Let $M$ be a closed complex manifold. Assume that for any family of closed complex manifolds over the unit disk containing $M$ as the central fiber, there exists a sufficiently small neighbourhood of 0 such that $h^{p, q}(M_t)=h^{p, q}(M)$ for all $t$ in the neighbourhood. What restrictions does this give on the geometry of $M$?

We know for a fact that this does impose some non-trivial restrictions (as there exist closed complex manifolds for which arbitrarily small deformations have different Hodge numbers). The class of the manifolds under consideration should be different from the class of locally rigid complex manifolds as well (since complex curves are not locally rigid but Hodge numbers are determined by the underlying topological space). It would also be nice if someone could point out a name for this class of complex manifolds (if they have been studied before).

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    $\begingroup$ We should rather find a name for those who do not have this property, which are extremely rare. You are certainly aware that for Kähler manifolds, for instance, the Hodge numbers are constant under deformations. $\endgroup$ – abx Feb 3 at 7:01
  • $\begingroup$ @abx it was a stupid way to put the question, I agree. On another hand, I think Kaehler manifolds are in some sense "the minority" of complex manifolds (but I do agree that regardless, the majority of complex manifolds should have the property in this question) $\endgroup$ – Aknazar Kazhymurat Feb 3 at 15:27

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