Kuranishi structure


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.

Definition

Let be a compact metrizable topological space. Let be a point. A Kuranishi neighborhood of is a 5-tuple
where
They should satisfy that.
If and, are their Kuranishi neighborhoods respectively, then a coordinate change from to is a triple
where
  • is an open sub-orbifold;
  • is an orbifold embedding;
  • is an orbifold vector bundle embedding which covers.
In addition, these data must satisfy the following compatibility conditions:
  • ;
  • .
A Kuranishi structure on of dimension is a collection
where
  • is a Kuranishi neighborhood of of dimension ;
  • is a coordinate change from to.
In addition, the coordinate changes must satisfy the cocycle condition, namely, whenever, we require that
over the regions where both sides are defined.

History

In Gromov–Witten theory, one needs to define integration over the moduli space of pseudoholomorphic curves. This moduli space is roughly the collection of maps from a nodal Riemann surface with genus and marked points into a symplectic manifold, such that each component satisfies the Cauchy–Riemann equation
If the moduli space is a smooth, compact, oriented manifold or orbifold, then the integration can be defined. When the symplectic manifold is semi-positive, this is indeed the case if the almost complex structure is perturbed generically. However, when is not semi-positive, 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, Yong-Geun Oh, Hiroshi Ohta, and Ono studied Lagrangian intersection Floer theory.