We present here the second in a sequence of three papers devoted to
the Gromov-Witten theory of nonsingular target curves X. Let ω ∈ H2(X,Q)
denote the Poincar´e dual of the point class. In the first paper , we considered
the stationary sector of the Gromov-Witten theory of X formed by the
descendents of ω. The stationary sector was identified in  with the Hurwitz
theory of X with completed cycle insertions.
The target P1 plays a distinguished role in the Gromov-Witten theory of
I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S g+1 (X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the GromovWitten invariants of a point. 1. Introduction Let X be a smooth projective variety. The genus 0 Gromov-Witten invariants of X satisfy relations which imply that they can be completely encoded in the structure of a Frobenius manifold on the cohomology H ∗ (X, C). ...
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).
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.