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Here is my biased view of a simple example: the two-torus. Everything I know about homological mirror symmetry stems from this example. Because the example is one-dimensional, a symplectic form is j …

Here are a few scattered observations: Our ability to construct examples (e.g. of CY manifolds) is limited, and the tools of algebraic geometry are perfectly suited to doing so (as has been noted). …

The curve you wrote in equations lies in C^2, while the "elliptic curve" of your text is presumably a compact projective variety -- meaning you imagine making your equations homogeneous (or even quasi …

I assume the question regards the coherent sheaves on these two CY's. These CY's should be regarded as the "same" complex manifold with two different choices of complexified symplectic forms ("Kahler …

Hi- Just saw this thread. Maybe I should comment. The conjecture can be viewed from the perspective of various categories: geometric, symplectic, topological. Since the argument is physical, it wa …

I think this is a stubborn case which does not fit into the general picture. For example, if you use the standard toric procedure to try to construct a differential equation for the log periods of th …

One thing missing from this discussion is the even-more-mysterious holomorphic ambiguity (not "anomaly"). BCOV is not deterministic, and should probably be thought of as part of a general schema for …

To define (as Kevin Lin does above) the B-model purely as the derived category of coherent sheaves is fine and rigorous, but it ignores the higher-genus aspects of mirror symmetry -- which was the ori …

Kevin Costello's mathematical definition of the B-model (math/0509264) is rigorous. It's an open problem whether this definition agrees with the BCOV construction, as far as I know.