Kuranishi structure

Summary

A Kuranishi structure on a compact space describes it locally as a quotient by a finite group. The thing divided is the zero set of a smooth section of a vector bundle. The bundle has finite rank and lives over a finite-dimensional manifold.

The local descriptions, called Kuranishi charts, are glued by coordinate changes that embed one chart into another. The notion was introduced by Fukaya and Ono to handle moduli spaces of pseudoholomorphic curves, which are usually singular and of the wrong dimension. A Kuranishi structure with an orientation has a virtual fundamental class, which is used to define Gromov–Witten invariants and Floer homology in general symplectic manifolds.