooth sections. It is relevant here to view the Riemann sphere as S 2 = CP 1 in order to use Grothendieck s lemma to obtain the decomposition u T X = j O(a j) for some a j Z. We then have the following regularity criterion: Lemma 3.9. Suppose that (X, ω, J) is Kähler and u : CP 1 X is a holomorphic sphere. If we have a j 1 in the above decomposition, then D u is surjective. Proof. As pointed out above, it is enough to consider D u restricted to smooth sections, obtaining the operator : A 0 (CP 1, n j=1 O(a j)) A 0,1 (CP 1, n j=1 O(a j)) having cokernel which we identify via Serre duality for sheaf cohomology groups: coker( O(aj)) = H 1 (CP 1, O(a j )) = H 0 (CP 1, O( a j 2)) where we have used K CP 1 = O( 2). If we use Serre duality again to get: H 0 (CP 1, O( a j 2)) = H 1 (CP 1, K CP 1 O(a j + 2)) we conclude using Kodaira vanishing theorem that coker( ) = 0 if CP 1 c 1 (O(a j + 2)) = a j + 2 > 0, i. Therefore the cokernel of D u will be trivial in that case. For generalizations of this criterion to the almost complex setting we refer the reader to the wonderful chapter 3.3 of McDuff and Salamon [22]. Let us go back to the situation when σ = (u, C, z) M 0,k (X, β; J) is a stable map modeled over a tree T. We write the nonlinear Cauchy-Riemann equation as the direct sum: J u := i J u i = 0 Γ(C, u T X J 0,1T C) 11

12 Therefore the operator D u is given by the composition: W 1,p (C, u T X) ı i W 1,p (C i, u i T X) D i u i L p (C, u T X J 0,1T C) We compute its index by noticing that the inclusion ı is a Fredholm operator with index(ı) = coker(ı) = 2n z sing = 2n( T 1). So we obtain using theorem 3.8 : ( T ) index(d u ) = index(du i i ) + index(ı) = 2n + 2 c 1 (T X) 2n( T 1) = 2n + 2 c 1 (T X) u ([C i]) β i=1 We will now construct the Kuranishi neighborhood of σ. Let E σ = coker(d u ) and enlarge it so that it is invariant under the action of Aut(σ), which is a finite group, and complex linear (still closed of finite dimension). We assume that we can find smooth representatives for the sections spanning E σ such their supports are contained in an Aut(σ)-invariant compact set K ob (σ) C \ z sing (see [6] page 979 and the references there). Let V m denote the kernel of the composition: Π Eσ D u : W 1,p (C, u T X) L p (C, u T X J 0,1T C)/E σ Now we deform the semistable curve (C, z). For each stable component C s (i.e. containing m s 3 special points) we take its neighborhood in the top stratum M 0,ms of the Deligne-Mumford compactification M 0,ms. We should view the space M 0,ms as a universal deformation of the stable component which exist by Kuranishi theorem 7.2. So we set V d to be a neighborhood of 0 in s Cms 3, s runs over the stable components of (C, z). We construct a deformation parametrized by V d of the whole semistable curve by deforming the complex structure on a compact set K d C \ (z sing z) (we can do this by the unique continuation theorem 2.3). Again by theorem 7.2 we obtain a versal family of curves over V d (we will explain how to fix unambiguously representatives later using rigidification). We also choose a family, constant outside K d, of Kähler metrics in the conformal class of the complex structures induced by the deformation, which are flat in a neighborhood of the singular points. We now resolve the whole semistable curve as we did with stable curves in section 2, the resolution parametrized by an open neighborhood V r of 0 in x z sing T x C i T x C j. We let K n be the closure of the domain of surgery, assuming that K ob K n = (shrinking V r if necessary). We hence obtain a (versal) family of semistable curves with fibrewise Kähler metrics, denoted C ζ for ζ = ((η s ), (α x )) V d V r where η is the deformation parameter, α the gluing parameter (x runs