I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:
- Mirror symmetry, as formulated on the first page of these notes
- Homological mirror symmetry (HMS)
- The SYZ conjecture
A first basic question: when people speak of the "mirror" of a CY variety, do they really always mean a mirror in the sense of point (1) above?
My main question is whether any of these conjectures actually imply each other? For example, HMS predicts an equivalence of categories, which is only applied, in heuristic arguments for SYZ, to skyscraper sheaves. So it seems that SYZ would be at most a (refinement of (skyscraper sheaves correspond to honest Lagrangians, not just any objects in the derived category) a) consequence of HMS. In particular, the two do not seem to imply eachother?