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).
$X$ quasi-projective.
$X$ proper.
$X$ projective (equivalently, both (1) and (2)).
$X$ affine.
$X$ smooth.
Are there simple (preferably low dimensional) counter examples to convergence of formal defomrations?