through the nodes) and C 0 = C. We assume we have an equivariant action of Aut(σ) on the family C ζ V d V r. We now put: V σ := V m V d V r This is a bad candidate for a Kuranishi neighborhood of σ because of the positive dimensional Aut 0 (C, z), the group of holomorphic reparametrizations of the unstable components fixing the special points, which gives rise to holomorphic vector fields on C in the kernel V m. We will fix this later by taking an appropriate slice; for now the program is to construct the Kuranishi map whose zero set maps onto a neighborhood of σ in M 0,k (X, β; J). We will be using the following version of the implicit function theorem proved in [22], appendix A, page 504. Theorem (Implicit function theorem). Let X and Y be Banach spaces, U X an open set and f : U Y a continuously differentiable map, i.e. x, y U df x L(X, Y ), the Banach space of bounded maps with the operator norm op, such that f(x + y) = f(x) + df x (y) + o( y ). Fix x 0 U such that D := df x0 has a bounded right inverse Q : Y X with Q op c 0. Suppose that there exists ε > 0 such that x B ε (x 0 ) U implies df x D op 1/(2c 0 ). Then if x 1 X satisfies: there exits a unique x X such that: Moreover, x x 1 2c 0 f(x 1 ). f(x 1 ) ε 4c 0, x 1 x 0 ε 8, f(x) = 0, x x 1 imq, and x x 0 ε 12

13 We are going now to build the Kuranishi map (see remark 3.6 concerning the regularity issues): Proposition Shrinking V σ if necessary, we can find a (continuous/smooth?) map s σ : V σ E σ induced by a family of smooth (outside nodal points) maps u ζ,ξ : C ζ X such that J u ζ,ξ = s(ξ, ζ) for all ζ V d V r and ξ V m, where we consider the obstruction space as: E σ L p (C ζ, u ζ,ξt X J 0,1T C ζ ) via an embedding to be constructed in the proof, and s σ (0) = J u = 0. Proof. We will sketch the construction of Fukaya and Ono [6] done in chapters 12 and 13, assuming all the estimates. The analysis is very similar to the one carried in chapter 10 of the book by McDuff and Salamon [22], modulo the fact that we have an obstruction space. Fix ζ = (η, α) V d V r and pick a section ξ V m. We first construct an approximate solution u a : C ζ X to the Cauchy-Riemann equation suitably perturbed by the obstruction space E σ on the curve C ζ and then use the implicit function theorem 3.10 to obtain the exact solution u ζ,ξ. Recall the surgery construction of section 2 : we let Rx 2 = α x for each x z sing and then C ζ can be obtained via the map: φ : (C i \ B(x, Rx 1 )) C ζ i x C i z sing which is two-to-one on B(x, Rx 1 ) and a diffeomorphism otherwise. We denote by B(x, Rx 1 ) := exp i x(b R 1 ) C i the image of the ball of the given radius and center 0 in the tangent plane at the x nodal point x z sing seen as belonging to C i (recall that exp i x : T x C i C i is an isometry there). We must know perform the surgery on the map u in X deforming it along the given vector field ξ. For, fix a small 1 > δ > 0 but such that δr x > Rx 1/2 and let: u a (φ(z)) := exp ui(z) (ξ i (z)), z C i \ x B(x, 2/δR x ) Since ξ i (x) = ξ j (x), we can define e x := exp ui(x)(ξ i (x)) if α x = 0. To glue the pieces together we use a cutoff function ρ : R 0 [0, 1] which is ρ(r) = 1 for r 2 and 0 if r 1. Then for each component C i and x C i z sing we let: { exp u a (φ(z)) := ex (ρ(δr x (exp i x) 1 (z) )) ) exp 1 e x (exp ui(z)(ξ i (z)) for z B(x, 2/δR x ) \ B(x, 1/δR x ) e x for z B(x, 1/δR x ) \ B(x, 1/R x ) This defines u a : C ζ X whose image is just the image of u restricted to a compact subset of C disjoint from the nodal points, translated by the vector field ξ, set to be constant where surgery is taking place and then glued together via a smooth cutoff function. We can prove the following estimate: J u a p L p (U) const./(δr x) 2 (3.1) where U = B(x, 2/δR x ) \ B(x, 1/δR x ) by noting that vol(u) = c/(δr x ) 2 for some constant c (the metric is flat there) and all the terms in the above definition of u a come with bounded derivatives. Identifying K d C and φ(k d ) C ζ and noticing that the volume form on C ζ is proportional to the one on C scaled by η, the deformation of the metric parameter, we have: J u Lp a (K d ) const.