Let $X$ be a smooth finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spec(\mathbb{C}[[x]])$ be a formal deformation of $X$ (flat family with special fiber $X$). Which of the following assumptions (or combinations thereof) are sufficient to imply that this deformation is convergent? (i.e. that it is pulled back from some family $\tilde{\mathcal{X}} \to Spec(\mathbb{C}\{x\})$).
$X$ quasi-projective.
$X$ proper.
$X$ projective (equivalently, both (1) and (2)).
$X$ affine.
Suppose we drop the condition that $X$ is smooth but instead require it to be (affine/proper/quasi-projective) are there simple (preferably low dimensional) counter examples to convergence of formal defomrations?