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
(
f
1
,
…
,
f
k
)
:
R
n
+
k
→
R
k
.
Kuranishi structure was introduced by Japanese mathematicians Kenji Fukaya and Kaoru Ono in the study of Gromov–Witten invariants in symplectic geometry.
Let
X
be a compact metrizable topological space. Let
p
∈
X
be a point. A Kuranishi neighborhood of
p
(of dimension
k
) is a 5-tuple
where
U
p
is a smooth orbifold;
E
p
→
U
p
is a smooth orbifold vector bundle;
S
p
:
U
p
→
E
p
is a smooth section;
ψ
p
:
S
p
−
1
(
0
)
→
X
is a continuous map and is homeomorphic onto its image
F
p
⊂
X
.
They should satisfy that
dim
U
p
−
rank
E
p
=
k
.
If
p
,
q
∈
X
and
K
p
=
(
U
p
,
E
p
,
S
p
,
ψ
p
,
F
p
)
,
K
q
=
(
U
q
,
E
q
,
S
q
,
ψ
q
,
F
q
)
are their Kuranishi neighborhoods respectively, then a coordinate change from
K
q
to
K
p
is a triple
where
U
p
q
⊂
U
q
is an open sub-orbifold;
ϕ
p
q
:
U
p
q
→
U
p
is an orbifold embedding;
ϕ
^
p
q
:
E
q
|
U
p
q
→
E
p
is an orbifold vector bundle embedding which covers
ϕ
p
q
.
In addition, they must satisfy the compatibility condition:
S
p
∘
ϕ
p
q
=
ϕ
^
p
q
∘
S
q
|
U
p
q
;
ψ
p
∘
ϕ
p
q
|
S
q
−
1
(
0
)
∩
U
p
q
=
ψ
q
|
S
q
−
1
(
0
)
∩
U
p
q
.
A Kuranishi structure on
X
of dimension
k
is a collection
where
K
p
is a Kuranishi neighborhood of
p
of dimension
k
;
T
p
q
is a coordinate change from
K
q
to
K
p
.
In addition, the coordinate changes must satisfy the cocycle condition, namely, whenever
q
∈
F
p
,
r
∈
F
q
, we require that
over the regions where both sides are defined.
In Gromov–Witten theory, one needs to define integration over the moduli space of stable maps
M
¯
g
,
n
(
X
,
A
)
. They are maps
u
from a nodal Riemann surface with genus
g
and
n
marked points into a symplectic manifold
X
, such that each component satisfies the Cauchy–Riemann equation
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
X
is semi-positive, this is indeed the case (except for codimension 2 boundaries of the moduli space) if the almost complex structure
J
is perturbed generically. However, when
X
is not semi-positive, the moduli space may contain configurations for which one component is a multiple cover of a holomorphic sphere
u
:
S
2
→
X
whose intersection with the first Chern class of
X
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, Ohta, Ono studied the Lagrangian intersection Floer theory.
