Let $X$ be a 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 the pullback of some flat analytic family $\tilde{\mathcal{X}} \to Spec(\mathbb{C}\{x\})$).

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?