---
tag: R0QABVGPRX
kind: concept
name: Kuranishi family
msc: [32G05, 32G13]
sources:
  - url: https://en.wikipedia.org/wiki/Kuranishi_structure
---

## Summary

The Kuranishi family of a compact [[complex manifold|JSGWAN2RJV]] $X$ is a semi-universal [[deformation|VKQSM3BCE4]] of $X$.
Every small deformation of $X$ is pulled back from it, and the pullback map is unique to first order.
Kuranishi constructed it in 1962 over a germ of [[complex analytic space|HXSZA44J5J]], the Kuranishi space.
That space is the zero set of a holomorphic obstruction map from a neighbourhood of $0$ in $H^1(X, \Theta_X)$ to $H^2(X, \Theta_X)$.
It is the local model for moduli of compact complex manifolds.

## Explanations

### L1 human

Every compact complex manifold has one "best" small family of deformations that contains all others.
It is best in the sense that any other small deformation of $X$ is obtained from it by a change of parameter.
The parameter space can be singular, because some first-order directions may fail to extend to actual deformations.
For an elliptic curve the Kuranishi family is a small disc of the family $\C / (\Z + \tau \Z)$; for projective space it is a single point, because projective space is rigid.