3
$\begingroup$

Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family?

The answer is manifestly positive in complex dimension 1.

It is known, I think, that if there a Kaehler fiber, then Hodge numbers must be constant in a small neighbourhood thereof.

Note that this question asks about topological invariance of Hodge numbers, while we are asking about deformation-equivalence invariance.

$\endgroup$
5
$\begingroup$

Nakamura has constructed a family of compact complex threefolds such that the Hodge number $h^{p,q}$ is not deformation invariant for $p+q>0$. They are obtained by deforming an "Iwasawa manifold" $\mathbb{C}^3/\Gamma $. See Complex parallelisable manifolds and their small deformations, J. Diff. Geom. 10 (1975), 95-112.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.