TITLE: Does the derived category of coherent sheaves determine the hodge theory?
QUESTION [21 upvotes]: 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?
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?

REPLY [6 votes]: I don't know the state of the art. In physics if two smooth Calabi-Yau varieties, $X$ and $X^\vee$, of dimension $n$ are mirror to each other then $h^{p,q}(X)=h^{n-p,q}(X^\vee)$. In general they will not smooth but the physics suggests that we take crepant resolutions $Y\to X$ and $Y^\vee\to X^\vee$ and hence we should have $h^{p,q}(Y)=h^{n-p,q}(Y^\vee)$. But, if a crepant resolution exists it is not necessarily unique. This motivated Maxim Kontsevich to introduce motivic integration and prove that if $X_1$ and $X_2$ are two crepant resolutions of $X$ then $h^{p,q}(X_1)=h^{p,q}(X_2)$.
As for the decomposition of Hochschild cohomology holding for derived equivalences, this is not true. In a paper by Oren Ben-Bassat, Jon Block, and Tony Pantev, they show that for a torus $X$ and its dual $X^\vee$ the Fourier-Mukai transfrom induces an isomorphism $H^0(X,\wedge^2T_X)\simeq H^2(X^\vee,\mathcal{O}_{X^\vee})$. In other words, a noncommutative deformation of $X$ goes to a gerby deformation of $X^\vee$.