Deformation of complex structure

Remarks

Two deformations of $X$ over the same pointed base are isomorphic if there is a biholomorphism of the total spaces over $B$ compatible with the identifications of $X_0$ with $X$. By Ehresmann's theorem, conditions 1 and 2 imply that $\varpi$ is a smooth fiber bundle, so all fibres are diffeomorphic to $X$. The fibres need not be biholomorphic to each other.

Example: let $\mathbb{H}$ be the upper half-plane and let $\Z^2$ act on $\C \times \mathbb{H}$ by $(m, n) \cdot (z, \tau) = (z + m + n\tau, \tau)$. The quotient $\mathcal{X}$ is a complex manifold, and the projection $\mathcal{X} \to \mathbb{H}$ is a proper submersion with fibre $\C / (\Z + \tau\Z)$ over $\tau$. So it is a deformation of the elliptic curve $\C/(\Z + i\Z)$ over $(\mathbb{H}, i)$.

Non-example: remove one point from the total space $\mathcal{X}$ of the previous family. The restricted projection is still a holomorphic submersion, but it is not proper, since one fibre is now a punctured torus, which is not compact.