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 1


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

  • 1
    $\begingroup$ 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? $\endgroup$
    – aglearner
    Oct 23, 2020 at 20:06

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