Technical details on Kuranishi structure and virtual fundamental chain

TL;DR

This paper provides an in-depth exposition of Kuranishi structures and virtual fundamental chains, including their abstract theory and applications to moduli spaces of pseudoholomorphic curves, with detailed analytic constructions.

Contribution

It offers a comprehensive, detailed explanation of Kuranishi structures, including new analytic details for their construction on moduli spaces, and addresses questions from prior discussions.

Findings

Complete analytic details of gluing construction

Construction of S^1-equivariant Kuranishi structures

Clarification of virtual fundamental chain theory

Abstract

This is an expository article on the theory of Kuranishi structure and is based on a series of pdf files we uploaded for the discussion of the google group named `Kuranishi' (with its administrator H. Hofer). There we replied to several questions concerning Kuranishi structure raised by K. Wehrheim. At this stage we submit this article to the e-print arXiv, all the questions or objections asked in that google group were answered, supplemented or confuted by us. We first discuss the abstract theory of Kuranishi structure and virtual fundamental chain/cycle. This part can be read independently from other parts. We then describe the construction of Kuranishi structure on the moduli space of pseudoholomorphic curves, including the complete analytic detail of the gluing construction as well as the smoothness of the resulting Kuranishi structure. The case of S^1 equivariant Kuranishi structure which appears in the study of time independent Hamiltonian and the moduli space of Floer's equation is included.