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?

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?

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 finite type analytic space $\tilde{\mathcal{X}} \to Spec(\mathbb{C}\{x\})$).

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?

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?

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 $\tilde{\mathcal{X}} \to Spec(\mathbb{C}\{x\})$).

1. $X$ quasi-projective.

2. $X$ proper.

3. $X$ projective (equivalently, both (1) and (2)).

4. $X$ 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?

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 finite type analytic space $\tilde{\mathcal{X}} \to Spec(\mathbb{C}\{x\})$).

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?