In the symplectic category there is a 'connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z = X#Y in terms of the relative GW invariants of X and Y . Several applications to enumerative geometry are given.Gromov-Witten invariants are counts of holomorphic maps into symplectic manifolds. To define them on a symplectic manifold (X, ω) one introduces an almost complex structure J compatible with the symplectic form ω and forms the moduli space of J-holomorphic maps from complex curves into X and the compactified moduli space, called the space of stable maps. One then imposes constraints on the stable maps, requiring the domain to have a certain form and the image to pass through fixed homology cycles in X. When the correct number of constraints is imposed there are only finitely many maps satisfying the constraints; the (oriented) count of these is the corresponding GW invariant. For complex algebraic manifolds these symplectic invariants can also be defined by algebraic geometry, and in important cases the invariants are the same as the curve counts that are the subject of classical enumerative algebraic geometry.In the past decade the foundations for this theory were laid and the invariants were used to solve several long-outstanding problems. The focus now is on finding effective ways of computing the invariants. One useful technique is the method of 'splitting the domain', in which one localizes the invariant to the set of maps whose domain curves have two irreducible components with the constraints distributed between them. This produces recursion relations relating the desired GW invariant to invariants with lower degree or genus. This paper establishes a general formula describing the behavior of GW invariants under the analogous operation of 'splitting the target'. Because we *The research of both authors was partially supported by the NSF. The first author was also supported by a Sloan Research Fellowship. 936ELENY-NICOLETA IONEL AND THOMAS H. PARKER work in the context of symplectic manifolds the natural splitting of the target is the one associated with the symplectic cut operation and its inverse, the symplectic sum.The symplectic sum is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n−2)-submanifold V . Given a similar pair (Y, V ) with a symplectic identification between the two copies of V and a complex anti-linear isomorphism between the normal bundles N X V and N Y V of V in X and in Y , we can form the symplectic sum Z = X# V Y . Our main theorem is a 'Symplectic Sum Formula' which expresses the GW invariants of the sum Z in terms of the relative GW invariants of (X, V ) and (Y, V ) introduced in [IP4].The symplectic sum is perhaps more naturally seen not as a single manifold but as a family depending on a 'squeezing parameter'. In Section 2 we construct a f...

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of 'V -stable' maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris.Gromov-Witten invariants are invariants of a closed symplectic manifold (X, ω). To define them, one introduces a compatible almost complex structure J and a perturbation term ν, and considers the maps f : C → X from a genus g complex curve C with n marked points which satisfy the pseudoholomorphic map equation ∂f = ν and represent a class A = [f ] ∈ H 2 (X). The set of such maps, together with their limits, forms the compact space of stable maps M g,n (X, A). For each stable map, the domain determines a point in the Deligne-Mumford space M g,n of curves, and evaluation at each marked point determines a point in X. Thus there is a mapThe Gromov-Witten invariant of (X, ω) is the homology class of the image for generic (J, ν). It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of M g,n and X n , the GW invariant can be viewed as a collection of numbers that count the number of stable maps satisfying constraints. In important cases these numbers are equal to enumerative invariants defined by algebraic geometry.In this article we construct Gromov-Witten invariants for a symplectic manifold (X, ω) relative to a codimension two symplectic submanifold V . These invariants are designed for use in formulas describing how GW invariants * The research of both authors was partially supported by the N.S.F. The first author was also supported by a Sloan Research Fellowship. 46ELENY-NICOLETA IONEL AND THOMAS H. PARKER behave under symplectic connect sums along V -an operation that removes V from X and replaces it with an open symplectic manifold Y with the symplectic structures matching on the overlap region. One expects the stable maps into the sum to be pairs of stable maps into the two sides which match in the middle. A sum formula thus requires a count of stable maps in X that keeps track of how the curves intersect V .Of course, before speaking of stable maps one must extend J and ν to the connect sum. To ensure that there is such an extension we require that the pair (J, ν) be 'V -compatible' as defined in Section 3. For such pairs, V is a J-holomorphic submanifold -something that is not true for generic (J, ν). The relative invariant gives counts of stable maps for these special V -compatible pairs. These counts are different from those associated with the absolute GW invariants.The restriction to V -compatible (J, ν) has repercussions. It means that pseudo-holomorphic maps f : C → V into V are automatically pseudo-holomorphic maps into X. Thus for V -compatible (J, ν), stable maps may have domain components whose image lies entirely in V . This creates problems beca...

