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.