( η + ξ L ) ξ W 1,p (3.2) We proceed now to the construction of the perturbed Cauchy-Riemann equation. We identify K ob C, the set containing the supports of the sections of the obstruction space E σ, with φ(k ob ) C ζ since they are not in the surgery domain. Now we restrict ourselves to an open neighborhood of 0 V m (denoted U m ) such that ξ U m the corresponding approximate solution u a,ξ satisfies sup z dist(u(z), u a,ξ (φ(z)) < κ. We choose κ > 0 such that the exponential map exp p : B κ B(x, κ) is a diffeomorphism (so less than the injectivity radius of X) and the complex linear parallel transport Φ x,y : T x X T y X along geodesics with respect to the connection := 1 2J J introduced earlier 13

14 is an isomorphism. Using Φ and the isomorphism Ψ : 0,1 T C Kd = 0,1 T C ζ Kd (existing for V d V r small enough), for any smooth map ũ : C ζ X such that sup z dist(u(z), ũ(φ(z)) < κ, we will identify E σ with its image via the isomorphism: E σ C (K ob, u T X J 0,1T C) = C (K ob, ũ T X J 0,1T C ζ ) (3.3) obtaining this way the required embedding in the statement of the proposition and which allows us to write the perturbed Cauchy-Riemann equation as: J ũ = 0 mod E σ L p (C ζ, ũ T X J 0,1T C ζ ) (3.4) We let K := K ob K n K d. By estimating pointwise J u a,ξ J const. ξ J ξ J for z / K and using the inequality (3.2) we admit the following estimates: J u a,ξ Lp (C ζ \K) const. ξ L ξ W 1,p J u a,ξ D u ξ L p (K ob ) const.( η + ξ L ) ξ W 1,p (3.5) We construct now the exact solution to the equation (3.4). Recall the bounded operator: Π Eσ D u : W 1,p (C, u T X) L p (C, u T X J 0,1T C)/E σ The kernel V m being finite dimensional, it has a closed complement V m := ker v i, v i basis for V m. So Π Eσ D u is bijective on V m and hence admit a bounded inverse by open mapping theorem: Q u : L p (C, u T X J 0,1T C)/E σ V m W 1,p (C, u T X) Using the parallel transport Φ, the isomorphism Ψ and the map φ we define the operator I ζ,ξ to be the composition below (the first arrow is an isomorphism, the second is obtained by extending the sections by 0): L p (C ζ, u a,ξt X J 0,1T C ζ ) i L p (C i \ x B(x, R 1 x ), u i T X J 0,1T C i ) L p (C, u T X J 0,1T C) Note that by definition I ζ,ξ (E σ ) E σ since K ob K n =. We next describe the gluing map: ι ζ,ξ : W 1,p (C, u T X) W 1,p (C ζ, u a,ξt X) Take the following cutoff function: β(r) := 1 for r δ, 0 for r 1 and log r/ log δ for δ < r < 1. For a section s W 1,p (C, u T X), if z i C i \ x B(x, Rx 1/2 ) or (δr x ) 1 exp 1 x (z) 1/2 R x, then we simply parallel transport the vector s(z) to the point u a,ξ (φ(z)): ι ζ,ξ (s(z)) := Φ u(z),ua,ξ (φ(z))(s(z)) Recall that if z i = exp i x(v) C i and z j = exp j x(w) C j are such that w = α x /v then they get identified during the surgery and so u a,ξ (φ(z i )) = u a,ξ (φ(z j )). We can require the surgery to take place on Rx 1 v (δr x ) 1 only. Taking the following cutoff function: β(r) := 1 for r δ, 0 for r 1 and log r/ log δ for δ < r < 1, we let: ι ζ,ξ (s(z i )) := Φ u(zi),u a,ξ (φ(z i)) [s(z i )] + (1 β( v Rx 1 )) [ Φ u(zj),u a,ξ (φ(z i)) [s(z j )] Φ u(x),ua,ξ (φ(z i)) [s(x)] ] This is well-defined on the boundary: we see that if v = R 1 x We hence obtain the following operator: ι ζ,ξ (s(z i )) = ι ζ,ξ (s(z j )) = w then β( ) = 0 and Q ζ,ξ := ι ζ,ξ Q u I ζ,ξ : L p (C ζ, u a,ξt X J 0,1T C ζ )/E σ W 1,p (C ζ, u a,ξt X) 14