Kuranishi theory

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 ${\displaystyle (f_{1},\ldots ,f_{k}):{\mathbb {R} }^{n+k}\to {\mathbb {R} }^{k}}$, or the quotient of such a zero set by a finite group. Kuranishi structure was introduced by Japanese mathematicians Kenji Fukaya and Kaoru Ono in the study of Gromov–Witten invariants in symplectic geometry and named after Masatake Kuranishi.

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

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