The virtual moduli cycle
Dusa McDuff · Translations - American Mathematical Society/Translations · 1999
additional remarks in 2004 This article is an attempt to describe one possible construction of the virtual moduli cycle that is used as a tool in the construction of Gromov–Witten invariants for a general symplectic manifold. There are many different versions of this construction. Here I will in the main follow Liu–Tian [LiuT1,2] since their approach (when modified by an idea of Seibert’s) seems to involve the least amount of analysis. However, it does involve quite a bit of topology, some of which they only outline. The aim here is to flesh out their picture and to explain the different ingredients that are needed to make the construction work. We do not try to give full proofs, nor do we work out all the ideas in full generality. Liu–Tian work in the category of partially smooth spaces (spaces with two topologies), and represent the virtual moduli cycle by the zero set of a suitable multi-section of a multi-bundle. We show explicitly how to assemble this zero set into an object that we call a branched labelled pseudomanifold. Our theory is quite general, and suggests that every finite dimensional orbifold has a “resolution ” of this form. An example is worked out in §4.4. Other approaches to this question have been developed by Fukaya–Ono [FO], Li–Tian [LiT], Ruan [R], Seibert [Sb] and Hofer–Salamon [HS]. We shall restrict here to curves of genus 0 but similar considerations