# Deformations of Hopf manifolds

Recall that a Hopf manifold is a quotient $$\mathbb C^n\setminus 0$$ by a free action of $$\mathbb Z$$ where the generator is acting by a holomorphic contraction.

Question 1. Is it true that any deformation of a Hopf manifold (as a complex manifold) is again a Hopf manifold for $$n\ge 3$$?

Question 2. Is there some kind of classification of Hopf manifolds and their deformations in dimension $$\ge 3$$ (for example $$n=3$$).

Note that for $$n=2$$ the answer to both questions is positive https://en.wikipedia.org/wiki/Hopf_surface

## 1 Answer

Both questions are answered affirmatively for sufficiently small deformations in this paper by Haefliger.

• YangMills, thanks a lot for both references! This leaves open just the question about all deformations (i.e. not small), as far as I understand? – aglearner Oct 23 at 20:06