Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities

This paper is concerned with two themes of symplectic topology. The first is the development of techniques to construct symplectic manifolds and, in particular, compact symplectic 4-manifolds. The second is the resolution of symplectic singularities and, in particular, the resolution of isolated singularities in symplectic 4-manifolds. On the first topic we prove a theorem which allows the gluing of two symplectic manifolds along a special class of hypersurfaces that we call o-compatible hypersurfaces. Let (X, og) be a symplectic 2n-manifold and M c X a hypersurface with a fixed point free S 1-action. M is called o2compatible if the orbits of the action lie in the null directions of og[u. An ~o-compatible hypersurface M has a canonical co-orientation. Hence, if M is a separating hypersurface, then M divides X into distinguished components X and X +. In dimension 4, our main gluing theorem is as follows.