Relative Gromov–Witten invariants

Abstract

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...

Open string instantons and relative stable morphisms

“…Hence we have (5)(6)(7)(8)(9)(10) where in the last equality we have used the fact that the moduli space M g,1 of DeligneMumford stable curves has dimension dim C M g,1 = 3g − 2. The Hodge integral above can be easily evaluated by using Faber and Pandharipande's generating function for Hodge integrals over the moduli space M g,1 [2].…”

“…It will become clear later in the paper that the theory of relative stable maps is tailor-made for studying topological open string theory. The construction of relative stable maps was first developed in the symplectic category (Li-Ruan [11], Ionel-Parker [6,7]). Recently in [13,12] the first author of the present paper has given an algebro-geometric definition of the moduli space of relative stable morphisms and has constructed relative Gromov-Witten invariants in the algebraic category.…”

“…According to the general principle of virtual moduli cycles, the expected (virtual) number of maps in M rel g,µ (W r ) should be (3)(4)(5)(6)(7)(8)(9)(10) [ Extending V to M rel g,µ (P 1 ) needs more work. There is an obvious extension as follows: Let f ∈ M rel g,µ (P 1 ) be any relative stable morphism with domain C and distinguished divisor D d as before.…”

Skeletons and Fans of Logarithmic Structures

“…In the mild setting of two smooth varieties meeting along a smooth divisor, such a theory has been developed by J. Li [Li01,Li02], following work in symplectic geometry by A. M. Li-Ruan [LR01] and Ionel-Parker [IP03,IP04].…”

Heegaard Diagrams and Holomorphic Disks

“…The third step follows from naturality properties of the Floer homology theories (using a holomorphic triangle construction which we return to in Subsection 2.5), and a direct calculation in a special case (where handle-slides are made over a g-fold connected sum of S 1 × S 2 ). The final step can be seen as an invariance of the theory under a natural degeneration of the (g + 1)-fold symmetric product of the connected sum of Σ with E, as the connected sum neck is stretched, compare also [58], [41].…”