Abstract:
Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of $X$ and $Y$. This formula has several applications to enumerative geometry. As one application, we obtain new relations in the cohomology ring of the moduli space of complex structures on a genus g Riemann surface with n marked points.

Abstract:
Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $M_g$ (and in fact that of $M_{g,2}$). One immediate consequence of these relations is that the classes $\kappa_1,...,\kappa_{[g/3]}$ generate the tautological ring of $M_g$, which has been conjectured by Faber, and recently proven at the level of {\em cohomology} by Morita.

Abstract:
In this paper we introduce a notion of symplectic normal crossings divisor V and define the GW invariant of a symplectic manifold X relative such a divisor. Our definition includes normal crossings divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [IP1], which covered the case V was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X, V) in the case V is a symplectic normal crossings divisor in X.

Abstract:
We show that any degree at least $g$ polynomial in descendant or tautological classes vanishes on $M_{g,n}$ when $g\ge 2$. This generalizes a result of Looijenga and proves a version of Getzler's conjecture. The method we use is the study of the relative Gromov-Witten invariants of $P^1$ relative 2 points combined with the degeneration formulas of [IP1]. At the end of the paper, we also included a quick proof of a very recent conjecture made by Vakil.

Abstract:
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

Abstract:
We correct an error and an oversight in [IP]. The sign of the curvature in (8.7) is wrong, requiring a new proof of Proposition 8.1. Also, several lemmas addressed only the basic case of maps with intersection multiplicity s=1; the general case follows by applying the pointwise estimates in [IP] with a modified Sobolev norm. These corrections do not affect the results of the paper. We thank M. Tehrani and A. Zinger for pointing out these issues.

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.

Abstract:
We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The virtual fundamental class is constructed using only Ruan-Tian perturbations by introducing stabilizing divisors, using Cech homology, and systematically applying naturality conditions. In high dimensions or low genus, no gluing theorems are needed.

Abstract:
Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants. This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants $Gr_{\delta}$, $\delta=0,1,2,...$. The first of these, $Gr_0$, generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants $Gr_{\delta}$ count immersed curves with $\delta$ double points. In particular, these results give an independent proof that Taubes' invariants are well-defined. They also show that some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants.

Abstract:
We define a notion of virtual fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov-Witten theory. For universal moduli spaces over a parameter space, the virtual fundamental class specifies an element of the Cech homology of the compactification of each fiber; it is defined if the compactification is "thin" in the sense that its boundary has homological codimension at least two.