In mathematics, especially in
topology, a Kuranishi structure is a smooth analogue of
scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified with the zero set of a smooth map , or the quotient of such a zero set by a finite group. Kuranishi structures were introduced by Japanese mathematicians
Kenji Fukaya and Kaoru Ono in the study of
Gromov–Witten invariants and
Floer homology in symplectic geometry, and were named after
Masatake Kuranishi.[1]
If the moduli space is a smooth, compact, oriented manifold or orbifold, then the integration (or a
fundamental class) can be defined. When the symplectic manifold is semi-positive, this is indeed the case (except for codimension 2 boundaries of the moduli space) if the
almost complex structure is perturbed generically. However, when is not semi-positive (for example, a smooth projective variety with negative first Chern class), the moduli space may contain configurations for which one component is a multiple cover of a holomorphic sphere whose intersection with the first
Chern class of is negative. Such configurations make the moduli space very singular so a fundamental class cannot be defined in the usual way.
The notion of Kuranishi structure was a way of defining a virtual fundamental cycle, which plays the same role as a fundamental cycle when the moduli space is cut out transversely. It was first used by Fukaya and Ono in defining the Gromov–Witten invariants and Floer homology, and was further developed when Fukaya,
Oh Yong-Geun, Hiroshi Ohta, and Ono studied
Lagrangian intersection Floer theory.[3]