Counting pseudo-holomorphic submanifolds in dimension 4 (Q1359249)

Due to the recent interaction with physics, a number of invariants of 4-manifolds was discovered, notably the Seiberg-Witten invariant which in turn has lead to the proof of Thom's conjecture by Kronheimer, Mrowka, Taubes and others. In a sequence of papers where the present article represents the first, Taubes identified the Seiberg-Witten invariant of a symplectic 4-manifold with the so-called ``Gromov invariant''. Roughly, this last invariant is a method of counting disjoint, connected, pseudo-holomorphic submanifolds inside a symplectic 4-manifold. Extending the work of Gromov, \textit{Y. Ruan} was the first to give a construction based on counting all those connected holomorphic submanifolds in [Symplectic topology and complex surfaces, Mabuchi, T. (ed.) et al., Geometry and analysis on complex manifolds, Singapore: World Scientific, 171-197 (1994)]. As Ruan's approach was incomplete, the present article made an important contribution to the subject by treating the more general and natural situation -- especially in its description of counting weights for multiply covered tori with trivial normal bundle. In typical Taubes style, this article is slightly long but full of glorious details -- definitely, a great paper to read for both beginners and experts alike!