I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:


*Find a pair of complex analytic families $\lbrace M\_t\rbrace$ and $\lbrace N\_t\rbrace$ with $|t|<1$ such that $M\_t=N\_0$ for $t\ne 0$ and $N\_t=M\_0$ for $t\ne0$, with $M\_0\ne N\_0$.* (Every manifold assumed compact.)


As of 1971 there were no known examples. **Up to date, is this possibly true?**


Naively it seems impossible. In the spirit of Lefschetz fibrations, we have $M\_0$ as a critical level set amongst the $M\_t$ (thinking of cylinders which get pinched via a vanishing cycle). It is then not generic, and any perturbation of this fiber (locally about the nodal point) would alter the topology. So to me the inverted family $\lbrace N\_t\rbrace$ couldn't exist, because it would force a generic $M\_{t\_0}$ to take over the role as the critical level set amongst the $N\_t$, but there is nothing "non-regular" about it.  

If the manifolds are allowed to be noncompact, perhaps it's easy, again via a rough sketch: take $\lbrace M\_t\rbrace$ to be the family of open cylinders which pinch the vanishing cycles at $t=0$, and $\lbrace N\_t\rbrace$ to be the family of "pinched" cylinders where the pinching point pushes off to infinity as $|t|\to 0$.


**Edit:** The problem is more subtle than I thought, and my example with Lefschetz fibrations doesn't apply. From the definition of a "complex analytic family", all $M\_t$ are diffeomorphic to each other (in particular, they're all nonsingular spaces), and so we can only distinguish them by their holomorphic structures. In other words, $M\_0\ne N\_0$ means diffeomorphic but not complex-analytically homeomorphic.



Inverted pair of complex analytic families
I'm going to assume that your objects are compact analytic spaces and show that then the situation you describe cannot arise. I don't have access to Morrow and Kodaira and cannot remember whether the "problem" concerns only compact manifolds.


Suppose that $\mathcal M\to S$ is a miniversal deformation of $M\_0$. Then the given family $\{M\_t\}$ defines a morphism $\mu:\Delta\to S$, where $\Delta$ is a complex disc. Since $M\_t=N\_0\ne M\_0$, the map $\mu$ is non-constant and $\mu(\Delta-\{0\})$ is contained in the stratum of $S$ that consists of points $s$ such that $\mathcal M\_s$ is isomorphic to $N\_0$. Now this stratum is smooth and its dimension, which we have just proved to be at least $1$, equals $\dim Aut\_{M\_0}-\dim Aut\_{N\_0}$; one reference for this, in the algebraic context (but in the current analytic context the result will be the same), is Theorem 2.7 of arXiv:1210.0342 by Hyland, Ekedahl and Shepherd-Barron. (In positive or mixed characteristic, where group schemes need not be reduced, this dimension becomes $\dim Lie(Aut\_{M\_0})-\dim Aut\_{N\_0}$.)


So $\dim Aut\_{N\_0}<\dim Aut\_{M\_0}$. From the assumptions, the opposite inequality is also valid, and we have a contradiction.


