Kuranishi structure

Definition

Let $X$ be a compact metrizable space. A Kuranishi neighbourhood of a point $p \in X$ is a tuple $(V_p, E_p, \Gamma_p, s_p, \psi_p)$ where:

1. $V_p$ is a smooth manifold and $\Gamma_p$ is a finite group with a smooth action on $V_p$.
2. $E_p \to V_p$ is a finite-dimensional real vector bundle with a $\Gamma_p$-action that lifts the action on $V_p$.
3. $s_p$ is a $\Gamma_p$-equivariant smooth section of $E_p$, the Kuranishi map.
4. $\psi_p \colon s_p^{-1}(0) / \Gamma_p \to X$ is a homeomorphism onto an open neighbourhood of $p$, where $s_p^{-1}(0) / \Gamma_p$ is the quotient space.

A Kuranishi structure on $X$ is a choice of a Kuranishi neighbourhood for every $p \in X$, together with coordinate changes, such that:

5. For each $q \in \psi_p(s_p^{-1}(0) / \Gamma_p)$ there is a coordinate change from the chart at $q$ to the chart at $p$. It consists of a $\Gamma_q$-invariant open neighbourhood $V_{pq} \subseteq V_q$ of a lift of $q$ and an injective homomorphism $h_{pq} \colon \Gamma_q \to \Gamma_p$. It also has an $h_{pq}$-equivariant smooth embedding $\phi_{pq} \colon V_{pq} \to V_p$ and an $h_{pq}$-equivariant bundle embedding $\hat{\phi}_{pq} \colon E_q|_{V_{pq}} \to E_p$ over $\phi_{pq}$. These satisfy $\hat{\phi}_{pq} \circ s_q = s_p \circ \phi_{pq}$, and $\psi_p \circ \phi_{pq} = \psi_q$ on $s_q^{-1}(0) \cap V_{pq}$.
6. The coordinate changes are compatible. Let $q$ lie in the image of the chart at $p$, and $r$ in the image of the chart at $q$. Then on the common domain $\phi_{pq} \circ \phi_{qr}$ agrees with $\phi_{pr}$ up to the action of an element of $\Gamma_p$, and likewise for $\hat{\phi}$ and $h$.
7. The number $\dim V_p - \operatorname{rank} E_p$ is the same for all $p$; it is the virtual dimension of $X$.
8. Let $N_{pq}$ be the normal bundle of $\phi_{pq}(V_{pq})$ in $V_p$. For each coordinate change, the derivative of $s_p$ induces an isomorphism from $N_{pq}$ onto $E_p / \hat{\phi}_{pq}(E_q)$ at the points of $\phi_{pq}(s_q^{-1}(0))$.