Looking at the construction of cohomology

In analogy with compact support as a direct limit of relative cohomology groups, one conceives that it is possible to do something similar the Hodge diagram for ordinary de Rham cohomologyand obtain a new sort , we should have some kind of Hodge 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.

Looking at the construction of cohomology with compact support as a direct limit of relative cohomology groups, one conceives that it is possible to do something similar for de Rham cohomology and obtain a new sort of Hodge diagram. 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.