  Consider the following uniformization problem. Take two holomorphic
(parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb
C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood
of $0$ in $\Bbb R^p$, for some $p>0$) deformation families of compact complex
manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of
the parameter space, the fiber over $t$ of the first family is biholomorphic to
the fiber over $t$ of the second family. Then, under which conditions are the
two families locally isomorphic at 0? In this article, we give a sufficient
condition in the case of holomorphic families. We show then that, surprisingly,
this condition is not sufficient in the case of differentiable families. We
also describe different types of counterexamples and give some elements of
classification of the counterexamples. These results rely on a geometric study
of the Kuranishi space of a compact complex manifold.
