Let $D=\mathop{Spec}(\mathbb C[[t]])$ be the algebraic small disk and let $\Delta= \{z\in \mathbb C: |z|<1\} $. Let $X_0$ be an algebraic surface over $\mathop{Spec}\mathbb C$

Suppose now that I have a deformation of $X_0\to\mathop{Spec}{\mathbb C}$ over $D$, that is a flat, proper (maybe projective) morphism $X\rightarrow D$ such that the special fibre is $X_0$.

**Question**: Under which conditions I can find a degeneration $Y\rightarrow \Delta$ with fibre at zero (central fibre) $X_0^{an}$ and with the property that we can recover $X\rightarrow D$ form $Y\rightarrow \Delta$ in the sense that if I start with the $$Y\rightarrow \Delta, \text{with central fibre }\ X_0^{an}$$ and we consider the composition $\Delta\rightarrow \mathbb C$, to get $Y\rightarrow \mathbb C$. If $Y$ is nice enough (projective, finite type etc) I can use GAGA to get an algebraic family $\mathcal X\rightarrow \mathop{Spec}\mathbb C[t]$ and if now I consider the inclusion $ \mathbb C[t]\rightarrow \mathbb C[[t]]$ and let $X$ be the base change I get a family $\mathcal X\otimes \mathop{Spec}(\mathbb C[[t]])\rightarrow D$ with special fibre $X_0$ and an isomorphism $\mathcal X\otimes \mathop{Spec}(\mathbb C[[t]])\simeq X$ over $D$?

If such a deformation exist, which properties are preserved? for example can I identify the Monodromy?

smoothover $D[1/t]$? It is also far too restrictive to expect an "algebraization" over $\mathbf{C}[t]_{(t)}$, since the completion cannot distinguish the genus of the curve arising from a given algebraic local ring. There is no "identification of $D$ and $\Delta$". You want results onopennessof loci in the base for fibral properties in thescheme-theoretic and analytic space cases; see Exp. XII of SGA1. $\endgroup$ – Marguax Oct 10 '13 at 2:45