Pseudoholomorphic curves in four-orbifolds and some applications

The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory (cf. [19, 34, 35, 44, 46]). They form the technical backbone of [9, 10] where it was shown that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. We believe that this theory has a broader interest and may find applications in other problems. One interesting feature is that existence of pseudoholomorphic curves gives certain restrictions on the singular points of the 4-orbifold contained by the pseudoholomorphic curves. There are four sections. The first one is concerned with the Fredholm theory for pseudoholomorphic curves in symplectic orbifolds, which is based on the theory of maps of orbifolds developed in [8], particularly the topological structure of the corresponding mapping spaces. In the second section, we discuss the orbifold version of adjunction formula and a formula expressing the intersection number of two distinct pseudoholomorphic curves in terms of contributions from each point in the intersection. Unlike the manifold case, the contribution from a singular point may not be integral, but it can be determined from the local structure of the pseudoholomorphic curves near the singular point, which is important in applications. The third section is occupied by a brief summary of the Seiberg-Witten theory for smooth 4-orbifolds and a discussion on the orbifold analog of the theorems of Taubes concerning existence of pseudoholomorphic curves. We also include here an example to illustrate how a combination of the various ingredients is put into use. In the last section, we discuss some applications and work in progress.