2
$\begingroup$

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.

However, the following question is still open:

For any $X$ in $\mathcal C$, does there always exists a deformation $\pi:\mathcal X\to B$ of $X$, such that for any small neighborhood $U$ of $X$, there exists a Kähler deformation $X_t:=\pi^{-1}(t)$ with $t\in U$?

Does anyone has any counterexample?

Improve this question
$\endgroup$
3
  • $\begingroup$ I think this already fails for the non-projective, proper complex schemes of Hironaka in the appendix of Hartshorne's "Algebraic geometry". $\endgroup$
    – Jason Starr
    Commented May 18, 2022 at 16:23
  • $\begingroup$ @JasonStarr , can you give more details? In Hironaka's example, the central fiber of the family is a non-Kähler Moishezon manifold, and the nearby fibers are all projective. I don't see why it provides a counterexample. $\endgroup$
    – Tom
    Commented May 18, 2022 at 16:49
  • $\begingroup$ You might mean a different example than I mean. At any rate, every complex manifold that is bimeromorphic to a Kaehler manifold is in Fujiki class $\mathcal{C}$. $\endgroup$
    – Jason Starr
    Commented May 18, 2022 at 21:55

0

Reset to default

You must log in to answer this question.