A new symplectic surgery: the 3-fold sum

We define a symplectic surgery along positively intersecting symplectic surfaces in 4-manifolds, the generalized symplectic sum, and prove an existence theorem for the special case of a 3-fold sum. With a slight generalization of the 3-fold sum, we show how to sum along immersed surfaces and indicate a relation between the sums and algebraic desingularization. We use images of the moment map for a torus acting in the neighborhood of intersection points to illustrate when it is possible to perform the proposed sums.