The symplectic sum formula for Gromov–Witten invariants

Abstract

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...

Flops, motives, and invariance of quantum rings

“…As the latter are obviously isomorphic, one is reduced to studying the local case. The degeneration formula [14], [15], [10] provides a rigorous formulation of the above naive picture.…”

Symplectic gluing and family Gromov-Witten invariants

“…In fact, the proof of the sum formula given in [IP3] holds, essentially without change, for the family invariants provided that all the maps that arise in the limit Z λ → Z 0 lie in compact relative moduli spaces. That is true for a fibration λ : Z → D whose generic fiber is E(n) and whose center fiber is E(n) ∪ V E(0) because the relative family moduli space is compact.…”

“…The "Symplectic Sum Formula" developed in [IP2] and [IP3] expresses the GW invariants of the symplectic sum Z λ in terms of invariants of X and Y . The derivation begins by considering what happens to J-holomorphic maps into Z λ as λ → 0.…”

“…For that reason, it is necessary to restrict to pairs (J, ν) that satisfy a certain "Vcompatibility" condition (see [IP3]). That condition implies, in particular, that V is a J-holomorphic submanifold.…”

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber’s Intersection Number Conjecture

“…We shall use the theory of stable relative maps to P 1 , following J. Li's algebro-geometric description in [Li1], and his description of their deformation-obstruction theory in [Li2]. (We point out earlier definitions of relative stable maps in the differentiable category due to A.-M. Li and Y. Ruan [LR], and Ionel and Parker [IP1,IP2], and Gathmann's work [Ga] in the algebraic category in genus 0.) We need the algebraic category for several reasons, most importantly because we shall use virtual localization, and an explicit description of the moduli space's deformation-obstruction theory.…”