Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture

Question

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:

1. Mirror symmetry, as formulated on the first page of these notes
2. Homological mirror symmetry (HMS)
3. 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?

Answer 1

Disclaimer: I am also not an expert.

According to Perutz (see 'Core homological mirror symmetry project'), it is expected that T-duality (SYZ, your 3) implies HMS (your 2), which should imply Hodge-theoretic mirror symmetry (essentially your 1). In fact, when Kontsevich put forward his HMS conjecture, he gave a heuristic argument why one should be able to deduce Hodge-theoretic ('numerical') mirror symmetry from HMS (see his ICM 1994 talk). As far as I know, this has never been proven, but recently there have been efforts in that direction by Ganatra, Perutz, Sheridan, where they prove that HMS implies Hodge-theoretic mirror symmetry, modulo a technical conjecture, and modulo the definition of the Fukaya category of a Calabi-Yau manifold. It is expected that such a definition is given in 'Quantum cohomology and split generation in Lagrangian Floer theory' by Abouzaid and the symplectic quartet usually referred to as 'FOOO', a work which has been in preparation for a long time, but it seems that there is no preprint yet.

when people speak of the "mirror" of a CY variety, do they really always mean a mirror in the sense of point (1) above?

I believe there is no 'always' in this subject (even for what is really meant by CY, CY means different things to different people - from holonomy $=SU(3)$ to 'noncommutative'); but a sufficient condition for a pair of CY's to be mirror is certainly that physicists say it is mirror in their sense (which is some relation between associated CFTs).

Answer 2

Another relation between 2) and 3) is that 2) is an algebraic statement while 3) is a geometric statement. So if one obtains an SYZ mirror by dualizing a Lagrangian torus fibration, then one obtains a pair of manifolds and can then try to prove HMS on that pair. Specifically, the input needed for SYZ mirror symmetry is a Lagrangian torus fibration structure on the original symplectic manifold, and its mirror complex manifold is the dual fibration. Then one can aim to compute the categories on each side and show they match to prove HMS for that pair. Here is a nice set of notes on SYZ mirror symmetry i.e. T-duality (T for torus): https://math.berkeley.edu/~auroux/papers/slagmirror.pdf