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Saal Hardali
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When is an unobstructed formal deformation convergent?

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 counter examples to convergence of formal defomrations?

Saal Hardali
  • 7.8k
  • 3
  • 43
  • 99