Symplectic Gluing and Family Gromov-Witten Invariants

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 pg > 0, for which the usual GW invariants vanish for most homology classes. This involves extending the Splitting Formula and the Symplectic Sum Formula to the family GW invariants introduced by the first author. We present applications the invariants of elliptic surfaces and to the Yau-Zaslow Conjecture. In both cases the results agree with the conjectures of algebraic geometers and yield a proof (to appear [LL]) of previously unproved cases of the Yau-Zaslow Conjecture. Gromov-Witten invariants are counts of holomorphic curves in a symplectic manifold X . To define them one chooses an almost complex structure J compatible with the symplectic structure and considers the set of maps f : Σ → X from Riemann surfaces Σ which satisfy the (nonlinear elliptic) J-holomorphic map equation ∂Jf = 0. (0.1) After compactifying the moduli space of such maps, one imposes constraints, counting, for example, those maps whose images pass through specified points. With the right number of constraints and a generic perturbation of the equation, the number of such maps is finite. That number is a GW invariant of the symplectic manifold X . The first part of this article focuses on cut-and-paste methods for computing GW invariants of symplectic four-manifolds. Two useful techniques are described in Sections 2–4. The first is the method of “splitting the domain”, in which one considers maps whose domains are pinched Riemann surfaces. This produces recursion relations, called TRR formulas, relating one GW invariant to invariants whose domains have smaller area or lower genus. The second author was partially supported by the N.S.F.. c ©0000 American Mathematical Society 1 2 Junho Lee and Thomas H. Parker One can also consider the behavior of GW invariants as one “splits the target” using the symplectic cut operation and its inverse, the symplectic sum. The resulting “Symplectic Sum Formula” of E. Ionel and the second author expresses the GW invariants of the symplectic sum of two manifolds X and Y in terms of the relative GW invariants of X and Y as described in Section 4. These splitting formulas for the domain and the target can sometimes be combined to completely determine a set of GW invariants. One such computation is given in Section 5. There we use a clever geometric argument of E. Ionel to derive an explicit formula for certain GW invariants of the rational elliptic surface E(1). That formula, explained in Section 5, is