I guess this is a well known fact/definition for many people. It is mentioned in many places that if $\Gamma$ is a lattice of a vector space(vector bundle/affine bundle) $V$, then there is a dual lattice $\check{\Gamma}$ in $V^\* $ and the torus $V/\Gamma$ has a dual torus $V^\*/\check{\Gamma}$. What does this mean? When is it meaningful? I want to know the answer because I hope that we can get an easy topological description of ("toy model" of) mirror manifold (I have to admit that I am not sure if I really understand what's "mirror manifold"). For example, if $V$ is a vector bundle and $\Gamma$ is a lattice in $V$ with some extra structure or information, how can we find a torus bundle dual to $V/\Gamma$? What kind of extra structure or information we need? If $V$ is described by local chart and transitive groups {$U\_\alpha,\mu\_{\alpha \beta}$}, can we find a dual set {$U^\* \_\alpha,\mu^\* \_{\alpha \beta}$} to describe $V^\* $? Is it unique or canonical?


The other question is about the canonical complex structure of tangent bundle. Of course, it relate to the above question. Again, like the symplectic structure on cotangent bundle, I guess it is well known, but I really can't find any reference. One source I found is a slide of Mark Gross (<http://math.mit.edu/~auroux/frg/mit08-notes/M.%20Gross%20-%20Slides%20-%20From%20affine%20manifolds%20to%20complex%20manifolds.pdf>). I am not sure if this is the standard one we understand. The complex structure looks much less natural then the canonical symplectic structure. 



What's dual torus and mirror manifold?
The usual answer is that the dual lattice is 
$\check{\Gamma}=\{f\in V^\* | f(\gamma)\in \mathbb{Z}\ \forall \gamma \in \Gamma\}$. It is defined for any lattice $\Gamma\subset V$ - no extra information needed. 


