Se p 20 01 Relative Gromov - Witten

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 nmarked points which satisfy the pseudo-holomorphic map equation ∂f = ν and represent a class A = [f ] ∈ H2(X). The set of such maps, together with their limits, forms the compact space of stable maps Mg,n(X,A). For each stable map, the domain determines a point in the Deligne-Mumford space Mg,n of curves and evaluation at each marked point determines a point in X. Thus there is a map Mg,n(X,A) → Mg,n ×X . (0.1) The 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 a basis of the homologies of Mg,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 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 ∗both authors partially supported by the N.S.F.