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  "body": "\n## Definition\n\nLet $X$ be a compact metrizable space.\nA *Kuranishi neighbourhood* of a point $p \\in X$ is a tuple $(V_p, E_p, \\Gamma_p, s_p, \\psi_p)$ where:\n\n1. $V_p$ is a [[smooth manifold|F45VQ47GPV]] and $\\Gamma_p$ is a finite group with a smooth [[action|0WCY42EV9V]] on $V_p$.\n2. $E_p \\to V_p$ is a finite-dimensional real [[vector bundle|P4DZM975BV]] with a $\\Gamma_p$-action that lifts the action on $V_p$.\n3. $s_p$ is a $\\Gamma_p$-equivariant smooth [[section|WWX7H5W1RP]] of $E_p$, the Kuranishi map.\n4. $\\psi_p \\colon s_p^{-1}(0) / \\Gamma_p \\to X$ is a [[homeomorphism|0QT00PB5G2]] onto an open neighbourhood of $p$, where $s_p^{-1}(0) / \\Gamma_p$ is the [[quotient space|0BR3FY5Q8M]].\n\nA *Kuranishi structure* on $X$ is a choice of a Kuranishi neighbourhood for every $p \\in X$, together with coordinate changes, such that:\n\n5. 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 $p$.\nIt consists of a $\\Gamma_q$-invariant open neighbourhood $V_{pq} \\subseteq V_q$ of a lift of $q$ and an injective homomorphism $h_{pq} \\colon \\Gamma_q \\to \\Gamma_p$.\nIt 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 $\\phi_{pq}$.\nThese satisfy $\\hat{\\phi}_{pq} \\circ s_q = s_p \\circ \\phi_{pq}$, and $\\psi_p \\circ \\phi_{pq} = \\psi_q$ on $s_q^{-1}(0) \\cap V_{pq}$.\n6. The coordinate changes are compatible.\nLet $q$ lie in the image of the chart at $p$, and $r$ in the image of the chart at $q$.\nThen 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 $h$.\n7. The number $\\dim V_p - \\operatorname{rank} E_p$ is the same for all $p$; it is the *virtual dimension* of $X$.\n8. Let $N_{pq}$ be the normal bundle of $\\phi_{pq}(V_{pq})$ in $V_p$.\nFor 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 $\\phi_{pq}(s_q^{-1}(0))$.\n\n## Remarks\n\nExample: let $V$ be a smooth manifold, $E \\to V$ a vector bundle and $s$ a section whose zero set $X = s^{-1}(0)$ is compact.\nTake for every $p$ the same chart $(V, E, \\set{1}, s, \\id)$, with identity coordinate changes.\nAll composites of coordinate changes are identities, the virtual dimension is $\\dim V - \\operatorname{rank} E$ everywhere, and the normal bundles in clause 8 are zero, as are the quotients $E / E$.\nSo this is a Kuranishi structure on $X$.\nIn particular a compact manifold $X$ has the Kuranishi structure with $V = X$ and $E = 0$.\n\nNon-example: let $X$ be a single point.\nOne Kuranishi neighbourhood is $V_A = \\R$, $E_A = \\R \\times \\R$, $\\Gamma_A$ trivial and $s_A(x) = x^2$, whose zero set is one point; another is $V_B$ a point with $E_B = 0$.\nBoth have virtual dimension $0$, so clause 7 holds.\nBut a coordinate change from $B$ to $A$ fails clause 8.\nThe normal space of the point $0$ in $\\R$ is $\\R$, the quotient $E_A / \\hat\\phi(E_B)$ is $\\R$, and the derivative $s_A'(0) = 0$ is not an isomorphism.\n\nSeveral variants of this definition are in use, due to Fukaya, Oh, Ohta and Ono, to Joyce, and to others.\nThey differ mainly in clauses 6 and 8 and in how the coordinate changes are organised.\n\nClause 8 is sometimes called the tangent bundle condition, and it is what makes orientations and virtual fundamental classes well defined.\n",
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