International Congress of Mathematicians

Symplectic Sums and Gromov-Witten Invariants 431called the S-matrix which keeps track of how the genus, homology class, and intersectionpoints with V change as the images of stable maps pass through the neckregion. One sees these quantities changing abruptly as the map passes through theneck — the maps are "scattered" by the neck. The scattering occurs when someof the stable maps contributing to the GT invariant of Z\ have components thatlie entirely in V in the limit as À —¥ 0. Those maps are not V-regular, so are notcounted in the relative invariants of X or Y. But this complication can be analyzedand related to the relative invariants of the ruled manifold P(A r x V ® C).Putting all these ingredients together, we can at last state the main result of[16].Theorem 2.1 Let Z be the symplectic sum of (X, V) and (Y, V) and fix a decompositionof the constraints a into ax on the X side and ay on the Y side. Thenthe GT invariant of Z is given in terms of the relative invariants of (X, V) and(Y,V) byGT z (a) = GTx'(ax) * S v * GT%(a Y ) (2.3)where * is the convolution operation and Sv is the S-matrix defined in [16].Several applications of this formula are described in the next two sections (seealso [16] for more applications). But the full strength of the symplectic sum theoremhas not yet been used.A.-M. Li and Y. Ruan also have a sum formula [18]. Eliashberg, Givental, andHofer are developing a general theory for invariants of symplectic manifolds gluedalong contact boundaries [3]. Jun Li has recently adapted our proof to the algebraiccase [19].3. Relations in H*(Aig, n )A smooth genus g curve with n marked points is stable if 2g — 2 + n > 0.The set of such curves, modulo diffeomorphisms, forms the moduli space A4 g , n .The stability condition assures that the group of diffeomorphisms acts with finitestabilizers, and so A4 g , n has a natural orbifold structure. Its Deligne-Mumfordcompactification A4 g , n is a projective variety. Elements of A4 g , n are called stablecurves; these are connected unions of smooth stable components G, joined at ddouble points with a total of n marked points and Euler characteristic \ = 2 — 2g+d.The compactification A4 g , n is also an orbifold, and in fact Looijenga proved thatit has a finite degree cover which is a smooth manifold. In any event, the rationalcohomology of A4 g , n satisfies Poincaré duality. Throughout this section we workonly with rational coefficients.There are several maps between moduli spaces of stable curves. First, there isa projection 7r, : M g: n+i —¥ A4 g , n that forgets the marked point xi (and collapsesthe components that become unstable). Second, we can consider the attaching mapsthat build a boundary stratum in A4 g , n . For each topological type of a stable curvewith d nodes, with components G, of genus gi and n, marked points the attachingmap £ at the d nodes takes UiA4 g^ni onto a boundary stratum of A4 g , n .