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… (More)

Let Σ be a compact Riemann surface. Any sequence fn : Σ → M of harmonic maps with bounded energy has a “bubble tree limit” consisting of a harmonic map f0 : Σ → M and a tree of bubblesfk : S → M . We… (More)

This paper develops the Riemannian geometry of classical gauge theories Yang-Mills fields coupled with scalar and spinor fields on compact four-dimensional manifolds. Some important properties of… (More)

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.… (More)

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… (More)

We prove a structure theorem for the Gromov-Witten invariants of compact Kähler surfaces with geometric genus pg > 0. Under the technical assumption that there is a canonical divisor that is a… (More)

where the Z action is generated by (x, s, θ) 7→ (f(x), s + 1, θ). In this paper we compute the Gromov invariants of the manifolds Xf and of fiber sums of the Xf with other symplectic manifolds. This… (More)

In [LP] the authors defined symplectic “Local Gromov-Witten invariants” associated to spin curves and showed that the GW invariants of a Kähler surface X with pg > 0 are a sum of such local GW… (More)

This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kähler surfaces with geometric genus… (More)