Kuranishi family

Summary

The Kuranishi family of a compact complex manifold $X$ is a semi-universal deformation 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, 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.