Abstract. The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.The Gopakumar-Vafa conjecture [GV] predicts that the Gromov-Witten invariants GW A,g of a Calabi-Yau 3-fold can be expressed in terms of some other invariants n A,h , called BPS numbers, by a transform between their generating functions:The content of the conjecture is that, while the GW A,g are rational numbers, the BPS numbers n A,h are integers. (Gopakumar and Vafa also conjectured that for each A ∈ H 2 (X, Z), the coefficients of (0.1) satisfy n A,h = 0 for large h; we do not address this finiteness statement here.) It is natural to enlarge the context by regarding this as a conjecture about the Gromov-Witten invariants of any closed symplectic 6-manifold X that satisfies the topological Calabi-Yau condition c 1 (X) = 0. Formula (0.1) can be viewed as a statement about the structure of the space of solutions to the Jholomorphic map equation. For a generic almost complex structure J, each J-holomorphic map is the composition f = ϕ • ρ of a multiple-cover ρ and an embedding ϕ. The embeddings are well-behaved: they have no nontrivial automorphisms, and the moduli space of J-holomorphic embeddings is a manifold. But multiply-covered maps cause severe analytical problems with transversality. In the symplectic construction of the GW invariants, these problems are avoided by lifting to a cover of the moduli space and turning on a lift-dependent perturbation ν of the equation; this destroys the multiple-cover structure and only shows that the numbers GW A,g are rational. But it also suggests an interpretation of the GV formula: the righthand side of (0.1) might be a sum over embeddings, with the sum over k counting the contributions of the multiple covers of each embedding.This viewpoint is very similar to C. Taubes' work [T] relating Gromov invariants to the SeibergWitten invariants of 4-manifolds, and our approach has been fundamentally influenced by Taubes. It is also similar to the 4-dimensional situation described by Lee and Parker in [LP1] and [LP2]. In both cases, the set of J-holomorphic embeddings in each homology class is discrete and compact for generic J -a simplifying circumstance that does not appear to be true in the context of formula (0.1). Rather, for generic J and with a fixed bound E on area and genus, the moduli space M emb (X) of embeddings is a countable set, possibly with accumulation points. With this picture in mind, our proof is based on three main ideas.The first is the observation that, again for fixed J and E, the full moduli space M(X) can be decomposed (in many ways) into finitely many "clusters" O j . Each cluster...

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n − 2)-submanifold V . Given a similar pair (Y, V ) with a symplectic identification V = V and a complex anti-linear isomorphism between the normal bundles of V and V , we can form the symplectic sum Z = X# V =V Y . This note announces a general formula for computing the Gromov-Witten invariants of the sum Z in terms of relative Gromov-Witten invariants of (X, V ) and (Y, V ).Section 1 is a review of the GW invariants for symplectic manifolds and the associated invariants, which we call T W invariants, that count reducible curves. The corresponding relative invariants of a symplectic pair (X, V ) are defined in section 2. The sum formula is stated in a special case in section 3, and in general as Theorem 4.1. The last section presents two applications: a short derivation of the Caporaso-Harris formula [CH], and new proof that the rational enumerative invariants of the rational elliptic surface are given by the "modular form" (5.3).Related results, involving symplectic sums along contact manifolds, are being developed by Li and Ruan [LR] and by Eliashberg and Hofer. Symplectic invariantsThe moduli space of (J, ν)-holomorphic maps from genus g curves with n marked points representing a class A in the free part of H 2 (X; Z) has a compactification M g,n (X, A). This comes with a mapwhere the first factor is the "stabilization" map st to the Deligne-Mumford moduli space (defined by collapsing all unstable components of the domain curve), the second factor records the images of the marked points, and the last one keeps track of the homology class A of the image. After perturbation (cf.[LT]), the image defines a rational homology class,for each choice g, n, A. The last term in (1.2) can be identified with the rational group ring RH 2 (X) of H free 2 (X; Z), that is, with finite sums c A t A over A ∈ H free 2 (X; Z) where c A ∈ Q and the t A are variables satisfying t A t B = t A+B .

C. Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants ([T1], [T2]). Independently, Y. Ruan and G. Tian defined symplectic invariants based on ideas of Witten ([RT]). While similar in spirit, these two sets of invariants are quite different in their details.In this note we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants (Theorem 4.5). This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants Gr δ , δ = 0, 1, 2 . . . . The first of these, Gr 0 , generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants Gr δ count immersed curves with δ double points.In particular, these results give an independent proof that Taubes' invariants are well-defined. Combined with Taubes' Theorem [T1], they also show that, for symplectic 4-manifolds with b + > 1, some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants. Gromov InvariantsFix a closed symplectic four-manifold (X, ω). Following the ideas of Gromov and Donaldson, one can define symplectic invariants by introducing an almost complex structure J and counting (with orientation) the number of J-holomorphic curves on X satisfying certain constraints. Unfortunately, technical difficulties make it necessary to modify the straightforward count in order to obtain an invariant. In this section we review the general construction and describe how the technicalities have led to two types of Gromov invariants.Given (X, ω), one can always choose an almost complex structure J tamed by ω, i.e. with ω(Z, JZ) > 0 for all tangent vectors Z. A map f : Σ → X from a topological surface Σ is called J-holomorphic if there is a complex structure j on Σ such thatwhere ∂ J f = 1 2 (df •j−J •df ). The image of such a map is a J-holomorphic curve. Conversely, each immersed J-holomorphic curve is uniquely specified by the equivalence class of a J-holomorphic * partially supported by a M.S.R.I. Postdoctoral Fellowship † partially supported by N.S.F. grant DMS-9626245