Given two smooth algebraic varieties (proper or not), if the two derived categories of the bounded complexes of coherent sheaves over them are equivalent (if necessary we assume there is a fully faithful equivalence between the two categories) , do they have the same hodge numbers, or even the same hodge decomposition in the hochschild homologies of the two categories?
Barannovsky
Baranovsky in arXiv:math/0206256 discussed this problem and gave a affirmative answer, but I think he only proved that the two varieties have the same odd and even part of homologies, but not the individual degrees of homologies, i.e. I don't know why they have the same betti numbers from his argument, let alone the hodge numbers. The point I feel is that, we can define hochschild homology from a dg-category (e.g. a dg-enhancement of the derived category), but can't define hodge decomposition of the hochschild homology from only the information of the derive category.
Orlov in arXiv:math/0512620v1 discussed this problem as a consequence of his theorem on motives and gave a sufficient condition which demands the support of the corresponding Fourier-Mukai transform to be equal to the dimension of the concerned varieties. I think this condition is too strong and difficult to verify.
A related problem is, does the homological mirror symmetry conjecture imply the coincedence of hodge numbers of two mirror duals?
