In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our space is a *noncompact* Calabi-Yau manifold, we get a reduced Hodge diamond, to which mirror symmetry probably applies. 


Unfortunately, I don't know anything about mirror symmetry. Do we still get meaningful geometric information (deformations, etc.)? I'd like to know what all the subtle obstructions are to defining things in the above way.



Mirror symmetry for noncompact Calabi-Yau manifolds
There is a version of mirror symmetry, called "local mirror symmetry", for certain non-compact Calabi-Yaus, for example the total space of the canonical bundle of P^2 (exercise: show this is CY). The mirror (or rather one possible mirror) of this non-compact Calabi-Yau is an affine elliptic curve in (C^\*)^2. I don't think that there is as yet a version of mirror symmetry for more general non-compact CYs, though I don't know too much about this story. In all of the papers that I've looked at on this stuff at least, the only non-compact CYs that have been considered in mirror symmetry so far are total spaces of vector bundles over compact (probably Fano) things.


I guess we should probably get some sort of "Hodge diamond symmetry" in local mirror symmetry, but the story becomes more complicated. One immediate thing to notice is that, at least in the example I've given, the dimensions of the manifolds aren't the same! So things will have to be modified.


The Hodge diamond symmetry in mirror symmetry for compact Calabi-Yaus should really be thought of as coming from a correspondence between certain deformations of a Calabi-Yau and certain deformations of its mirror. In the non-compact case, the deformations that we should consider will be somewhat different from the deformations that we should consider in the compact case.


A somewhat recent point of view, due to Kontsevich, is that this correspondence between deformations can be gotten from homological mirror symmetry. In homological mirror symmetry for compact Calabi-Yau manifolds, we consider a derived category of coherent sheaves on one side and a Fukaya category on the other side. Then we should have an equivalence of categories, plus an equivalence of a certain structure on their Hochschild cohomologies -- in particular their Hochschild cohomologies should be equivalent at least as vector spaces. These Hochschild cohomologies should be thought as the appropriate deformation spaces (or maybe rather the tangent spaces to the appropriate deformation spaces?), and an appropriate identification of the Hochschild cohomologies (plus the extra structure that I mentioned) should give in particular the Hodge diamond symmetry. There should also be homological mirror symmetry for non-compact Calabi-Yau manifolds, but we must define the analogues of derived category and Fukaya category in this situation appropriately. Then there should be an analogous story on the Hochschild cohomologies of the categories, and an appropriate analogue of the Hodge diamond symmetry. See the paper "Hodge theoretic aspects of mirror symmetry" by Katzarkov-Kontsevich-Pantev for more details.


