This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper http://arxiv.org/abs/hep-th/0010293 which further directed me to an older paper that I had forgot about, namely http://arxiv.org/abs/math/9812003 (Mirror Symmetry for abelian varieties). My question is about some results in this latter paper.
The main question is about Proposition 9.6.1, which I will reproduce here:
Let $A$ be a complex torus of dimension $n$. Let $\phi ∈ NS^0_A$, i.e. $\phi \in Hom(A, \hat{A})$ is an isogeny. Let $\tau = a + ib ∈ \mathbb{C}, b \neq 0$. Consider the element $\omega_A := \tau\phi ∈ NS_A(\mathbb{C})^0$ and the weak pair $(A, \omega_A)$. Then there exist isogeneous elliptic curves $E_1, ..., E_n$ and an element $\omega_E ∈ NS_E(\mathbb{C})^0$, where $E = E_1 \times ... \times E_n$, such that the weak pair $(E, \omega_E)$ is mirror symmetric to the weak pair $(A, \omega_A)$.
Notation aside, what this seems to be saying is that every abelian variety is mirror symmetric to a product of elliptic curves. This seems very surprising for me, and very much not what I would expect from the case of K3 surfaces. Moreover, from Proposition 9.2.6 in this paper, it would follow then that the derived category of an abelian variety $A$ is always equivalent to the derived category of a product of elliptic curves, which also surprises me.
What I was hoping for (at least, for abelian surfaces) is something akin to the notion of Mirror Symmetry for lattice polarized K3 surfaces, but this seems to not be the case.
Am I missing something? Or is it just that this is one notion of mirror symmetry, but not necessarily the only one?
