Take the 2-minute tour ×  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.
share|improve this question add comment  1 Answer  up vote 4
down vote accepted  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 f