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 $(f_1, \ldots, f_k): {\mathbb R}^{n+k} \to {\mathbb R}^k.$ Kuranishi structure was introduced by Japanese mathematicians Kenji Fukaya and Kaoru Ono in the study of Gromov–Witten invariants in symplectic geometry.^[1]

Definition

Let $X$ be a compact metrizable topological space. Let $p \in X$ be a point. A Kuranishi neighborhood of $p$ (of dimension $k$) is a 5-tuple

$K_p = (U_p, E_p, S_p, \psi_p, F_p)$

where

• $U_p$ is a smooth orbifold;
• $E_p \to$ is a smooth orbifold vector bundle;
• $S_p: U_p \to E_p$ is a smooth section;
• $\psi_p: S_p^{-1}(0) \to X$ is a continuous map and is homeomorphic onto its image $F_p \subset X$.

They should satisfy that $\dim U_p - \operatorname{rank} E_p = k$.

If $p, q \in X$ and $K_p = (U_p, E_p, S_p, \psi_p, F_p)$, $K_q = (U_q, E_q, S_q, \psi_q, F_q)$ are their Kuranishi neighborhoods respectively, then a coordinate change from $K_q$ to $K_p$ is a triple

$T_{pq} = ( U_{pq}, \phi_{pq}, \hat\phi_{pq})$

where

• $U_{pq} \subset U_q$ is an open sub-orbifold;
• $\phi_{pq}: U_{pq} \to U_p$ is an orbifold embedding;
• $\hat\phi_{pq}: E_q|_{U_{pq}} \to E_p$ is an orbifold vector bundle embedding which covers $\phi_{pq}$.

In addition, they must satisfy the compatibility condition:

• $S_p \circ \phi_{pq} = \hat\phi_{pq} \circ S_q|_{U_{pq}}$;
• $\psi_p \circ \phi_{pq}|_{S_q^{-1}(0) \cap U_{pq}} = \psi_q|_{S_q^{-1}(0)\cap U_{pq}}$.

A Kuranishi structure on $X$ of dimension $k$ is a collection

$\Big( \{ K_p = (U_p, E_p, S_p, \psi_p, F_p) \ |\ p \in X \},\ \{ T_{pq} = (U_{pq}, \phi_{pq}, \hat\phi_{pq} ) \ |\ p \in X,\ q \in F_p\} \Big)$

where

• $K_p$ is a Kuranishi neighborhood of $p$ of dimension $k$;
• $T_{pq}$ is a coordinate change from $K_q$ to $K_p$.

In addition, the coordinate changes must satisfy the cocycle condition, namely, whenever $q\in F_p,\ r \in F_q$, we require that

$\phi_{pq} \circ \phi_{qr} = \phi_{pr},\ \hat\phi_{pq} \circ \hat\phi_{qr} = \hat\phi_{pr}$

over the regions where both sides are defined.

History

In Gromov–Witten theory, one needs to define integration over the moduli space of stable maps $\overline{\mathcal M}_{g, n} (X, A)$(see for example ^[2]). 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

$\overline\partial_J u = 0$.

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 $u: S^2 \to X$ 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.^[3]

References

1. Fukaya, K. and Ono, K., "Arnold Conjecture and Gromov–Witten Invariant", Topology 38 (1999), no. 5, 933–1048
2. McDuff, D and Salamon, D. "J-holomorphic curves and symplectic topology", American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2004, ISBN 0-8218-3485-1
3. Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., "Lagrangian intersection Floer theory: anomaly and obstruction, Part I and Part II", AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. ISBN 978-0-8218-4836-4