Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. **Which of the following assumptions (or combinations thereof) are sufficient to imply that this deformation is convergent?** (i.e. that it comes from some flat analytic family $\tilde{\mathcal{X}} \to \mathcal{D}$ - where $\mathcal{D}$ means a closed analytic disk of some non-zero radius).


1. $X$ quasi-projective.
2. $X$ proper.
3. $X$ projective (equivalently, both (1) and (2)).
4. $X$ affine.
5. $X$ smooth.


Are there simple (preferably low dimensional) counter examples to convergence of formal defomrations?



When is a formal deformation convergent?
**EDIT.** New version, addressing questions in the comments.


(1) Affine and smooth implies what you want. 


Indeed, suppose $\mathcal{X}$ is smooth and that $H^1(X\_0, T\_{X\_0/\mathbb{C}}) = 0$ where $X\_0/\mathbb{C}$ is the special fiber and where $T\_{X\_0/\mathbb{C}}$ is the tangent bundle. This is of course satisfied if $X\_0$ is affine. 


I claim that in this case $\mathcal{X}$ is actually constant, i.e. $\mathcal{X}$ is the $t$-adic completion of $X\_0 \times {\rm Spec}\, \mathbb{C}[[t]]$; if this is true then the constant family $X\_0 \times (\text{unit disc})$ is the desired extension. 


The claim follows by usual deformation theory: one shows by induction on $n$ that $X\_n = \mathcal{X}\otimes\_{\mathbb{C}[[t]]} \mathbb{C}[t]/(t^{n+1})$ is isomorphic over $\mathbb{C}[[t]]$ to $X\_0 \otimes\_\mathbb{C} \mathbb{C}[t]/(t^{n+1})$. For the induction step, basic deformation theory (e.g. Fantechi's and Illusie's articles in "FGA explained") tells you that liftings of $X\_n \cong X\_0 \otimes\_\mathbb{C} \mathbb{C}[t]/(t^{n+1})$ to $\mathbb{C}[t]/(t^{n+2})$ are a torsor under $H^1(X\_0, T\_{X\_0/\mathbb{C}})$. Since this group vanishes, there is only one such lifting, up to a non-canonical isomorphism. In the limit, we obtain the desired isomorphism.


See also the results of Renee Elkik *Solutions d’équations à coefficients dans un anneau hensélien* Annales scientifiques de l’É.N.S. 4e série, tome 6, no 4 (1973), p. 553-603. She shows (see Theorem 6, p. 580) that a smooth algebra over a ring $R\_0= R/I$ can always be lifted (not only formally) over $R$ if $R$ is noetherian and henselian along $I$.


(2) Smooth affines really are the only case when the answer is positive.


Indeed, take an elliptic curve over $\mathbb{C}[[t]]$ with divergent $j$-invariant, for example
$$ E\colon y^2 z = x(x-z)(x-\lambda z), \quad \lambda = \sum\_{n\geq 0} n! t^n. $$ 
Then the $j$-invariant 
$$ j(\lambda) = 258 \frac{(1-\lambda(1-\lambda))^3}{(\lambda(1-\lambda))^2} \in \mathbb{C}[[t]] $$
is likely not convergent. If $E$ was the completion of an analytic family (I am identifying $E$ with the corresponding formal scheme which does not make much difference since $E$ is proper), the $j$-invariant would have to have positive radius of convergence.


The same example works in the affine but singular case by taking the affine cone, i.e. 
$$ \mathcal{X} = {\rm Spf}\, \mathbb{C}[[t]]\{x,y,z\} / (y^2 z - x(x-z)(x-\lambda z), $$
or the same with $[[x,y,z]]$.


Perhaps the simplest example is the plane curve singularity with four lines meeting at a point whose cross-ratio $\lambda$ is divergent, i.e.
$$ xy(x+y)(x-\lambda y).$$


