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.