Pseudoholomorphic curves in four-orbifolds and some applications

摘要

The main purpose of this paper is to summarize the basic ingredients,illustrated with examples, of a pseudoholomorphic curve theory for symplectic4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubeson symplectic 4-manifolds concerning pseudoholomorphic curves andSeiberg-Witten theory. They form the technical backbone of the proof that asymplectic s-cobordism of elliptic 3-manifolds (with a canonical contactstructure on the boundary) is smoothly a product. One interesting feature ofthe theory is that existence of pseudoholomorphic curves gives certainrestrictions on the singular points of the 4-orbifold contained by thepseudoholomorphic curves. In the last section, we discuss applications (orpotential ones) of the theory in some other problems such as symplectic finitegroup actions on 4-manifolds, symplectic circle actions on 6-manifolds, andalgebraic surfaces with quotient singularities.