Pseudoholomorphic curves in four-orbifolds and some applications

Details Pseudoholomorphic curves in four-orbifolds and some applications Pseudoholomorphic curves in four-orbifolds and some applications

2004-10-28

arxiv.org/abs/math/0410608v2

Mathematics

Symplectic Geometry

27 pages, no figures, expository article, improved exposition, accepted for publication in Proceedings of Conference on Geomet

Scientific paper

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. They form the technical backbone of the proof that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. One interesting feature of the theory is that existence of pseudoholomorphic curves gives certain restrictions on the singular points of the 4-orbifold contained by the pseudoholomorphic curves. In the last section, we discuss applications (or potential ones) of the theory in some other problems such as symplectic finite group actions on 4-manifolds, symplectic circle actions on 6-manifolds, and algebraic surfaces with quotient singularities.

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